Controls and Fractional Order Systems

#### **Chapter 1**

## An Optimal Control Approach to Portfolio Diversification on Large Cap Stocks Traded in Tokyo Stock Exchange

*Muhammad Jaffar Sadiq Abdullah and Norizarina Ishak*

#### **Abstract**

In this chapter, Markowitz mean-variance approach is proposed for examining the best portfolio diversification strategy within three subperiods which are during the global financial crisis (GFC), post-global financial crisis, and during the noncrisis period. In our approach, we used 10 securities from five different industries to represent a risk-mitigation parameter. In this way, the naive diversification strategy is used to serve as a comparison for the approach used. During the computation process, the correlation matrices revealed that the portfolio risk is not well diversified during non-crisis periods, meanwhile, the variance-covariance matrices indicated that volatility can be minimized during portfolio construction. On this basis, 10 efficient portfolios were constructed and the optimal portfolios were selected in each subperiods based on the risk-averse preference. Performance-wise that optimal portfolio dominated the naïve strategy throughout the three subperiods tested. All the optimal portfolios selected are yielding more returns compared to the naïve portfolio.

**Keywords:** naïve diversification, mean-variance, Sharpe ratio, efficient frontier, portfolio optimization

#### **1. Introduction**

Investment decision and capital allocation in the stock market is subjected to the variability of risk and return. Thus, the investors, as well as the portfolio manager, must find the best solutions to allocate the various assets efficiently. Constructing a portfolio of a stock based on the investor's risk tolerance is a difficult task. In this situation, the portfolio manager needs to have some background knowledge in the economy of the market and mathematical modeling to create a portfolio based on the market participant's appetite.

In the year 1952, Markowitz's has introduced mean-variance portfolio optimization, where the investors are called rational investors by analyzing the mean and variance to determine the value of expected return and risk preference for investors [1]. This modern portfolio theory suggested that an investor can perform diversification in allocating assets in their portfolios by concerning how much risk they are willing to bear due to the outcome uncertainty in the stock market.

On the risk-return spectrum, some of the investor's favor seeking an opportunity to invest in the large-cap stock due to its stability and safer investment during turbulent times. In this study, the Tokyo Stock Exchange (TSE) is chosen as a medium to implement modern portfolio theory since the TSE has remained the focal point for investors looking to invest in Asia's largest stock market, ranking third in the world behind the New York Stock Exchange and NASDAQ in 2018 [2]. Japan has also been recognized as a member of the G7, which includes the world's six major advanced economies: Canada, France, Italy, the United Kingdom, the United States, and Germany. However, in the last decade, Japan has undergone an economic crisis period and the emergence of China has become a threat to Japan's economy. At some point, the systematic risk that occurred might affect the investor's portfolio thus leaving the greatest risk if there is no proper plan in constructing the best portfolio strategy.

Hence, the mean-variance portfolio optimization remains widely used among investors in analyzing their portfolio investment due to its simplicity and ease of derivation [3]. Therefore, there are many previous researchers had done their study using this modern portfolio theory that was first introduced by Harry Markowitz. Kulali [1] claimed that individuals are mainly aiming to select the best diversification through these two related strategies as maximizing return or minimizing risk despite given the characteristics of the country's stock market.

So, is the optimal portfolio being always the best portfolio diversification strategy? A lot of researchers have a debate regarding the issues of the mean-variance approach versus the 1/N strategy. Few researchers such as [4–7] conducted to determine the best portfolio strategy on the various samples of the equity market. Similar results are found that naïve diversification has performed better compared to the optimization model. Ramilton [8], on the other hand, refuted DeMiguel et al. [6]'s claim that a naive portfolio is preferable to an ex-ante portfolio was preferable over the ex-ante optimal portfolio. He found out that the optimal portfolios significantly outperform the naïve portfolio strategy in terms of the Sharpe ratio. Other findings such as [9] explained that if the naïve model outperforms a more sophisticated model, it is a clear indicator that the modeling of the data generating process is not accurate enough. Other research [10–12] used the mean-variance model to optimize asset allocation, therefore, the result showed the mean-variance model outperformed the naïve diversification strategy.

The aim of this paper is briefly to construct the optimal portfolio by using by Markowitz mean-variance model and compare to the naïve diversification strategy in the context of large-cap stock in Tokyo Stock Exchange (TSE) over the three subperiods which are during the global financial crisis (GFC) (2008–2010), the post-global financial crisis (2011–2013), and the non-crisis period (2014–2018).

#### **2. Materials and methods**

This study mainly focuses on the model parameters in determining the portfolio optimization construction by using mean-variance analysis on the large-cap stocks in Tokyo Stock Exchange (TSE). Then, we determine the best portfolio diversification strategy in the three subperiods tested.

#### **2.1 Data**

The historical stock prices of the Japanese stocks are collected from the Bloomberg Terminal web page and treated as a primary source of data for this study. About 571 data of the stock prices of each stock were picked from the last trading day of every week and were then transformed to weekly return (adjusted *An Optimal Control Approach to Portfolio Diversification on Large Cap Stocks Traded… DOI: http://dx.doi.org/10.5772/intechopen.100613*

price for Japanese Yen). The analysis based on the weekly return is conducted to avoid the non-synchronous trading effect [4].

#### *2.1.1 Stock selection*

Tokyo Stock Exchange has remained the largest stock exchange in Asia in terms of market capitalization. Thus, **Table 1** shows about 10 large-cap stocks were chosen randomly upon the stocks that listed in the TOPIX Core 30 + 70 Large and are traded in Tokyo Stock Exchange. The selections of the stocks are the companies that already been existed during the period of this study.

#### *2.1.2 Time periods*

The construction of the portfolio of the stocks will be based on 10 years. To look for the workability and superiority of the portfolio strategy in a different timeframe, the 10 years will be divided into three subperiods which are during the global financial crisis (GFC) (2008–2010), the post-global financial crisis (2011–2013), and during the non-crisis period (2014–2018).


#### **Table 1.**

*Data description for large-cap stock (TOPIX Core 30 + 70 Large).*

#### **2.2 Model framework of portfolio optimization**

The rate of return determines whether the investors gain or lose money from their investment [12]. In this study, we calculate the weekly stock return for stock *i* at time *t*, as shown in the Eq. (1).

$$
\dot{m} = \frac{\text{Sit} - \text{Sit} - 1}{\text{Sit} - 1} \tag{1}
$$

where *Sit* is the closing price of stock *i* at time *t*. *Sit* � 1 be the closing price at time *t* � 1. We assume that stock *i* pays dividends.

We calculate the average return of stocks based on the following equation:

$$\mu i = E(ri) = \frac{1}{M} \sum\_{t=1}^{M} ri \,\tag{2}$$

where is an average return on stock *i*, is a market return of stocks *i* at time *t*, *M* is the number of weeks. Then, García et al. [13] defines the expected return of a portfolio is the weighted average of the expected returns of individual stocks. Thus, the equation of expected return for *n* asset is as the following:

$$E(rp) = \sum\_{i=1}^{n} wiE(ri)\tag{3}$$

where is the proportion of the funds invested in stock *i*, *n* is the number of stocks, and are the return of *i*th stock and the return of portfolio *p*, respectively.

Garcia et al. [10] consider risk as uncertainty through the variability of future returns. So, we calculate the variance of stock *i* on the weekly return and the index return using the historical volatility formula as in Eq. (4). The higher value in variance for an expected return, the higher the dispersion of expected returns, and the greater the risk of the investment [14].

$$
\sigma\_i^2 = Var(r\_i) = \frac{\sum\_{i=1}^n (r\_i - \mu\_i)^2}{M - 1} \tag{4}
$$

where,

*Var r*ð Þ*<sup>i</sup>* is a variance of weekly stock return,

*ri* is a weekly stock return,

*μ<sup>i</sup>* is an average weekly return,

*M* is the sample size.

Ivanovic et al. [12] were able to measure how the stocks vary together by determining the covariance of each stock. The dimensions of risk are organized in the covariance matrix which is denoted by Ω*<sup>n</sup>*�*<sup>n</sup>*. This matrix contains variance in its main diagonal and covariances between all pairs of stocks.

$$
\Omega\_{n \times n} = \begin{pmatrix}
\sigma\_1^2 & \sigma\_{12} & \dots & \sigma\_{1n} \\
\sigma\_{21} & \sigma\_2^2 & \dots & \sigma\_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\sigma\_{n1} & \sigma\_{n2} & \dots & \sigma\_n^2
\end{pmatrix} \tag{5}
$$

where,

$$\sigma\_{\vec{\eta}} = \text{Cov}(r\_i, r\_j) = \frac{\sum\_{i=1, j=1}^{n} (r\_i - \mu\_i) \left(r\_j - \mu\_j\right)}{n - 1} \tag{6}$$

Equation (6) can be interpreted as the sum of the distance for each value and from the mean is divided by the number of observations minus one. Then, the covariance enables us to calculate the correlation coefficient, shown as:

$$
\rho = \frac{\sigma\_{\vec{\eta}}}{\sigma\_{\vec{\imath}} \sigma\_{\vec{\jmath}}} \tag{7}
$$

where *σ* is the standard deviation of each asset. This measure is useful to determine the degree of portfolio risk [1].

In this study, the standard deviation is used to measure the risk of the portfolio. The standard deviation of the portfolio is the most common statistical indicator of an asset's risk which measures the dispersion around the expected value [12]. By means, the higher the risk, the higher value of standard deviation. Hence, the equation for standard deviation is defined as follows:

*An Optimal Control Approach to Portfolio Diversification on Large Cap Stocks Traded… DOI: http://dx.doi.org/10.5772/intechopen.100613*

**Figure 1.** *Efficient frontier [15].*

$$\sigma\_p = \sqrt{\sum\_{i=1}^{n} (w\_i^2 \cdot \sigma\_i^2) + 2 \left(\sum\_{i=1}^{n} \sum\_{j=1}^{n} w\_i \cdot \sigma\_i \cdot w\_j \cdot \sigma\_j \cdot \rho\right)} \tag{8}$$

where,

*σ<sup>p</sup>* is the standard deviation of a portfolio,

*σ<sup>i</sup>* is the standard deviation of stocks,

*wi* is the weight of stocks in a portfolio,

*ρ* is the correlation coefficient between stock *i* and *j*.

Based on the parameters used in the Markowitz mean-variance model, we can get different combinations of expected return and risk. Every possible asset combination is known as an attainable set that can be plotted in risk-return space, and the collection of all such possible portfolios defines a region in this space. **Figure 1** shows that the line along the upper edge of this region is known as the efficient frontier. It is defined as the best return of a portfolio with the lowest risk.

#### **2.3 Performance evaluation ratio**

In this study, the Sharpe ratio will be used to measure the portfolio performance that provides risk premium per unit of total risk, which is measured by the portfolio's standard deviation of return. The risk premium is defined as the difference between portfolio return and the risk-free rate. It can be expressed in the following equation:

$$S\_p = \frac{\mu\_p - R}{\sigma\_p} \tag{9}$$

where *R* is the return on the risk-free asset.

#### **3. Results and discussion**

In this study, excel functions and data solvers are used for all calculations. The construction of portfolio optimization using the Markowitz mean-variance approach involves the following steps:

Step 1: Determining return and standard deviation of 10 stocks during three subperiods (**Tables 2**–**4**).

Step 2: Creating correlations matrix (**Tables 5**–**7**).

Step 3: Creating variance-covariance matrices during three subperiods (**Tables 8**–**10**).

Step 4: Calculating the volatility and return of the portfolio with equal weight (**Table 11**).

Step 5: Calculating the volatility and return of the portfolio with difference weights (**Tables 12**–**14** (all three tables in the Appendices).

Step 6: Creating an efficient frontier (**Figure 2A**–**C**).

**Table 2** reflects during GFC from 2008 to 2010, most of the stock were volatile at high risk and yielding negative returns since the bearish market except Honda Motor, Tokyo Electron, and Mitsui Fudosan which annual returns vary from 0.08 to 6.24%. The risks of 10 stocks vary from 30.05 to 51.63% and Mizuho Financial Group, Inc. has the highest risk recorded during the crisis. As seen from **Table 3**, all


**Table 2.**

*Risk and return of stocks during global financial crisis (GFC) (2008–2010).*


**Table 3.** *Risk and return of stocks during post-global financial crisis (2011–2013).* **Stocks Average weekly return (post-GFC) Weekly variance Weekly standard deviation Average annual return Annual variance Annual standard deviation** 7201.T 0.001 0.001 0.035 0.030 0.062 0.249 7267.T �0.001 0.002 0.041 �0.049 0.087 0.296 8411.T �0.001 0.001 0.038 �0.028 0.076 0.275 8306.T 0.000 0.002 0.040 0.000 0.082 0.286 8035.T 0.004 0.002 0.048 0.217 0.119 0.345 6702.T 0.002 0.002 0.048 0.101 0.121 0.349 8801.T �0.001 0.002 0.050 �0.041 0.129 0.359 8802.T �0.002 0.002 0.049 �0.079 0.125 0.353 9342.T 0.002 0.001 0.027 0.115 0.039 0.198 9433.T 0.001 0.002 0.040 0.071 0.084 0.290

*An Optimal Control Approach to Portfolio Diversification on Large Cap Stocks Traded… DOI: http://dx.doi.org/10.5772/intechopen.100613*

#### **Table 4.**

*Risk and return of stocks during non-crisis period (2014–2018).*


#### **Table 5.**

*Correlation matrix during GFC (2008–2010).*

stocks are yielding positive returns ranging from 4.53 to 33.73% during the period of post-GFC. Mitsui Fudosan lead the market with the highest annual return and the lowest annual return is Fujitsu. Despite the fact that the annual returns show a difference, both stocks are significantly risky, with risk levels ranging between 35.33 and 33.95%. **Table 4** shows during the non-crisis period, about four stocks have poor performance with a negative rate of return. The remaining stocks are ranging from 0.00 to 21.7%. Tokyo Electron is dominating the market with the highest return and the Mitsubishi UFJ Financial Group, Inc. is recovering with the lowest return. The risks of each stock are varying from 19.8 to 35.3% which is dominated by the Mitsui Fudosan. The lowest risk recorded is from Nippon Telegraph and Telephone Corporation. Thus, apart from the results of the risk and return of the individual stock from the three subperiods, the variation of risks per weekly and annual basis has indicated that there is a chance of high variability of investment.


#### **Table 6.**

*Correlation matrix during post-GFC (2010–2013).*


#### **Table 7.**

*Correlation matrix during the non-crisis period (2014–2018).*


#### **Table 8.**

*Variance-covariance matrix during GFC.*

*An Optimal Control Approach to Portfolio Diversification on Large Cap Stocks Traded… DOI: http://dx.doi.org/10.5772/intechopen.100613*


#### **Table 9.**

*Variance-covariance matrix during post-GFC.*


#### **Table 10.**

*Variance-covariance matrix during non-crisis period.*


#### **Table 11.**

*Portfolio with equal weight (naïve diversification).*

Correlation between assets determines the degree of portfolio risk [11]. Negative or small correlation between assets, the risk of the portfolio is low whereas the positive or large correlation between the assets, the risk of the portfolio is high. As seen in **Table 5**, only 9342.T–8306.T, 9433.T–8801.T, and 9433.T–8802.T have a negative correlation between the stocks and others are greater than zero. **Table 6** reflects that only two pairs of stocks which are 9433.T–8035.T and 9433.T–9342.T have a negative correlation and **Table 7** has all positive correlation between the

stocks and not low enough. In that sense, only during the non-crisis period the portfolio constructed will not be well diversified.

When there are more than two assets, covariance can best be calculated by using matrix algebra based on Eq. (6). The covariance calculation which involved the excess return of stocks and the number of observations in each subperiod allows us to compute by using few functions in excel such as @MMULT( … ) and TRANS-POSE( … ). The procedure has already allowed us to determine the variancecovariance matrix as per **Tables 8**–**10**. The values of the variance-covariance matrices have shown that in the three subperiods tested, all the stocks are move in the same direction with a lower degree. Thus, this indicates that the volatility can be reduced during the construction of the portfolios.

The variance-covariance matrix helps in a simple way of measuring portfolio variance. At this step, the portfolio return and portfolio variance can be calculated. In this sense, the naïve allocation of weight whereby the equal proportion in the portfolio is used to calculate the portfolio return and portfolio variance. Since we have 10 stocks, each stock will have about 1/10 or 10% weight.

So, **Table 11** shows that the portfolio returns and portfolio variance of the naïve diversification strategy. Among the three subperiods, during GFC, the portfolio indicates poor performance by having a negative return of �0.12% and the highest standard deviation of 4.35%. In contrast, the portfolio return is recorded high during post-GFC which is about 0.36% with the lowest standard deviation of 3.10%.

Note that, Markowitz's mean-variance model stated that the investor must be rational and risk-averse [16]. Thus, in constructing the efficient portfolios, two conditions need to be satisfied which are maximum return for varying levels of risk and minimum risk for varying levels of expected return. In the context of three subperiods tested, this study proposed rational investors for looking at optimal portfolio with a minimum level of risk. Thus, this optimal portfolio analysis can be shown as the subject function according to the formula:

$$\text{Min}\sigma^2 \sum\_{j=1}^n w\_i w\_j \text{Cov}\_{ij} \tag{10}$$

where *wi* and *w <sup>j</sup>* are weights of stocks in the portfolio and *Covij* is the covariance value between stock *i* and *j*.

Then, three main constrains in Markowitz mean-variance portfolio optimization are included for the optimization problem. The formulas are written as:

$$\sum\_{i=1}^{n} w\_i E(r\_i)^3 \ge E \ast \tag{11}$$

$$\sum\_{i=1}^{n} w\_i = 1 \tag{12}$$

and

$$wi \ge 0; i = 1, \dots, N \tag{13}$$

where *E R*ð Þ*<sup>i</sup>* is the target expected return, *<sup>E</sup>*<sup>∗</sup> is an expected return and *wi* is the weight of the stock *i*. The third constraint is added to restrict the short sell to happen.

Hence, Excel Solver is used to optimizing the weight by including all the constraints according to Eqs. (10)–(13). About 10 iterations were done in Solver to

*An Optimal Control Approach to Portfolio Diversification on Large Cap Stocks Traded… DOI: http://dx.doi.org/10.5772/intechopen.100613*

**Figure 2.**

*(a) Efficient frontier of portfolios during GFC, (b) efficient frontier of portfolios during post-GFC, and (c) efficient frontier of portfolios during the non-crisis period.*

generate 10 portfolios. The 10 portfolios that are generated produce 10 different sets of weights for the level of return which minimizes the standard deviation of the portfolio. All the results are shown in **Tables 12**–**14** (in Appendices) indicate the composition of stocks, portfolio return, and portfolio risk. All the set of possible expected return and risk combinations is then called attainable set [1]. The attainable set is plotted on the risk-return space as shown in **Figure 2A**–**C**.

**Figure 2A**–**C** demonstrate the efficient frontier with minimum variance portfolios. Ivanova et al. [11] states that given expected return, investors could choose an optimal portfolio based on risk preference since all the efficient portfolios are located over the efficient frontier curve. Many portfolios are existed on the efficient frontier curve with different risk and return combinations. Then, the optimal portfolio is selected on the efficient frontier curve in each subperiods then compare with the portfolio return and risk for naïve diversification strategy.

**Table 15** reflects the analysis on the risk, return and performance of the portfolio constructed by using naïve diversification strategy and Markowitz meanvariance portfolio optimization. In the three subperiods tested, the optimal portfolio outperformed the naïve diversification strategy.

In terms of portfolio risk-return, optimal portfolios selected during global financial crisis tend to have a high standard deviation of 4.71% compared to naïve diversification which is about 4.35%, respectively. But, the portfolio with equal weight yields an abnormal return of �0.12% compared to the optimal portfolio of 0.06%. This is not a surprising result since the turbulence of economic recession contributes to the high-risk investment. During the post-GFC, again the optimal portfolio yields a higher return with 0.5% with lower risk at 2.67% compared to the naïve strategy. Furthermore, there is not much difference in standard deviation for both strategies in the non-crisis period, but the optimal portfolio still dominates with a high return at 2.5% and low risk at 3.44%.

Sharpe ratio classifies during financial crisis both strategies produce negative Sharpe ratio which indicates the portfolio's return is less than the risk-free rate. Thus, the interpretation of the negative Sharpe ratio will be put aside since it does not convey any useful meaning. For the rest two periods, the Sharpe ratio is undoubtedly the highest for the optimal portfolio compare to the naïve portfolio. This shows the reward-to-volatility ratio when investing in the optimal portfolio 6 and portfolio 7 are 0.1032 and 0.0582.

#### **4. Conclusion**

This study was aimed to help investors to plan for the best investment strategy in maximizing return with the given level of risk or minimizing risk. In this study, there are subperiods been tested for this whole study, during the global financial crisis (GFC) 2008–2010, the post-financial crisis 2011–2013 and non-financial crisis 2014–2018. Then, we followed the Markowitz mean-variance model which involved the best possible combination of expected return and risk to construct efficient portfolios in each period. The portfolio which did not contain short sale was achieved by a data solver. We made a choice to choose the optimal portfolio from this efficient set based on rational investors.

Meantime, we also constructed a portfolio with equal weight as a control parameter to determine the best diversification strategy. It was figured out during the period of study; the optimal portfolio was chosen from the efficient frontier formed by Markowitz model perform better than the portfolio with equal weight in all three subperiods. The result reiterates the previous study conducted by [1, 9, 11]. Hence, the Markowitz framework is successful since the variance-covariance and correlation played an important role in determining the optimal portfolio. So, if the Japanese and foreign investors know properly how to apply the Markowitz mean-variance model in their investment. We believe this is the best solution in many alternatives.

But, the limitation of this study is there is no out-of-sample data tested. For future research, a robust optimization approach can be considered with the out-ofsample data tested to construct the optimal portfolio. The portfolio also needs to be rebalanced again to ensure a more accurate result.

#### **Acknowledgements**

I would to express my deepest appreciation to all those who provided me the possibility to complete this research paper. A special gratitude I give to my supervisor Dr. Norizarina binti Ishak and my friend Nur Fathin Shaida Binti Muhammad Nadhirin whose contribution in stimulating suggestions and encouragement, helped me to coordinate my research especially in writing this research paper.


*An Optimal Control Approach to Portfolio Diversification on Large Cap Stocks Traded … DOI: http://dx.doi.org/10.5772/intechopen.100613*

**Appendices**  *Composition of weight and risk of different expected return portfolios during GFC period.*



**Table**

*Control Systems in Engineering and Optimization Techniques*


*An Optimal Control Approach to Portfolio Diversification on Large Cap Stocks Traded … DOI: http://dx.doi.org/10.5772/intechopen.100613*

**Table 14.**

*Composition of weight and risk of different expected return portfolios during non-crisis period.*

#### *Control Systems in Engineering and Optimization Techniques*


**Table 15.**

*Comparison of portfolio risk, return, and performance evaluation.*

#### **Author details**

Muhammad Jaffar Sadiq Abdullah and Norizarina Ishak\* Faculty of Science and Technology, Universiti Sains Islam Malaysia, Nilai, Malaysia

\*Address all correspondence to: norizarina@usim.edu.my

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*An Optimal Control Approach to Portfolio Diversification on Large Cap Stocks Traded… DOI: http://dx.doi.org/10.5772/intechopen.100613*

#### **References**

[1] Kulali I. Portfolio optimization analysis with Markowitz quadratic mean-variance model. European Journal of Business and Management. 2016; **8**(7):73-79

[2] Shukla V. Top 10 Largest Stock Exchanges in the World by Market Capitalization. Valuewalk [Internet]. 2019. Available from: https://www.va luewalk.com/2019/02/top-10-largeststock-exchanges/

[3] Shalit H, Yitzhaki S. The mean-Gini efficient portfolio frontier. Journal of Financial Research. 2005; **28**(1):59-75

[4] Baumöhl E, Lyócsa Š. Constructing weekly returns based on daily stock market data: A puzzle for empirical research? In: MPRA Paper 43431. Germany: University Library of Munich; 2012

[5] Brown SJ, Hwang I, In F. Why optimal diversification cannot outperform naive diversification: Evidence from tail risk exposure. SSRN Electronic Journal. 2013:1-55. DOI: 10.2139/ssrn.2242694

[6] DeMiguel V, Garlappi L, Uppal R. Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Review of Financial Studies. 2009;**22**(5):1915-1953

[7] Gupta M, Aggarwal N. Naïve versus mean-variance diversification in Indian capital markets. Asia-Pacific Journal of Management Research and Innovation. 2015;**11**(3):198-204

[8] Ramilton A. Should you optimize your portfolio?: On portfolio optimization: The optimized strategy versus the naïve and market strategy on the Swedish stock market. 2014. Available from: http://urn.kb.se/ resolve?urn=urn:nbn:se:uu:diva-218024

[9] Pflug GC, Pichler A, Wozabal D. The 1/N investment strategy is optimal under high model ambiguity. Journal of Banking & Finance. 2012;**36**(2):410-417. DOI: 10.1016/j.jbankfin.2011.07.018

[10] Garcia T, Borrego D. Markowitz efficient frontier and capital market line—Evidence from the Portuguese. Portuguese Journal of Management Studies. 2017;**22**(1):3-23

[11] Ivanova M, Dospatliev L. Application of Markowitz portfolio optimization on Bulgarian stock market from 2013 to 2016. International Journal of Pure and Applied Mathematics. 2017;**117**(2): 291-307. DOI: 10.12732/ijpam.v117i2.5

[12] Ivanovic Z, Baresa S, Bogdan S. Portfolio optimization on Croatian capital market. UTMS Journal of Economics. 2013;**4**(3):269-282

[13] García F, González-Bueno JA, Oliver J. Mean-variance investment strategy applied in emerging financial markets: Evidence from the Colombian stock market. Intellectual Economics. 2015;**9**(1):22-29

[14] Sun Y. Optimization stock portfolio with mean-variance and linear programming: Case in Indonesia stock market. Binus Business Review. 2010; **1**(1):15. DOI: 10.21512/bbr.v1i1.1018

[15] Chen WP, Chung H, Ho KY, Hsu TL. Portfolio optimization models and mean–variance spanning tests. In Handbook of quantitative finance and risk management. Boston, MA: Springer. 2010;165-184

[16] Markowitz H. Portfolio Selection\*. The Journal of Finance. 1952;**7**:77-91. DOI: 10.1111/j.1540-6261.1952.tb01525.x

#### **Chapter 2**

## Existence, Uniqueness and Stability of Fractional Order Stochastic Delay System

*Sathiyaraj Thambiayya, P. Balasubramaniam, K. Ratnavelu and JinRong Wang*

#### **Abstract**

This chapter deals with the problem of fractional higher-order stochastic delay systems. A solution representation is given by using sin and cos matrix functions for different delay intervals. Further, existence and uniqueness results are proved through fixed point theorem. Moreover, finite-time stability criteria are obtained using fractional Gronwall-Bellman inequality lemma. Finally, numerical simulation is carried out to check the proposed theoretical results.

**Keywords:** existence and uniqueness of solution, fixed point theorem, fractional differential equations (FDEs), stochastic differential system

#### **1. Introduction**

Fractional derivatives (FD) initiative concept is quite old and its history spans three centuries. The variety of papers dedicated to FD is multiplied swiftly in the mid-twentieth century and later decades. One of the motives for the full-size interest within the discipline of FD is that it's far feasible to describe the variety of physical [1], synthetic [2], and organic [3] occurrence with fractional differential equations (FDEs). As a new branch of applied mathematics, the field of FD can be seen in many applications. Nevertheless, more and more compelling implementations have been found in various engineering and science fields over the past few decades (see [4]). It is noted that the existing theory of FDEs is committed to a larger part of the research works (see [5–8]). While modeling functional structures, ambient noise and time delays need to be taken into account, which might be very beneficial in building extra sensible fashions of sciences, and so on [9]. It is referred to as the pattern direction houses of the stochastic fractional partial differential gadget powered by way of area time white noise [10].

The problems in a stochastic environment replicate the modeling of single-sever *m*-mode random queues in computer networks [11], the spatial distribution of mobile users in the telecommunications network coverage area [12], and other anomalies that occurred naturally in many disciplines [13]. Authors in [14] investigated the existence, uniqueness, and large deviation principle solutions to stochastic evolution equations of jump type. Among the many meaningful properties of stochastic stability results describe the maximum vital feature of fractional order stochastic systems and have been investigated in Refs. [15–18]. The notion of

finite-time stability for fractional stochastic delay systems occurs a matter of course in stochastic control systems. Without any doubt that this type of fractional stochastic stability is most important in both theory and applications.

However, only few introductions and discussions exist on the definition of finite-time stability in stochastic finite space using fixed point theorem approach. Burton [19] started to analyze the stability characters of dynamical systems broadly using fixed point theorems. Subsequently, few authors applied fixed point approach to establish sufficient conditions for stability of the differential systems (see [20–24]). Based on the above discussions, this chapter provides finite-time stability of the Caputo sense FDEs via fixed point theorems.

The primary contribution of this chapter is defined as follows:


Novelties and challenges of this chapter are described through the subsequent statements:


Organization of this chapter is as follows: system description and preliminaries are provided in Section 2. Existence and uniqueness of solution are provided in Section 3. Finite-time stability result is proved in Section 4 and Section 5 consists of a numerical example.

**Notations:** *CD<sup>q</sup>* �*κ*<sup>þ</sup> represent respectively the Caputo derivative with *<sup>q</sup>*∈ð Þ 0, 1 ; *<sup>R</sup><sup>n</sup>* and *<sup>R</sup><sup>n</sup>*�*<sup>n</sup>* represent the *<sup>n</sup>*�dimensional Euclidean space and *<sup>n</sup>* � *<sup>n</sup>* real matrix; *E*ð Þ� denotes the mathematical expectation with some probability measure; <sup>Ω</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> *<sup>F</sup>*<sup>0</sup> ½ � 0, *<sup>b</sup>* , *<sup>R</sup><sup>n</sup>* ð Þ, <sup>∥</sup> � <sup>∥</sup> � �; for any *<sup>y</sup>*<sup>∈</sup> *Rn*, we define the norm

$$\|\mathcal{y}(\eta)\| = \sqrt{\sup\_{\eta \in \left[-\kappa, b\right]} \left\{ e^{-2N\eta} \left|\mathcal{y}(\eta)\right|^2 \right\}};$$

define a column-wise matrix sum

$$\|M\| = \max\left\{ \sum\_{k=1}^{j} |m\_{k1}|, \sum\_{k=1}^{j} |m\_{k2}|, \dots, \sum\_{k=1}^{j} |m\_{kjn}| \right\}.$$

Further, let as define the matrix norm

$$\max\_{\eta \in [-\kappa, 0]} \left\{ \varepsilon^{-2N\eta} |\psi(\eta)|^2 \right\} = \left\| \left| \mu \right| \right\|^2, \max\_{\eta \in [-\kappa, 0]} \left\{ \varepsilon^{-2N\eta} |\nu'(\eta)|^2 \right\} = \left\| |\nu'| \right\|^2.$$

#### **2. System description and preliminaries**

Consider the following system:

$$\begin{cases} \,^C D\_{-\kappa^+}^q \left( ^C D\_{-\kappa^+}^q y \right)(\eta) + M^2 y(\eta - \kappa) = F(\eta, y(\eta)) + \int\_0^\eta \Delta(\zeta, y(\zeta)) d\nu(\zeta), & \eta \in [0, b], \ \kappa > 0, \\\\ y(\eta) = y(\eta), \quad y'(\eta) = y'(\eta), \quad \eta \in [-\kappa, 0], \ \kappa > 0, \end{cases} \tag{1}$$

where *<sup>y</sup>*ð Þ*<sup>η</sup>* <sup>∈</sup>*R<sup>n</sup>* is a state vector. Here, *<sup>M</sup>* <sup>∈</sup>*R<sup>n</sup>*�*<sup>n</sup>* is taken as a nonsingular matrix. *<sup>F</sup>* is mapping from 0, ½ �� *<sup>b</sup> <sup>R</sup><sup>n</sup>* to *Rn* and <sup>Δ</sup> is a mapping from 0, ½ �� *<sup>b</sup> <sup>R</sup><sup>n</sup>* to *<sup>R</sup><sup>n</sup>*�*<sup>d</sup>* are nonlinear continuous function and *<sup>ψ</sup>* <sup>∈</sup>C<sup>1</sup> ½ � �*κ*, 0 , *<sup>R</sup><sup>n</sup>* ð Þ is an initial value function. *<sup>w</sup>* denotes *d*�dimensional Wiener process.

**Definition 2.1.** ([5]) *The Caputo derivative for f* : ½ Þ! �*κ*, ∞ *R*, *is*

$$\left( \,^C D\_{-\kappa}^q f \right)(\eta) = \frac{1}{\Gamma(\mathfrak{1} - q)} \int\_{-\kappa}^{\eta} (\eta - \zeta)^{-q} f'(\zeta) d\zeta, \; q \in (0, 1], \; \eta > -\kappa, \; f'(\eta) = \frac{df}{d\eta}.$$

**Definition 2.2.** *(see* [5]*) Mittag-Leffler function is*

$$E\_{q,p}(u) = \sum\_{k=0}^{\infty} \frac{u^k}{\Gamma(kq+p)} \quad \text{for } q, p > 0.$$

*In particular, for p* ¼ 1*,*

$$E\_{q, \mathbf{1}}(\theta u^q) = E\_q(\theta u^q) = \sum\_{k=0}^{\infty} \frac{\theta^k u^{kq}}{\Gamma(qk+\mathbf{1})}, \quad \theta, u \in \mathbb{C}.$$

**Definition 2.3.** *(see* [25]*) The* 2*kq degree of polynomial for delayed fractional* cos *matrix is given at η* ¼ *kκ*, *k* ¼ 0, 1, ⋯

$$\cos\_{\kappa q}M\eta^{q} = \begin{cases} \Theta, & -\infty < \eta < -\kappa, \\ I, & -\kappa \le \eta < 0, \\ \vdots & \vdots \\ I - M^{2} \frac{\eta^{2q}}{\Gamma(2q+1)} + \dots + (-1)^{k}M^{2k} \frac{(\eta - (k-1)\kappa)^{2k\kappa}}{\Gamma(2kq+1)}, & (k-1)\kappa \le \eta < k\kappa, \\ \vdots & \vdots \end{cases}$$

where Θ and *I* represent the zero and identity matrices.

**Definition 2.4.** *(*[25]*) The* ð Þ 2*k* þ 1 *q degree of polynomial for a delayed fractional* sin *matrix is given at η* ¼ *kκ*, *k* ¼ 0, 1, ⋯

$$\sin \kappa\_{\boldsymbol{\eta}} M \eta^{\boldsymbol{\eta}} = \begin{cases} \Theta, & -\infty < \boldsymbol{\eta} < -\kappa, \\ M \frac{(\boldsymbol{\eta} + \kappa)^{q}}{\Gamma(q+1)}, & -\kappa \le \boldsymbol{\eta} < 0, \\ \vdots & \vdots \\ M \frac{(\boldsymbol{\eta} + \kappa)^{q}}{\Gamma(q+1)} + \dots + (-1)^{k} M^{2k+1} \frac{(\boldsymbol{\eta} - (k-1)\kappa)^{(2k+1)q}}{\Gamma[(2k+1)q+1]}, & (k-1)\kappa \le \boldsymbol{\eta} < k\kappa, \\ \vdots & \vdots \end{cases}$$

We have the following square norm estimations:

$$\begin{aligned} \text{i. } \|\cos\_{\kappa,q}M\eta^q\|^2 &= \left(\sum\_{k=0}^\infty \frac{\left(\|M\|\nu^{2q}\right)^k}{\Gamma(2kq+1)}\right)^2 \le \left[\mathbb{E}\_{2q}\left(\|M\|^2\eta^{2q}\right)\right]^2, \quad \eta \in [(k-1)\kappa, k\kappa),\\ k = \mathbf{0}, \mathbf{1}, \mathbf{2}, \cdots & \mathbf{n} \end{aligned}$$

ii.

$$\begin{split} & \| \sin \chi\_{\kappa} M \eta^{q} \|^{2} \\ &= \sum\_{k=0}^{\infty} \frac{\left( \| M \| (\eta + \kappa)^{q} \right)^{2k}}{\left( \Gamma(kq + 1) \right)^{2}} + \sum\_{k=0}^{\infty} \frac{\left( \| M \| ^{2} (\eta + \kappa)^{2q} \right)^{2k}}{\left( \Gamma(2kq + 1) \right)^{2}} \\ & \quad + 2 \sum\_{k\_{1}=0}^{\infty} \sum\_{k\_{2}=0}^{\infty} \frac{\left( \| M \| (\eta + \kappa)^{q} \right)^{k\_{1}}}{\Gamma(k\_{1}q + 1)} \frac{\left( \| M \| ^{2} (\eta + \kappa)^{2q} \right)^{k\_{2}}}{\Gamma(2k\_{2}q + 1)} \\ & \leq \left[ \mathbb{E}\_{\mathbb{Q}} (\| M \| (\eta + \kappa)^{q}) \right]^{2} + \left[ \mathbb{E}\_{\mathbb{Q}} \left( \| M \| ^{2} (\eta + \kappa)^{2q} \right) \right]^{2} \\ & \quad + 2 \mathbb{E}\_{\mathbb{Q}} (\| M \| (\eta + \kappa)^{q}) \mathbb{E}\_{\mathbb{Q}} \Big{(} \| M \| ^{2} (\eta + \kappa)^{2q} \Big{)}, \quad \eta \in [(k - 1)\kappa, k\kappa), \ k = 0, 1, 2, \cdots \\ \end{split}$$

**Definition 2.5.** *System* (1) *satisfying y*ð Þ� *η ψ η*ð Þ *and y*<sup>0</sup> ð Þ� *η ψ*<sup>0</sup> ð Þ*η for* �*κ* ≤*η*≤0 *is finite-time stable in mean square with respect to* f g 0, 0, ½ � *b* , *δ*, *ε*, *κ* , *if and only if <sup>δ</sup>*<sup>1</sup> <sup>&</sup>lt;*δ δ*ð Þ <sup>&</sup>gt; <sup>0</sup> *implies E*∥*y*ð Þ*<sup>η</sup>* <sup>∥</sup><sup>2</sup> <sup>&</sup>lt; *ε ε*ð Þ <sup>&</sup>gt; <sup>0</sup> , <sup>∀</sup> *<sup>η</sup>*∈½ � 0, *<sup>b</sup> where <sup>δ</sup>*<sup>1</sup> <sup>¼</sup> max ∥*ψ*∥<sup>2</sup> , ∥*ψ*<sup>0</sup> ∥<sup>2</sup> � � *denotes the initial time of observation of the system.*

**Lemma 2.1.** *[26](Generalized Gronwall-Bellman inequality) Let v*ð Þ*η* , *b*ð Þ*η be nonnegative and locally integrable on* 0≤*η*<*b and let h*ð Þ*η be a nonnegative, nondecreasing continuous function defined on* 0≤*η*<*b*, *h*ð Þ*η* ≤ *M*, *and let M be a real constant, q*>0 *with*

$$\nu(\eta) \le b(\eta) + h(\eta) \int\_0^\eta (\eta - \zeta)^{q-1} \nu(\zeta) d\zeta$$

and then

$$v(\eta) \le b(\eta) + \int\_0^\eta \left[ \sum\_{k=1}^\infty \frac{(h(\eta)\Gamma(q))^k}{\Gamma(kq)} (\eta - \zeta)^{kq-1} v(\zeta) d\zeta \right].$$

*Moreover, if b*ð Þ*η is a nondecreasing function on* ½ � 0, *b : Then*

$$\nu(\eta) \le b(\eta) E\_{q,1}(h(\eta)\Gamma(q)\eta^q), \quad \eta \in [0, b], \eta$$

where *Eq*,1ð Þ� is the one parameter Mittag-Leffler function.

**Assumption 1:** Let *x*, *y* ∈*Rn*, then we take

$$\sup\_{\eta \in [-\kappa, b]} e^{-2N\eta} \left| \varkappa(\eta) - \jmath(\eta) \right|^2 = E \left\| \varkappa(\eta) - \jmath(\eta) \right\|^2.$$

**Lemma 2.2.** *For a nonsingular matrix M*, *the solution of the inhomogeneous system is*

$$\begin{cases} \,^C D\_{-\kappa^+}^q \left( ^C D\_{-\kappa^+}^q \boldsymbol{\nu} \right)(\eta) + M^2 \boldsymbol{\jmath}(\eta - \kappa) = \boldsymbol{f}(\eta), \ \eta \in [0, b], \quad \kappa > 0, \\\\ \boldsymbol{\nu}(\eta) = \boldsymbol{\nu}(\eta), \\\\ \boldsymbol{\nu}'(\eta) = \boldsymbol{\nu}'(\eta), \quad \eta \in [-\kappa, 0], \end{cases} \tag{2}$$

for zero initial value has the below form:

$$\mathcal{Y}(\eta) = \int\_0^{\eta} \cos\_{\kappa, q} M(\eta - \kappa - \zeta)^q f(\zeta) d\zeta, \quad \eta \in [0, b].$$

*Proof.* Consider

$$\mathcal{Y}(\eta) = \int\_0^{\eta} \cos\_{\kappa, q} \mathcal{M}(\eta - \kappa - \zeta)^q \mathcal{C}(\zeta) d\zeta$$

where *<sup>C</sup>*ð Þ*<sup>ζ</sup>* (unknown) *<sup>ζ</sup>* <sup>∈</sup>½ � 0, *<sup>η</sup> :* By applying *CD<sup>q</sup>* �*κ*<sup>þ</sup> *CD<sup>q</sup>* �*κ*<sup>þ</sup> � � on both sides of the above equation one can obtain

$$\begin{aligned} \, ^C D\_{-\kappa^+}^q ( ^C D\_{-\kappa^+}^q \chi )(\eta) &= \left( \cos\_{\kappa;q} M \eta^q \right) C(\eta) - M^2 \Big\vert\_0^\eta \cos\_{\kappa;q} M (\eta - 2\kappa - \zeta)^q C(\zeta) d\zeta \\ &= C(\eta) - M^2 \Big\vert\_0^\eta \cos\_{\kappa;q} M (\eta - 2\kappa - \zeta)^q C(\zeta) d\zeta \\ &+ M^2 \int\_0^{\eta - \kappa} \cos\_{\kappa;q} M (\eta - 2\kappa - \zeta)^q C(\zeta) d\zeta. \end{aligned}$$

Substitute the above expression into (2), one can get

$$C(\eta) - M^2 \int\_0^{\eta} \cos\_{\kappa \eta} M(\eta - 2\kappa - \zeta)^q C(\zeta) d\zeta + M^2 \int\_0^{\eta - \kappa} \cos\_{\kappa \eta} M(\eta - 2\kappa - \zeta)^q C(\zeta) d\zeta = f(\eta),$$

since Ð *<sup>η</sup> <sup>η</sup>*�*<sup>κ</sup>* cos *<sup>κ</sup>*,*qM*ð Þ *<sup>η</sup>* � <sup>2</sup>*<sup>κ</sup>* � *<sup>ζ</sup> <sup>q</sup> <sup>C</sup>*ð Þ*<sup>ζ</sup> <sup>d</sup><sup>ζ</sup>* <sup>¼</sup> <sup>0</sup>*:* Hence the proof. □ Using [25] and Lemma 2.2, the solution of (1) is

$$\begin{split} \mathcal{Y}(\eta) &= \left( \cos\_{\kappa,q} M \eta^q \right) \mathfrak{y}(-\kappa) + M^{-1} \Big( \sin\_{\kappa,q} M (\eta - \kappa)^q \big) \mathfrak{y}'(0) + \int\_{-\kappa}^0 \cos\_{\kappa,q} M (\eta - \kappa - \zeta)^q \mathfrak{y}'(\zeta) d\zeta \\ &+ \int\_0^\eta \cos\_{\kappa,q} M (\eta - \kappa - \zeta)^q F(\zeta, \mathfrak{y}(\zeta)) d\zeta \\ &+ \int\_0^\eta \cos\_{\kappa,q} M (\eta - \kappa - \zeta)^q \Big( \int\_0^\zeta \Delta(\lambda, \mathfrak{y}(\lambda)) d\nu(\lambda) \Big) d\zeta. \end{split}$$

$$\begin{split} (\mathcal{P}\mathfrak{y})(\eta) &= \left(\cos\_{\kappa,q}M\eta^{q}\right)\mathfrak{y}(-\kappa) + M^{-1}\Big(\sin\_{\kappa,q}M(\eta-\kappa)^{q}\big)\mathfrak{y}'(\mathbf{0}) \\ &+ \int\_{-\kappa}^{0} \cos\_{\kappa,q}M(\eta-\kappa-\zeta)^{q}\mathfrak{y}'(\zeta)d\zeta + \int\_{0}^{\eta} \cos\_{\kappa,q}M(\eta-\kappa-\zeta)^{q}F(\zeta,\mathfrak{y}(\zeta))d\zeta \\ &+ \int\_{0}^{\eta} \cos\_{\kappa,q}M(\eta-\kappa-\zeta)^{q}\left(\int\_{0}^{\zeta} \Delta(\lambda,\mathfrak{y}(\lambda))d\mathfrak{w}(\lambda)\right)d\zeta, \quad \eta \in [0b]. \end{split}$$

$$K := 2\left[E\_{2q}\left(\|M\|^2 (b+\kappa)^{2q}\right)\right]^2 \left[\left(\frac{e^{-Nb}-1}{Nb}\right)^2 + 4b\left(\frac{e^{-2Nb}-1}{2Nb}\right)\right] < 1.1$$

$$\begin{split} \left| (\mathcal{P}\mathbf{x})(\eta) - (\mathcal{P}\mathbf{y})(\eta) \right|^2 &\leq 2 \left| \int\_0^\eta \cos \kappa\_{\kappa,\eta} M(\eta - \kappa - \zeta)^q [F(\zeta, \mathbf{x}(\zeta)) - F(\zeta, \mathbf{y}(\zeta))] d\zeta \right|^2 \\ &+ 2 \left| \int\_0^\eta \cos \kappa\_{\kappa,\eta} M(\eta - \kappa - \zeta)^q \left( \int\_0^\zeta [\Delta(\lambda, \mathbf{x}(\lambda)) - \Delta(\lambda, \mathbf{y}(\lambda))] dw(\lambda) \right) d\zeta \right|^2. \end{split}$$

$$\begin{aligned} &e^{-2N\eta} \left| (\mathcal{P}\mathbf{x})(\eta) - (\mathcal{P}\mathbf{y})(\eta) \right|^2 \\ &\leq 2 \left| \int\_0^\eta \cos\_{\kappa\bar{\eta}} \mathcal{M}(\eta - \kappa - \zeta)^q e^{-N\zeta} [F(\zeta, \mathbf{x}(\zeta)) - F(\zeta, \mathbf{y}(\zeta))] d\zeta \right|^2 \\ &\quad + 2 \left| \int\_0^\eta \cos\_{\kappa\bar{\eta}} \mathcal{M}(\eta - \kappa - \zeta)^q e^{-N\zeta} \left( \int\_0^\zeta [\Delta(\lambda, \mathbf{x}(\lambda)) - \Delta(\lambda, \mathbf{y}(\lambda))] d\mathbf{w}(\lambda) \right) d\zeta \right|^2 := 2[(i) + (ii)]. \end{aligned} \tag{3}$$

$$\begin{split} & \left| \int\_{0}^{\eta} \cos\_{\kappa, q} M(\eta - \kappa - \zeta)^{q} e^{-N\zeta} [F(\zeta, \kappa(\zeta)) - F(\zeta, \jmath(\zeta))] d\zeta \right|^{2} \right| \\ & \leq \left( \int\_{0}^{\eta} |\cos\_{\kappa, q} M(\eta - \kappa - \zeta)^{q}|^{2} e^{-N(\eta - \zeta)} d\zeta \right) \\ & \qquad \times \left( \int\_{0}^{\eta} e^{-N(\eta - \zeta)} e^{-2N\zeta} |F(\zeta, \kappa(\zeta)) - F(\zeta, \jmath(\zeta))|^{2} d\zeta \right) \\ & \leq \left[ E\_{2q} \left( \|M\|^{2} (b + \kappa)^{2q} \right) \right]^{2} \left( \int\_{0}^{\eta} e^{-N(\eta - \zeta)} d\zeta \right) \\ & \qquad \times \left( \int\_{0}^{\eta} e^{-N(\eta - \zeta)} d\zeta \right) e^{-2N\eta} |F(\eta, \kappa(\eta)) - F(\eta, \jmath(\eta))|^{2} \\ & \leq \left[ E\_{2q} \left( \|M\|^{2} (b + \kappa)^{2q} \right) \right]^{2} \left( \int\_{0}^{\eta} e^{-N(\eta - \zeta)} d\zeta \right)^{2} E \|\varkappa(\eta) - \jmath(\eta)\|^{2} \\ & \leq \left[ E\_{2q} \left( \|M\|^{2} (b + \kappa)^{2q} \right) \right]^{2} \left( \frac{e^{-Nb} - 1}{Nb} \right)^{2} E \|\varkappa(\eta) - \jmath(\eta)\|^{2} . \end{split}$$

By Burkholder-Davis-Gundy inequality and Assumption 1, the estimate for (ii) is given by

$$\begin{split} & \left| \int\_{0}^{\eta} \cos\_{\kappa q} \mathsf{M}(\eta - \kappa - \zeta)^{q} e^{-N\zeta} \left( \int\_{0}^{\zeta} [\Delta(\lambda, \mathsf{x}(\lambda)) - \Delta(\lambda, \mathsf{y}(\lambda))] d\mathsf{w}(\lambda) \right) d\zeta \right|^{2} \\ & \leq 4b \int\_{0}^{\eta} |\cos\_{\kappa q} \mathsf{M}(b - \kappa - \zeta)^{q}|^{2} e^{-2N(b - \zeta)} e^{-2N\zeta} |\Delta(\zeta, \mathsf{x}(\zeta)) - \Delta(\zeta, \mathsf{y}(\zeta))|^{2} d\zeta \\ & \leq 4b \left[ E\_{2q} \left( \|\mathsf{M}\|^{2} (b + \kappa)^{2q} \right) \right]^{2} \left( \int\_{0}^{b} e^{-2N(b - \zeta)} d\zeta \right) e^{-2N\eta} |\Delta(\eta, \mathsf{x}(\eta)) - \Delta(\eta, \mathsf{y}(\eta))|^{2} \\ & \leq 4b \left[ E\_{2q} \left( \|\mathsf{M}\|^{2} (b + \kappa)^{2q} \right) \right]^{2} \left( \frac{e^{-2Nb} - 1}{2Nb} \right) E \|\mathsf{x}(\eta) - \mathsf{y}(\eta)\|^{2}. \end{split}$$

From the above two estimates of (i) and (ii), Eq. (3) becomes

$$\begin{split} &E\|\left(\mathcal{P}\mathbf{x}\right)(\eta)-\left(\mathcal{P}\mathbf{y}\right)(\eta)\|^{2} \\ &\leq 2\left[E\_{2q}\left(\|M\|^{2}(b+\kappa)^{2q}\right)\right]^{2}\left(\frac{e^{-Nb}-1}{Nb}\right)^{2}E\|\mathbf{x}(\eta)-\mathbf{y}(\eta)\|^{2} \\ &\qquad+ 8b\left[E\_{2q}\left(\|M\|^{2}(b+\kappa)^{2q}\right)\right]^{2}\left(\frac{e^{-2Nb}-1}{2Nb}\right)E\|\mathbf{x}(\eta)-\mathbf{y}(\eta)\|^{2} \\ &\leq 2\left[E\_{2q}\left(\|M\|^{2}(b+\kappa)^{2q}\right)\right]^{2}\left[\left(\frac{e^{-Nb}-1}{Nb}\right)^{2}+4b\left(\frac{e^{-2Nb}-1}{2Nb}\right)\right]E\|\mathbf{x}(\eta)-\mathbf{y}(\eta)\|^{2} .\end{split}$$

This implies that

$$E\left\|(\mathcal{P}\mathbf{x})(\eta) - (\mathcal{P}\mathbf{y})(\eta)\right\|^2 \le KE\left\|\mathbf{x}(\eta) - \mathbf{y}(\eta)\right\|^2.$$

Hence, from statement of the Theorem 3.1, the nonlinear operator (P) is a contraction. Hence the nonlinear operator (P) has a unique solution *<sup>y</sup>*ð Þ� <sup>∈</sup> *Rn*, which is nothing but solution of Eq. (1). Hence the proof. □

#### **4. Finite-time stability**

**Theorem 4.1.** *If Assumption 1 hold and provided that*

$$\left[\mathfrak{S}\left[K\_1+\left|M^{-1}\right|^2\left(K\_3+K\_2-2\sqrt{K\_3}\sqrt{K\_2}\right)+\kappa^2K\_2\right]E\_{q,1}\left[\mathfrak{S}\frac{b^{2-q}}{2-q}K\_2\Gamma(q)\eta^q\right]\right]\leq\frac{\varepsilon}{\delta}.$$

*Then the system (1) is finite-time stable in mean square.*

According to Lemma 2.1, let as take

$$b(\eta) = \mathbf{5} \left[ K\_1 + \left| \boldsymbol{M}^{-1} \right|^2 \left( K\_3 + K\_2 - 2\sqrt{K\_3}\sqrt{K\_2} \right) + \kappa^2 K\_2 \right] \delta^2$$

and

$$h(\eta) = \mathfrak{H} \frac{b^{2-q}}{2-q} K\_2.$$

Moreover, *b*ð Þ*η* is a nondecreasing function on 0, ½ � *b* , then

$$\begin{aligned} \|\nu(\eta) \le & b(\eta) E\_{q,1}[h(\eta) \Gamma(q) \eta^q] \\ \|\|\boldsymbol{y}(\eta)\|\|^2 \le & \left[ K\_1 + \left| \boldsymbol{M}^{-1} \right|^2 \left( K\_3 + K\_2 - 2\sqrt{K\_3}\sqrt{K\_2} \right) + \kappa^2 K\_2 \right] \delta E\_{q,1} \left[ 5 \frac{\boldsymbol{b}^{2-q}}{2-q} K\_2 \Gamma(q) \eta^q \right] \end{aligned}$$

Then from the statement of Theorem 4.1, we get

<sup>∥</sup>*y*ð Þ*<sup>η</sup>* <sup>∥</sup><sup>2</sup> <sup>≤</sup>*ε:*

Hence the system (1) is finite-time stable in mean square. Hence the proof. □ **Remark 4.1.** *By using fixed-point rule, existence and uniqueness of solution, and controllability results have been investigated in* [27]*. Some well-known results on relative controllability of semilinear delay differential system with linear parts defined by permutable matrices are studied in* [28]*. In this chapter, we proved some new results of finite-time stability criteria in finite-dimensional space by employing Generalized Gronwall-Bellman inequality and suitable assumption on nonlinear terms.*

#### **5. An example**

Consider Eq. (1) in the below matrix form:

$$\begin{cases} ^C D\_{-0.75^+}^{0.5} \left( ^C D\_{-0.75^+}^{0.5} \varphi\_1 \right)(\eta) + 0.01 \mathfrak{y}\_1(\eta - 0.75) = -(3 - \eta) \frac{\mathfrak{y}\_1^2(\eta)}{\mathfrak{1} - \eta} + \int\_0^\eta (\zeta \mathfrak{y}\_1(\zeta) \Delta\_1 d\mathcal{B}\_1(\zeta), \eta) \\ \qquad \qquad \qquad \qquad \qquad \eta \in [-0.75, 0]; \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \eta \in [-0.75, 0]; \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \zeta \not\equiv [0.75, 0]; \\ \qquad \qquad \qquad \qquad \zeta \not\equiv \eta, \mathcal{Y}\_2(\eta) = 4, \quad \eta \in [-0.75, 0], \end{cases} (5\eta^2)$$

where *q* ¼ 0*:*5, *κ* ¼ 0*:*75, Δ<sup>1</sup> ¼ 0*:*3, Δ<sup>2</sup> ¼ 0*:*5

$$A = \begin{pmatrix} \mathbf{0.1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0.1} \end{pmatrix}, \quad F(\eta, \mathbf{y}(\eta)) = \begin{pmatrix} -(\mathbf{3} - \eta) \frac{y\_1^2(\eta)}{\mathbf{1} - \eta} e^{2N\eta} \\\\ -(\mathbf{3} - \eta) \frac{y\_2^2(\eta)}{\mathbf{1} - \eta} e^{2N\eta} \end{pmatrix}.$$

$$\Delta(\eta, \mathbf{y}(\eta)) = \begin{pmatrix} -\eta \nu\_1(\eta) e^{2N\eta} \sigma\_1 d B\_1 \\\\ -\eta \nu\_2(\eta) e^{2N\eta} \sigma\_2 d B\_2 \end{pmatrix}, \quad \boldsymbol{\psi}(\eta) = \begin{pmatrix} 2\eta \\\\ 4\eta \end{pmatrix}, \quad \boldsymbol{\psi}'(\eta) = \begin{pmatrix} 2 \\\\ 4 \end{pmatrix}.$$

Further, we have the following fractional delayed cos matrices:

$$\cos \circ\_{0.75, 0.65} \left( M \eta^{0.65} \right) = \begin{cases} \Theta, & -\infty < \eta < -0.75, \\ I, & -0.75 \le \eta < 0, \\\ I - M^2 \frac{\eta^{1.3}}{\Gamma(2.3)}, & 0 \le \eta < 0.75, \\\ I - M^2 \frac{\eta^{1.3}}{\Gamma(2.3)} + M^4 \frac{\left( \eta - 0.75 \right)^{2.6}}{\Gamma(3.6)}, & 0.75 \le \eta < 1.5. \end{cases} \tag{5}$$

From Eq.(5) and using basic calculation, one can get *E*2*<sup>q</sup>* ∥*M*∥<sup>2</sup> ð Þ *b* þ *κ* <sup>2</sup>*<sup>q</sup>* h i � � <sup>2</sup> ¼ 0*:*9712, *<sup>e</sup>*�*Nb*�<sup>1</sup> *Nb* � �<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*2683 and 4*<sup>b</sup> <sup>e</sup>*�2*Nb*�<sup>1</sup> <sup>2</sup>*Nb* � � ¼ �1*:*9004*:* Using the above-obtained values, one can easily verify that

$$2\left[E\_{2q}\left(\|M\|^2(b+\kappa)^{2q}\right)\right]^2\left[\left(\frac{e^{-Nb}-1}{Nb}\right)^2+4b\left(\frac{e^{-2Nb}-1}{2Nb}\right)\right]<1.$$

**Figure 1.** *The system* ð Þ 4 *is stable at q* ¼ 0*:*5*.*

**Figure 2.** *The system* ð Þ 4 *is stable at q* ¼ 0*:*7*.*

Hence we verified Theorem 3.1. Further, it is easy to verify that for any *<sup>x</sup>*ð Þ*<sup>η</sup>* , *<sup>y</sup>*ð Þ*<sup>η</sup>* <sup>∈</sup>*R*<sup>2</sup> .

$$\begin{aligned} \left| e^{-2N\eta} |F(\eta, \mathfrak{x}(\eta)) - F(\eta, \mathfrak{y}(\eta))|^2 \right| &\leq - (\mathfrak{Z} - \eta) \mathbb{E} \|\mathfrak{x}(\eta) - \mathfrak{y}(\eta)\|^2 \\\ \left| e^{-2N\eta} |\Delta(\eta, \mathfrak{x}(\eta)) - \Delta(\eta, \mathfrak{y}(\eta))|^2 \right| &\leq - \mathbf{0}. \mathsf{5}\eta \, \, \, \mathbb{E} \|\mathfrak{x}(\eta) - \mathfrak{y}(\eta)\|^2 \end{aligned}$$

Hence, *F* and Δ satisfies Assumption 1. In **Figures 1** and **2**, we showed the stable response of the system (4) with fractional order *q* ¼ 0*:*5 and *q* ¼ 0*:*7, respectively. From the above verification, one can conclude that the system (4) is finite-time stable in mean square.

#### **6. Conclusion**

In this chapter, we have derived some meaningful and general results for finitetime stability of nonlinear fractional stochastic delay systems. Existence, uniqueness of solution and stability analysis of FSDS have been proved in finite-dimensional stochastic fractional higher-order differential system. Finally, a numerical simulation test is carried out to validate the obtained theoretical results. Derived result generalizes many existing results with integer and fractional-order systems.

#### **AMS subject classifications (2010)**

34A08; 43A15; 37C25; 37A50

#### **Author details**

Sathiyaraj Thambiayya<sup>1</sup> , P. Balasubramaniam<sup>2</sup> \*, K. Ratnavelu<sup>3</sup> and JinRong Wang<sup>4</sup>

1 Institute of Actuarial Science and Data Analytics, UCSI University, Kuala Lumpur, Malaysia

2 Department of Mathematics, The Gandhigram Rural Institute (Deemed to be University), Gandhigram, Tamil Nadu, India

3 Institute of Computer Science and Digital Innovation, UCSI University, Kuala Lumpur, Malaysia

4 Department of Mathematics, Guizhou University, Guiyang, China

\*Address all correspondence to: balugru@gmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Scientific; 2000

[2] Oldham KB. Fractional differential equations in electrochemistry. Advances in Engineering Software. 2010;**41**: 1171-1183

[3] Magin RL. Fractional calculus models of complex dynamics in biological tissues. Computers and Mathematics with Applications. 2010;**59**:1586-1593

[4] Ortigueira MD. Fractional Calculus for Scientists and Engineers. New York: Springer Science & Business; 2011

[5] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: North-Holland Mathematics Studies, Elsevier; 2006

[6] Nieto JJ. Solvability of an implicit fractional integral equation via a measure of noncompactness argument. Acta Mathematics Scientia. 2017;**37**: 195-204

[7] Singh J, Kumar D, Nieto JJ. Analysis of an el nino-southern oscillation model with a new fractional derivative. Chaos, Solitons and Fractals. 2017;**99**:109-115

[8] Tian Y, Nieto JJ. The applications of critical-point theory discontinuous fractional-order differential equations. Proceedings of the Edinburgh Mathematical Society. 2017;**60**: 1021-1051

[9] Mao X. Stochastic Differential Equations and Applications. Chichester: Horwood Publishing, Cambridge; 1997

[10] Wu D. On the solution process for a stochastic fractional partial differential equation driven by space-time white noise. Statistics and Probability Letters. 2011;**81**:1161-1172

[11] Seo D, Lee H. Stationary waiting times in m-node tandem queues with production blocking. IEEE Transactions on Automatic Control. 2011;**56**:958-961

[12] Taheri M, Navaie K, Bastani M. On the outage probability of SIR-based power-controlled DS-CDMA networks with spatial Poisson traffic. IEEE Transactions on Vehicular Technology. 2010;**59**:499-506

[13] Applebaum D. Levy Processes and Stochastic Calculus. Cambridge: Cambridge University Press; 2009

[14] Rockner M, Zhang T. Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principle. Potential Analysis. 2007;**26**:255-279

[15] Ahmed E, El-Sayed AMA, El-Saka HAA. Equilibrium points, stability and numerical solutions of fractional-order predatorprey and rabies models. Journal of Mathematics Analysis and Applications. 2007;**325**:542-553

[16] Gao X, Yu J. Chaos in the fractional order periodically forced complex duffing oscillators. Chaos, Solitons and Fractals. 2005;**24**:1097-1104

[17] Odibat ZM. Analytic study on linear systems of fractional differential equations. Computers and Mathematics with Applications. 2010; **59**:1171-1183

[18] Wang J, Zhou Y, Fečkan M. Nonlinear impulsive problems for fractional differential equations and Ulam stability. Computers and Mathematics with Applications. 2012; **64**:3389-3405

[19] Burton TA, Zhang B. Fractional equations and generalizations of Schaefers and Krasnoselskii's fixed point theorems. Nonllinear Analysis: Theory,

Methods and Applications. 2012;**75**: 6485-6495

[20] Balasubramaniam P, Sathiyaraj T, Priya K. Exponential stability of nonlinear fractional stochastic system with Poisson jumps. Stochastics. 2021; **93**:945-957

[21] Fečkan M, Sathiyaraj T, Wang JR. Synchronization of Butterfly fractional order chaotic system. Mathematics. 2020;**8**:446

[22] Ren Y, Jia X, Sakthivel R. The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion. Applicable Analysis. 2017;**96**:988-1003

[23] Shen G, Sakthivel R, Ren Y, Mengyu L. Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process. Collectanea Mathematica. 2020;**71**:63-82

[24] Sathiyaraj T, Wang JR, Balasubramaniam P. Ulam's stability of Hilfer fractional stochastic differential systems. The European Physical Journal Plus. 2019;**134**:605

[25] Liang C, Wang J, O'Regan D. Representation of a solution for a fractional linear system with pure delay. Applied Mathematics Letters. 2018;**77**: 72-78

[26] Ye H, Gao J, Ding Y. A generalized Gronwall inequality and its application to a fractional differential equation. Journal of Mathematical Analysis and Applications. 2007;**328**:1075-1081

[27] Sathiyaraj T, Balasubramaniam P. Fractional order stochastic dynamical systems with distributed delayed control and Poisson jumps. The European Physical Journal Special Topics. 2016;**225**:83-96

[28] Wang J, Luo Z, Fečkan M. Relative controllability of semilinear delay

differential systems with linear parts defined by permutable matrices. European Journal of Control. 2017;**30**: 39-46

## **Chapter 3** Control of Supply Chains

*Kannan Nilakantan*

#### **Abstract**

This chapter aims to apply control principles in the discrete-time control of Supply Chains. The primary objective of the control is to keep the inventory levels (state variables) steady at their predetermined values and reduce any deviations to zero in the shortest possible time. The disturbances are induced by demand deviations from the planned/anticipated levels. The replenishment flows are the control variables. Thus, the control action is very similar to a "Linear Regulator with zero set-point". A novel development in this chapter is the use of direct Operator methods to solve the system Difference Equations, thereby obviating the need for Z-Transforms, block diagrams and transfer functions of classical control theory. This chapter provides a novel application of control theory as well as an easier method of solution.

**Keywords:** Supply Chain Dynamic Modeling, Supply Chain Control, Feedback Control, Linear Regulator Problem, Direct Operator Methods

#### **1. Introduction**

In the field of business, one of the most important constituents of a business system, and one which can give a business a cutting-edge over the others, is the supply chain (SC) of the business, and its effective operation and management.

A fundamental strategy associated with supply chains is that of a 'Responsive Chain'. A responsive chain focuses on its ability to respond quickly to demand changes and meet them within the shortest possible time. A responsive chain, by its very definition, has *necessarily* to be able to respond swiftly to unanticipated changes in demand, and this is determined largely by the dynamics of the system. Under normal circumstances, good responsiveness can be achieved through the maintenance of *adequate pipeline inventories* throughout the system, and one of the factors impacting this decision in a major way, is again the dynamics of the system. Hence, given the impact of the dynamics of the SC system on its design and operation as mentioned above, it is imperative to include the tenets of dynamic analysis in its design and operation. Thus system-dynamic methods are of significant value and utility in both SC design and SC operation. The design aspects pertain to the setting of inventory levels and the operational aspects to choosing the appropriate system controls for good operational performance.

Thus, we are led naturally to the use of system-dynamic methods and concepts of control theory for effective control of a supply chain. In the following sections we explore the dynamic modeling and analysis concepts in effectively controlling a supply chain that would yield good performance characteristics.

#### **2. Supply chain dynamics and notation**

Supply chain dynamics essentially deals with the dynamic behavior and temporal variation of the inventories and flows in the system over time when subjected to demand disturbances.

We now look at a simple three stage serial supply chain, a schematic diagram of which is given below in **Figure 1**. Each stage represents an inventory storage facility in the chain. Stage 1 represents the raw material input storage to a manufacturing plant, and stage 2 the finished goods inventory storage at the manufacturing plant. Stage 3 represents the finished goods warehouse (W/H) at the downstream end of the chain which meets the demand for the finished product, and from which material is shipped out to customers. The inventory levels at each stage are the state variables of the system, and the material flows between the stages the control variables.

In constructing a dynamic model of a supply chain system, we adopt the standard schematic diagram (**Figure 1**), system notation, and model variables and equations below, as commonly found in control theory literature [1–9].

We use a discrete time representation of the SC system, as this is more in keeping with the prevailing practices in the SC industry, wherein the inventory levels (state variables) are recorded at the end of each period or day. In most cases the state variable records are updated at the end of each day. Hence, we take our period to be a single day. However, we need to also emphasize that this need not always be so and would be dependent entirely on the convenience of the SC practitioners. And hence the state variables are recorded at epochs corresponding to the end of each period. The flow variables however are aggregated and taken to occur within and up to the end of each period.

The inventory and flow variables are subscripted to indicate the stages in the chain. We now give the detailed representation below:

*yi* ð Þ*k* : inventory at stage i of the chain at time epoch k

with: i = 1, representing (the raw material) the upstream end of the production facility.

i = 2 representing (the finished goods) the downstream end of the production facility.

i = 3 representing (the finished goods) the warehouse.

*q*0 *i* ð Þ*k* : material flow in period (k-1,k] into stage i of the chain

*r*0 <sup>3</sup>ð Þ*k* : finished goods demand at warehouse in period (k-1,k]

The dynamic equations of the system are written using deviation variables, as under:

$$x\_i(k) = y\_i(k) - y\_i^0(k) \text{ for } \mathbf{i} = \mathbf{1}, 2, 3$$

$$q\_i(k) = q\_i'(k) - q\_i^0(k) \text{ for } \mathbf{i} = \mathbf{1}, 2, 3$$

$$r\_3(k) = r\_3'(k) - r\_3^0(k)$$

**Figure 1.** *A three-stage serial supply chain.* where,

*y*0 *<sup>i</sup>* ð Þ*k* is the desired or planned inventory level at time k

*q*0 *<sup>i</sup>* ð Þ*k* is the desired or planned flow in period (k-1,k]

*r*0 <sup>3</sup> ð Þ*k* is the anticipated or forecasted demand at the warehouse in period (k-1,k], based on which the desired inventory levels in the chain have been set.

*xi*ð Þ*k* is the inventory deviation at time k.

*qi* ð Þ*k* is the material flow deviation in period (k-1,k].

*r*3ð Þ*k* is the deviation in demand observed at the warehouse in (k-1, k].

Under consistent units (in equivalent units of finished goods) for all inventory and flows in the system, the dynamic equations of the system are written as:

$$\varkappa\_i(k+1) = \varkappa\_i(k) + q\_i(k+1) - q\_{i+1}(k+1) \tag{1}$$

where for i = 3, *q*4ð Þ¼ *k r*3ð Þ*k* , the demand outflow from the warehouse. The system behavior is controlled by the replenishment policies followed in the system.

Now, since the demand is a stochastic variable, it can be split into two components, viz., a mean demand component, and a stochastic component which represents the random variations over and above the mean demand. The mean demand is what is predicted using forecasting techniques and is the planned offtake and the demand that the system is designed to meet in the normal course. Since prediction can never be exact, the residual variation is the stochastic component.

In such stochastic systems, the demand has an additional stochastic term, which is a white noise term, given by *<sup>ε</sup>*ð Þ� *<sup>k</sup> WN* 0, *<sup>σ</sup>*<sup>2</sup> ð Þ for all k. The sequence of stochastic disturbance components over each period is an independent and identically distributed sequence of *<sup>N</sup>* 0, *<sup>σ</sup>*<sup>2</sup> ð Þ random variables.

The standard initial conditions for the system are: *xi*ð Þ¼ *k* 0 for all *k*≤0 and *r*3ð Þ¼ *k* 0 for all *k*≤1, i.e. the system is at zero deviation at time k = 0, and the first deviation in demand is felt at the end of the first period, at k = 1.

The demand disturbance could be of the following Types:

1.a sudden shock demand increase, represented by a Dirac delta input function.

2.a sudden and sustained increase in demand, represented by a step input.

3.a demand with an increasing trend, represented by a ramp input function.


The first five components pertain to and define *the mean demand disturbance*, while the last represents the *residual random variations over and above the mean demand disturbance*.

The demand can have any combination of, or even all of the above components in the mean demand, and, of course, the additional stochastic component represented by a White Noise process. Such demands are often seen in supply chain warehouses.

Thus, a demand disturbance at the downstream end (at the warehouse) provides the perturbation to the system. The system is controlled through regulation/control of the replenishment flows which are the control variables in the system.

Now from the above, we can see that the system response will have two components, the *Mean Response*, and a stochastic component of the response. It is the *mean response that characterizes the system behavior*, while the stochastic component is characterized by the inventory variance.

#### **3. Dynamic supply chain performance metrics and control action triggers**

#### **3.1 Dynamic performance metrics**

The performance metrics of a supply chain that would be of interest to us from a dynamic performance point of view and control system design are explained below.

Since the demand input to the system suddenly increases, we can expect the system response to lag the demand and to fluctuate, as the system scrambles to catch up with the increased demand. Accordingly, the key performance indicators that would be of importance and interest to us from the point of view of both design and operation are the following [1–9]:


The first indicator shows whether the system is able to catch up with the demand and whether it is ultimately restored to its original level, or not. The second indicator, the trough value, represents the lowest point or value that the inventory level is likely to touch, and impacts the *base stock levels* that would have to be carried to maintain uninterrupted material flows in the system, (adequate pipeline inventories). The third defines the magnitude of fluctuations of the inventory levels, and high values would naturally be undesirable; and we would like to damp them down to zero as quickly as possible. The fourth indicates the center-line about which fluctuations occur and is indicative of the *average inventory level*s, which we would like to keep at positive levels for comfortable operation. While the fifth can be taken to be a surrogate measure of the *stock-out risk*, and the higher the fraction of time in the negative region (with a depleted inventory level), the higher could be taken to be the *implied stock-out risk*. The last, the inventory variance is the variation that we could expect even after the system has been restored to its original operating levels and is commonly taken as a measure of the robustness of the system to random demand variations.

#### **3.2 Control action triggers**

The two most common triggers for initiation of replenishment control action in the system are [1–9]:

1.The inventory levels at the W/H, as found in logistic systems and warehouses,

2.The demand variations at the W/H, as in electronic-data-interchange systems.

Thus, the control flows into the W/H are set to a function of the latest available inventory deviations and latest available Demand deviations. Thus, we have,

$$q\_3(k+1) = f[\mathbf{x}\_3(k-1), \mathbf{x}\_3(k-2), \dots, \mathbf{x}\_3(k-r); r\_3(k-1), r\_3(k-2), \dots r\_3(k-p)] \tag{2}$$

We discuss these points in more detail for a single stage system first.

#### **3.3 Single stage control notation**

To differentiate between the standard abbreviations in conventional control theory, we adopt the following notation used to indicate the type of control used.

P, PI, PID would represent Proportional, Proportional-integral, and Proportional-integral-derivative control as usual. Additionally, MA denotes a 'Moving Average' type of control (explained in detail subsequently).

An 'I' within parenthesis would represent 'inventory-triggered' control, while 'D' within parenthesis would denote a 'Demand-triggered' control. Also, 'ID' within parenthesis would denote a control which would have both inventory-triggered and Demand-triggered components. We give examples below.

Thus P(D) (pronounced as "P of D") denotes Demand-triggered Proportional control, PI(I) denotes Inventory-triggered PI control. Similarly, PID(ID) (pronounced as "PID of ID") denotes PID control with inventory-triggered and Demand-triggered components.

We also could have cases of multiple inventory triggers and multiple-demand triggers. These will be denoted as follows:

*X I*ð Þ *nDm* control would indicate a control of type X (P, PI, PID, MA, Composite) with n-inventory trigger terms and m-demand trigger terms. Thus, the control *PD I*ð Þ <sup>5</sup>*D*<sup>3</sup> (pronounced as "PD of I5D3") would be Proportional-derivative control with 5 inventory triggers and 3 demand triggers.

We now first look at a single stage system below which we take to be the warehouse end of a SC.

#### **4. Single stage control**

#### **4.1 The two response components**

Since the demand has both a deterministic component as well as a stochastic component, the response of the system can also be broken down into two components, viz., a deterministic part which is *the mean response*, and a random or stochastic part characterized by the *inventory variance*. It is the mean response that portrays the behavior of the system and yields the performance indicators of the system. Whereas, the stochastic part, represents the random fluctuations that have to be accounted for even after the system is brought under full control.

From the above discussion we can see the close parallel with conventional feedback control theory. Here the information about the warehouse inventory (the state variable) is fed back to the system for initiating replenishment flow control action. Additionally, we can also have controls wherein the disturbance is also directly fed back to the system for initiation of control action.

#### **4.2 Single stage system controls and transportation lags**

In close parallel with classical feedback control theory, the control flows are functions of the state variables and the demand or input perturbation to the system. The types of functions used also closely parallel classical control theory. And hence we have P, PI, and PID controls.

Additionally, we also have Moving Average (MA) controls, wherein the control flow is set to a weighted Moving Average of the latest available inventory deviations of up to 'r' periods back. The parameter 'r' is termed the 'Order' of the Moving Average.

We first look at Proportional Controls.

Now, for Proportional Controls of the P(I) type, the control flows into the warehouse in stage 3 of the chain would be given by:

$$q\_3(k+1) = K\_3 \varkappa\_3(k-1-l\_3) \tag{3}$$

where *l*<sup>3</sup> is the transportation lag in the flows into the warehouse from the upstream unit of the system, i.e., the finished goods inventory at the manufacturing plant in our case. And *K*<sup>3</sup> is the constant of proportionality between the control flow and the *latest available inventory deviation* based on which the order is initiated (and hence 'Proportional' to the error in conventional control theory).

The development herein corresponds to the 'Regulator Problem' with 'set point' of zero. And hence control of a Responsive Chain parallels the regulator problem in conventional control theory.

In conventional modeling of SCs, the lag is taken as the number of periods strictly between the period of order initiation and the period of arrival of the consignment, not including the period of arrival of the consignment. Thus, the lag is taken to be zero if the replenishment consignment arrives in the period immediately succeeding the period of order initiation. Instantaneous replenishment is not envisaged and is very rare in SC contexts. Arrival of the consignment within the same period of order initiation is also not envisaged and is very rare in such contexts. The earliest arrival of an ordered consignment is taken to be the immediately succeeding period, for which we take the lag as zero.

This convention is based on what is normally followed in the industry and practice, as well as the literature on dynamic modeling of SCs.

Hence in our further development of dynamic models of SCs we take the lag as zero if an order initiated in period k i.e., in the interval (k – 1, k] arrives in period (k + 1), i.e., in the interval (k, k + 1].

For a transportation lag of 'l' periods, an order placed in the interval (k – 1, k] would arrive in the interval (k + l, k + l + 1].

Hence for the warehouse under *zero lag* the control flows for P(I) control would be set as:

$$q\_3(k+1) = K\_3 \mathbb{1}\_3(k-1) \tag{4}$$

Now it is to be noted that the order is initiated in period k i.e., in the interval (k – 1, k] based on the *latest fully observed inventory deviation* which in this case would be that at the start of period k, which is the inventory recorded at the end of period (k-1) i.e., *x*3ð Þ *k* � 1 . We take it that since the inflows in period k would be the aggregate of flows in the interval (k – 1, k], and the order is initiated *within* the period k (*and not at time point k*), the latest available fully observed inventory level would be *x*3ð Þ *k* � 1 and not *x*3ð Þ*k* because *x*3ð Þ*k* would not be available to the order initiator within period k. The value of *x*3ð Þ*k* is recorded at the closure of period k, after the

*Control of Supply Chains DOI: http://dx.doi.org/10.5772/intechopen.100523*

cessation of all activities including the action of ordering, of period k. Thus, replenishment orders need to be initiated within and before the closure of a period and not at the end of a period.

Thus, warehouse records would be updated at the closure of the day's operations and would show the *closing inventory* and the replenishment orders placed *during* the day. The replenishment orders would have been placed based on the previous day's closing stock and would arrive during the course of the next day if the lag is zero.

This is the convention that we will follow in the further development of the models.

We next look at the next type of control, which is the PI(I) control.

For the Proportional-Integral (PI(I) type) control case with *zero lag*, the control flows at the warehouse would be set as under:

$$q\_{\mathfrak{J}}(k+1) = K\_{\mathfrak{J}} \mathfrak{x}\_{\mathfrak{J}}(k-1) + K\_{\mathfrak{c}} \sum\_{m=0}^{k-1} \mathfrak{x}\_{\mathfrak{J}}(m) \tag{5}$$

where the second term is the integral term in our discrete-time system, and Kc is the proportionality constant (gain term) factor of the integral of the error.

Next we have the PID(I) control wherein the control flows into the warehouse under *zero lag* would be set to:

$$q\_{\mathfrak{J}}(k+1) = K\_{\mathfrak{J}}\mathfrak{x}\_{\mathfrak{J}}(k-1) + K\_{\mathfrak{c}}\sum\_{m=0}^{k-1} \mathfrak{x}\_{\mathfrak{J}}(m) + K\_{\mathfrak{d}}(\mathfrak{x}\_{\mathfrak{J}}(k-1) - \mathfrak{x}\_{\mathfrak{J}}(k-2)) \tag{6}$$

where the last term represents the derivative term in our discrete-time system, and Kd is the proportionality constant factor (gain term) of the derivative component of the control.

Next we have the Moving Average (MA) type of control (MA(I) type), for which the control flows into the warehouse under *zero lag* would be set to:

$$q\_3(k+1) = K\_{\mathfrak{F}}^1 \mathfrak{x}\_3(k-1) + K\_{\mathfrak{F}}^2 \mathfrak{x}(k-2) + K\_{\mathfrak{F}}^3 \mathfrak{x}(k-3) + \dots \\ \dots \\ \dots \\ + K\_{\mathfrak{F}}^r \mathfrak{x}(k-r) \tag{7}$$

where the r is the order of the moving average, the Ks are the control parameters (the weights) of the MA terms. Thus, the control flow is set to a weighted moving average of the latest available fully observed inventory levels up to r period back.

The above controls discussed above are the conventional inventory-triggered schemes. In all these types of controls, we could additionally have demandtriggered terms also, like the PI(ID), PID(ID), MA(ID) etc.

We discuss some of them below for single stage systems.

#### **5. Solution of single stage systems**

Firstly, we note herein that in solving for the response of a supply chain system, we will not use the Z-transform nor block diagrams and transfer functions as in conventional control theory. Rather we will work directly on the system difference equation in the time domain itself. And instead, we will use direct Operator methods to obtain the system solution and response (rather than the transformed equations and inverse transforms). This is one of the advantages of this modeling paradigm.

Another advantage of the type of discrete time modeling taken up here is that *transportation lags do not result in differential-delay equations* as in conventional

continuous-time control theory. Rather, transportation lags would increase the order of the resulting system difference equation, which is expected to be easier to solve than differential-delay equations.

We now take up the simplest form of control which is the P(I) control and illustrate the formulation of the system equation and its solution method.

#### **5.1 P(I) control under zero lag**

The control is an inventory-triggered Proportional control. And we take the Replenishment Lag = 0 in the simplest case.

The flow balance equation for the warehouse is given by:

$$\mathbf{x}\_3(k+1) = \mathbf{x}\_3(k) + q\_3(k+1) - r\_3(k+1) \tag{8}$$

The control flows into the warehouse are given by:

$$q\_3(k+1) = K\_3 \varkappa\_3(k-1) \tag{9}$$

It is to be noted that the value of *K*<sup>3</sup> < 0, since the control flows and inventory deviation have to be of opposite sign, i.e., when the inventory deviation is (�)ve (inventory is at a depleted level) then the flow deviation has to be (+)ve to enable more/extra flow into the warehouse to make up the inventory shortfall. Likewise, if the inventory deviation is (+)ve (extra inventory in stock) then the flow deviation has to be (�)ve (i.e., reduced flow) to reduce the inflow into the warehouse to maintain inventory levels.

Thus, we can clearly see that the controls are of the 'feedback' type, and they seek to keep the inventory deviation at zero level, which is just the' Linear Regulator with zero set-point' in standard control theory.

Thus, substituting for the control flow into the flow balance eqn. Above, yields the system-dynamic eqn. For the warehouse as:

$$\varkappa\_{\mathfrak{I}}(k+\mathbf{1}) \equiv \varkappa\_{\mathfrak{I}}(k) + K\_{\mathfrak{I}}\varkappa\_{\mathfrak{I}}(k-\mathbf{1}) - r\_{\mathfrak{I}}(k+\mathbf{1}) \tag{10}$$

$$\text{Or equivalently,}\\\text{x}\_3(k+1) - \text{x}\_3(k) - K\_3\\\text{x}\_3(k-1) \equiv -r\_3(k+1) \tag{11}$$

The above is the deterministic part of the system equation. Since demand is a stochastic variable with a stochastic component, the complete system equation is given by:

$$
\varkappa\_3(k+1) - \varkappa\_3(k) - K\_{\mathfrak{P}}\varkappa(k-1) \equiv -r\_3(k+1) - \varepsilon(k+1) \tag{12}
$$

which has both parts. To solve the system equation completely, we split it into its two components and solve for each of the components separately. We hence solve the following two equations, one each for the deterministic part and the stochastic part.

$$\mathbf{x}\_{3}^{\det}(\mathbf{k}+\mathbf{1}) - \mathbf{x}\_{3}^{\det}(\mathbf{k}) - \mathbf{K}\_{3}\mathbf{x}\_{3}^{\det}(\mathbf{k}-\mathbf{1}) \equiv -r\_{3}(\mathbf{k}+\mathbf{1}) \text{ for the deterministic part, and,} \tag{13}$$

$$\mathfrak{x}\_3^{\text{tot}}(k+1) - \mathfrak{x}\_3^{\text{tot}}(k) - K\_3 \mathfrak{x}\_3^{\text{tot}}(k-1) \equiv -\varepsilon(k+1) \text{ for the stochastic part.} \tag{14}$$

The first is a deterministic Linear Difference Equation, and the second, a Stochastic Linear Difference Equation (SDE).

*Control of Supply Chains DOI: http://dx.doi.org/10.5772/intechopen.100523*

Both are second-order Linear Difference Equations (LDEs) in the state variable *x*3ð Þ*k* and can be solved by direct Operator methods for any type of input or disturbance term on the RHS of the system LDE.

An excellent treatment of difference calculus and solution methods for LDEs is given in [10]. We follow the methods given therein.

In order to solve the LDE, we first introduce the Forward Shift Operator E as under: *Ex*3ð Þ� *<sup>k</sup> <sup>x</sup>*3ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> , with the important property that: *E Ex* ½ �¼ <sup>3</sup>ð Þ*<sup>k</sup> <sup>E</sup>*<sup>2</sup> *x*3ð Þ¼ *k x*3ð Þ *k* þ 2 . Before using the Operator, we first write the LDE in standard form as under: (the lowest time value is taken as k):

$$
\varkappa\_3(k+2) - \varkappa\_3(k+1) - K\_3\varkappa\_3(k) \equiv -r\_3(k+2) \tag{15}
$$

Now using the forward Shift Operator E, we can write the LDE in Operator form as:

$$\{E^2 - E - K\_3\} \mathfrak{x}\_3^{\text{det}}(k) \equiv -r\_3(k+2) \text{ for the deterministic part, and},\tag{16}$$

$$\{E^2 - E - K\_3\} \kappa\_3^{\text{tot}}(k) \equiv -\epsilon(k+2) \text{ for the stochastic part.} \tag{17}$$

#### *5.1.1 The mean response: solution of the deterministic LDE*

We first look at the deterministic part of the solution below, which will yield the *mean response.* For notational convenience, we drop the superscript on *x*3ð Þ*k* .

$$\left[\mathbf{E}^2 - \mathbf{E} - \mathbf{K}\_3\right] \mathbf{x}\_3(\mathbf{k}) \equiv -\mathbf{r}\_3(\mathbf{k} + \mathbf{2})$$

Now this is an LDE of order two (the order being the highest power of the Operator E).

We write the Characteristic Equation of the LDE as [10]:

$$a^2 - a - K\_3 = 0\tag{18}$$

to determine the characteristic roots of the LHS Operator.

$$\text{From which we get the roots as } : a\_{1/2} = \frac{1 \pm \sqrt{1 + 4K\_3}}{2}. \tag{19}$$

The stability of the system is entirely controlled by the roots of the LHS Operator. And elementary analysis leads to the following stability conditions:

*K*<sup>3</sup> ≥ 0: instability

�1*=*4≤*K*<sup>3</sup> <0: stability with non-oscillatory response

�1≤ *K*<sup>3</sup> < � 1*=*4: stability with oscillatory behavior

*K*<sup>3</sup> < � 1: instability with oscillatory behavior

Now we take up the solution of the system LDE for a unit step increase in demand, i.e.,

$$r\_3(k+1) = \mathbf{1}, \forall k \ge 0 \tag{20}$$

Substituting for the demand disturbance in the system equation yields the LDE:

$$[E^2 - E - K\_3] \varkappa\_3(k) \equiv -\mathbf{1} \tag{21}$$

which we can call the "Original Non-Homogeneous Eqn." (O-NHE).

We first look at the solution of the homogeneous LDE (i.e., with RHS = 0). The homogeneous LDE is:

$$\left[\left[E^2 - E - K\_3\right]\omega\_3(k) \equiv 0\tag{22}$$

which upon factoring the LHS Operator can be written as: (*λ*1, *λ*<sup>2</sup> being the distinct roots of the LHS Operator):

$$[(E - \lambda\_1)(E - \lambda\_2)]\varkappa\_3(k) \equiv \mathbf{0} \tag{23}$$

which has the solution as: *<sup>x</sup>*3ð Þ� *<sup>k</sup> <sup>C</sup>*1*λ<sup>k</sup>* <sup>1</sup> <sup>þ</sup> *<sup>C</sup>*2*λ<sup>k</sup>* <sup>2</sup> for the case of distinct roots. For a repeated root, the solution is given by: *<sup>x</sup>*3ð Þ� *<sup>k</sup>* ð Þ *<sup>C</sup>*<sup>0</sup> <sup>þ</sup> *<sup>C</sup>*1*<sup>k</sup> <sup>λ</sup><sup>k</sup>* for a repeated root of algebraic multiplicity two.

Now that we have the solution of the homogeneous LDE, we next look for a particular solution of the Original Non-Homogeneous LDE (O-NHE).

A standard method of solution of the Non-homogeneous eqn. is by the 'Annihilator Method' [10].

We look for the Operator that annihilates the RHS terms of the O-NHE, say A (E). Then operating by the Annihilator on both sides of the O-NHE yields:

$$A(E)[(E-\lambda\_1)(E-\lambda\_2)]\mathbf{x}\_3(k) \equiv A(E)r\_3(k+2) \equiv \mathbf{0} \tag{24}$$

which is a homogeneous LDE albeit of a higher order, but which can be solved by factorizing the LHS Operator. As an example, in our case of a unit step disturbance, *r*3ð Þ�� *k* 1, ∀*k*≥ 1.

And the Annihilator is given by:

$$A(E) \equiv (E - \mathbf{1}) , \hfil\!\!\!\! (E - \mathbf{1}) r\_3(k) \equiv -(E - \mathbf{1})(\mathbf{1}) = \mathbf{0}$$

And hence the equivalent Homogeneous LDE is given by:

$$[(E - \mathbf{1})(E - \lambda\_1)(E - \lambda\_2)]\mathbf{x}\_3(k) \equiv \mathbf{0}$$

which has the solution:

$$\mathbf{x}\_3(k) \equiv D\left(\mathbf{1}^k\right) + \mathbf{C}\_1\mathbf{i}\_1^k + \mathbf{C}\_2\mathbf{i}\_2^k \equiv D + \mathbf{C}\_1\mathbf{i}\_1^k + \mathbf{C}\_2\mathbf{i}\_2^k \text{ for the case of distinct roots.} \tag{25}$$
 
$$\dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \tag{26}$$

*<sup>x</sup>*3ð Þ� *<sup>k</sup> <sup>D</sup>*ð Þ<sup>1</sup> *<sup>k</sup>* <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>0</sup> <sup>þ</sup> *<sup>C</sup>*1*<sup>k</sup> <sup>λ</sup><sup>k</sup>* for the case of repeated roots (26)

Now we note that the O-NHE being of order two, will admit only two undetermined constants. The above solution, however, has three undetermined constants. The third was introduced by us due to the Annihilator. Hence to determine the extra constant D in the solution, we substitute the solution into the O-NHE to determine D. Thus, and noting that the terms involving the roots of the LHS operator of the O-NHE are precisely the homogeneous solution terms of the O-NHE, we only need substitute the extra terms introduced by the Annihilator into the O-NHE. Hence, we have:

$$[(E - \lambda\_1)(E - \lambda\_2)]D \equiv [E^2 - E - K\_3]D = -\mathbf{1}$$

Noting that E(D) = D itself (∵*E D*ð Þ� *D k*ð Þ� þ 1 *D*, ∀*k*), we obtain *D k*ð Þ� þ 2 *D k*ð Þ� þ 1 *K*3*D* � *D* � *D* � *K*3*D* ¼ �1 from which we readily obtain

$$D = \mathbf{1}/K\_3 \text{ which is } (-) \text{ve due to the } (-) \text{ve sign of } K\_3. \tag{27}$$

Thus, the full solution is given by:

$$\mathbf{x}\_3(k) \equiv \mathbf{1}/K\_3 + \mathbf{C}\_1 \boldsymbol{\lambda}\_1^k + \mathbf{C}\_2 \boldsymbol{\lambda}\_2^k \text{ for the case of distinct roots, and} \tag{28}$$

$$\mathbf{x}\_3(k) \equiv \mathbf{1}/K\_3 + (\mathbf{C}\_0 + \mathbf{C}\_1 k) \boldsymbol{\lambda}^k \text{ for the case of repeated roots.} \tag{29}$$

Now, for stable solutions, we will have j j *λ*<sup>1</sup> < 1, j j *λ*<sup>2</sup> <1, and hence the terms involving the roots of the LHS Operator decay to zero for large k, leaving the 'Offset' term as 1*=K*<sup>3</sup> which is the Offset value. The Offset value is (�)ve because of the (�)ve sign of *K*3.

The Damping rate, which is the rate at which the fluctuations decay to zero are given by the magnitudes of the roots of the LHS Operator j j *λ*<sup>1</sup> *k* , j j *λ*<sup>2</sup> *<sup>k</sup>* � �, which can clearly be seen to decrease in magnitude with decrease in magnitude of j j *K*<sup>3</sup> . However, the Offset value increases in magnitude with decrease of *K*3.

We take the case of distinct roots first, say for a value of *K*<sup>3</sup> ¼ �1*=*2, in the stable region.

Hence, we have the system equation as:

$$[E^2 - E - K\_3] \varkappa\_3(k) \equiv [E - E - (-1/2)] \varkappa\_3(k) \equiv -1 \tag{30}$$

The roots of the LHS Operator are readily obtained as: *λ* ¼ ð Þ� 1*=*2 ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>j</sup>*, j j *<sup>λ</sup>*<sup>1</sup> <sup>&</sup>lt;1, j j *<sup>λ</sup>*<sup>2</sup> <sup>&</sup>lt;1, *<sup>j</sup>* <sup>¼</sup> ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> , which can be written in the form.

*<sup>λ</sup>* <sup>¼</sup> *<sup>ρ</sup>e*�*j<sup>θ</sup>* � <sup>1</sup>*<sup>=</sup>* ffiffi <sup>2</sup> � � <sup>p</sup> *<sup>e</sup>*�*jπ<sup>=</sup>*4, which hence yields the homogeneous solution as: f g *<sup>A</sup>*Cosð Þþ *<sup>k</sup>π=*<sup>4</sup> *<sup>B</sup>*Sinð Þ *<sup>k</sup>π=*<sup>4</sup> <sup>1</sup>*<sup>=</sup>* ffiffi <sup>2</sup> � � <sup>p</sup> *<sup>k</sup>* , and hence the full solution as:

$$\mathbf{x}\_3(k) \equiv (\mathbf{1}/K\_3) + \{A\mathbf{C}\cos(k\pi/4) + B\mathbf{S}\sin(k\pi/4)\} \left(\mathbf{1}/\sqrt{2}\right)^k \tag{31}$$

$$\equiv -2 + \left\{ \text{ACcos}(k\pi/4) + B\text{Sin}(k\pi/4) \right\} \left( \mathbf{1}/\sqrt{2} \right)^{k},\tag{32}$$

which shows a Damping Rate of the Order of *O* 1*=* ffiffi <sup>2</sup> � � <sup>p</sup> *<sup>k</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ <sup>0</sup>*:*<sup>707</sup> *<sup>k</sup>* , with an Offset of - 2 units. The response if plotted in **Figure 2**, and the 'Undershoot' is obtained as �2.5 units.

We next take a value of *K*<sup>3</sup> ¼ �1*=*4 for the case of repeated roots, again in the stable region. The roots of the LHS Operator are obtained as: *λ*<sup>1</sup> ¼ *λ*<sup>2</sup> ¼ *λ* ¼ 1*=*2, and the full solution as:

**Figure 2.** *P(I) control.*

$$\mathbf{x}\_3(k) \equiv (\mathbf{1}/K\_3) + \{\mathbf{C}\_0 + \mathbf{C}\_1 k\} (\mathbf{1}/2)^k \text{ which for } K\_3 = -\mathbf{1}/4 \text{ yields } \tag{33}$$

$$\varkappa\_3(k) \equiv -4 + \{\mathcal{C}\_0 + \mathcal{C}\_1 k\} \left(\mathbf{1}/2\right)^\mu,\tag{34}$$

which shows a Damping Rate of the Order of *<sup>O</sup>*ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>k</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ <sup>0</sup>*:*<sup>5</sup> *<sup>k</sup>* , but with a high Offset value of - 4 units, and an Undershoot of - 4 units.

Thus, while we would like the oscillations to be damped out rapidly, this would compromise on the Offset value, which would impact the base stock requirements of the system.

Thus, for the practitioner, the implications are quite clear: *there is a trade-off between stability and rapid damping on the one hand, and the base stock requirements of the system (for low stock-out risk) on the other. The higher the damping (stability) required, the higher would be the base-stock requirements to keep stock-out risk low; and alternatively, the higher the (stability) damping achieved, the higher would be the stockout risk at fixed base-stock levels.*

We next obtain the undetermined constants in the general solutions above, using the Initial Conditions (ICs) of the system, as under:

The standard ICs of the system are as: f g *x*3ð Þ� *k* 0,*r*3ð Þ� *k* 0, ∀*k*≤0 .

Now the system LDE: *<sup>E</sup>*<sup>2</sup> � *<sup>E</sup>* � *<sup>K</sup>*<sup>3</sup> � �*x*3ð Þ�� *<sup>k</sup> r k*ð Þ�� <sup>þ</sup> <sup>2</sup> 1, <sup>∀</sup>*k*≥0 is valid for ∀*k*≥0.

And hence the ICs for our warehouse system can be obtained from the system equation itself using the standard system ICs and yields: f g *x*3ð Þ¼ 0 0, *x*3ð Þ¼� 1 1 . Substituting these ICs into the solutions above yields the full solutions as:

$$\text{For } K\_3 = -1/2: \varkappa\_3(k) \equiv -2 + 2\text{Cov}(k\pi/4) \left(1/\sqrt{2}\right)^k, k \ge 0 \tag{35}$$

$$\text{For } K\_3 = -\mathbf{1}/4 : \varkappa\_3(k) \equiv -4 + (4 + 2k)(\mathbf{1}/2)^k, k \ge 0. \tag{36}$$

We additionally examine the case for *K*<sup>3</sup> ¼ �1 (the maximum possible magnitude for stability), the response is given by

*<sup>x</sup>*3ð Þ�� *<sup>k</sup>* <sup>1</sup> <sup>þ</sup> ½ � *<sup>A</sup>*Cosð Þþ *<sup>k</sup>π=*<sup>3</sup> *<sup>B</sup>*Sinð Þ *<sup>k</sup>π=*<sup>3</sup> ð Þ<sup>1</sup> *<sup>k</sup>* , *k*≥0, which using the LDE ICs yields:

$$\mathbf{x}\_3(k) \equiv -\mathbf{1} + \mathbf{C} \mathbf{s}(k\pi/3) + \left(\mathbf{1}/\sqrt{3}\right) \mathbf{S} \mathbf{n}(k\pi/3), k \ge \mathbf{0},\tag{37}$$

which can be simplified to.

$$\mathbf{x}\_{3}(k) = \left\{-\mathbf{1} - \left(2/\sqrt{3}\right) \text{Sim}((k-1)\pi/3) \right\} \mathbf{H}(k-1), k \ge \mathbf{1}, \mathbf{x}\_{3}(0) = \mathbf{0}.\tag{38}$$

where H(.) is the unit Heaviside step function and yields a sinusoidal pattern with a center-line of – 1 and constant amplitude of 2*=* ffiffiffi <sup>3</sup> <sup>p</sup> and the maximum negative deviation in inventory, the undershoot, equal to �2.

This last case is that of *marginal stability characterized by constant amplitude perpetual oscillations which never die down to zero.*

The responses for the three cases above are plotted in **Figure 2**.

#### *5.1.2 The limiting inventory variance: solution of the SDE*

We next look at the determination of the limiting inventory variance, which is a measure of the variation that we could expect even after the system (mean inventory level) has been restored to its original value.

Also, for the behavior of the mean response as well as our inferences from it to be meaningful, it is necessary that the limiting inventory variance be bounded and finite. We can then expect the inventory levels to be within the band given by: *<sup>x</sup>*3ð Þ*<sup>k</sup>* det � <sup>3</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lim *<sup>t</sup>*!<sup>∞</sup>varð Þ *<sup>x</sup>*3ð Þ*<sup>k</sup>* <sup>p</sup> , where *<sup>x</sup>*3ð Þ*<sup>k</sup>* det is the mean response that has been determined above from the deterministic system LDE.

In order to determine the limiting inventory variance, we make use of the stochastic component of the system equation and determine the stochastic part of the response. For our system the Stochastic LDE (or SDE) is as under:

$$\{E^2 - E - K\_3\} \varkappa\_3(k) \equiv -\varepsilon(k+2),\tag{39}$$

where the term on the RHS of the SDE is the random variation represented by a White Noise Process, with *<sup>ε</sup>*ð Þ� *<sup>k</sup> WN* 0, *<sup>σ</sup>*<sup>2</sup> ð Þ.

We can note that the LHS Operator is again the same as in the deterministic part of the system LDE that we have solved for above. Hence the roots of the LHS Operator remain unaltered in the SDE also.

Now following the method used in [11], we can note that if *unity is not a root of the LHS Operator*, then the SDE admits as a solution, an infinite weighted Moving Average representation in terms of the White Noise disturbance terms as under:

$$\left(\varkappa\_3(k)\right)^{\text{tot}} \equiv \sum\_{l=0}^{k-1} \beta\_l \varepsilon(k-l) \tag{40}$$

where the *β<sup>l</sup>* s are the weights.

And hence the stochastic part of the solution of the system eqn. can be written as an infinite linear combination of the white noise disturbance terms.

Now since the individual white noise terms are Uncorrelated and Normal, i.e.,

with *<sup>ε</sup>*ð Þ� *<sup>k</sup> <sup>N</sup>* 0, *<sup>σ</sup>*<sup>2</sup> ð Þ, and, with *Cov*ð Þ¼ *<sup>ε</sup>*ð Þ*<sup>i</sup>* , *<sup>ε</sup>*ð Þ*<sup>j</sup> <sup>δ</sup>ijσ*2, *<sup>δ</sup>ij* <sup>¼</sup> 0, *<sup>i</sup>* 6¼ *<sup>j</sup>* 1, *i* ¼ *j* � , the limiting variance of x(k) is given by:

$$\lim\_{k \to \infty} \text{var}(\mathbf{x}\_3(k)) = \lim\_{k \to \infty} \text{var}\left(\left\{\sum\_{l=0}^{k-1} \beta\_l e(k-l)\right\}\right) = \lim\_{k \to \infty} \left\{\sum\_{l=0}^k \beta\_l^2\right\} \sigma^2 \tag{41}$$

In order to solve for the weighting terms, the βs, we substitute the solution into the system SDE above, as under:

$$\mathbb{E}\left[E^2 - E - K\_3\right] \mathbb{1}\_3(k)^{\text{stoc}} \equiv -\epsilon(k+2), \text{ valid for all } k \ge 0,\tag{42}$$

which is

$$\left\{ \left[ E^2 - E - K\_3 \right] \left\{ \sum\_{l=0}^{k-1} \beta\_l e(k-l) \right\} \equiv -\varepsilon (k+2), k \ge 0. \tag{43}$$

Another and more convenient way to write the SDE is to use the Backward Shift Operator *L*, defined by:

$$L\mathbf{x}(\mathbf{k}) = \mathbf{x}(\mathbf{k} - \mathbf{1}), \text{i.e.,} \\ L = E^{-1}, E = L^{-1}. \tag{44}$$

The SDE for the βs becomes:

$$\mathbb{E}\left[\mathbf{1} - L - K\_3 L^2\right] \left\{ \sum\_{l=0}^{k-1} \beta\_l \varepsilon(k - l) \right\} \equiv -\varepsilon(k), k \ge 0,\tag{45}$$

which is

$$\begin{cases} \left[\mathbf{1} - L - K\_3 L^2\right] \left\{\beta\_0 \varepsilon(k) + \beta\_1 \varepsilon(k-1) + \beta\_2 \varepsilon(k-2) + \beta\_3 \varepsilon(k-3) \dots \right\} \equiv -\varepsilon(k), k \ge 0 \end{cases} \tag{46}$$

Now comparing coefficients of ε(k) for each k, yields the system of equations as below:


The above set of equations yields: *β*<sup>0</sup> ¼ �1, *β*<sup>1</sup> � *β*<sup>0</sup> ¼ 0, &, *β<sup>k</sup>* � *β<sup>k</sup>*�<sup>1</sup> � *K*3*β<sup>k</sup>*�<sup>2</sup> ¼ 0, ∀*k*≥2, which is an LDE for the βs, as.

$$\{E^2 - E - K\_3\} \beta\_k \equiv \mathbf{0}, \forall k \ge \mathbf{0}, \text{with } \text{ICs}: \{\beta\_0 = -\mathbf{1} = \beta\_1\}. \tag{47}$$

We can note that the LHS Operator is the same as for the system LDE, and hence has the same characteristics and form of solution.

We first illustrate the computation for the stable case *K*<sup>3</sup> ¼ �1*=*2, for which the solution form has already been obtained as: ð Þ *<sup>A</sup>*Cosð Þþ *<sup>k</sup>π=*<sup>4</sup> *<sup>B</sup>*Sinð Þ *<sup>k</sup>π=*<sup>4</sup> <sup>1</sup>*<sup>=</sup>* ffiffi <sup>2</sup> � � <sup>p</sup> *<sup>k</sup>* , and hence plugging in the ICs, we have the solution for βs as:

$$\beta\_k \equiv (-\text{Cov}(k\pi/4) - \text{Sim}(k\pi/4)) \left(1/\sqrt{2}\right)^k, k \ge 0.1$$

To obtain the limiting inventory variance, we firstly note that:

$$\beta\_k^2 = (-\text{Cov}(k\pi/4) - \text{Sim}(k\pi/4))^2 (1/2)^k = (1 - 2\text{Sim}k\pi/2)(1/2)^k \le 3(1/2)^k. \tag{48}$$

And hence,

$$\begin{aligned} \text{Min}\_{k \to \infty} & \sum\_{l=0}^{k} \beta\_l^2 \le 1 + 1 + 3(1/2)^2 \left\{ 1 + (1/2) + (1/2)^2 + (1/2)^3 + \dots \right\} = 2 + 3/2 \\ &= 3.5. \end{aligned} \tag{49}$$

Hence, we have: *Limk*!<sup>∞</sup>varð Þ *<sup>x</sup>*3ð Þ*<sup>k</sup>* <sup>≤</sup>3*:*5*σ*2, showing that the inventory variance is bounded and finite for this case.

We next illustrate the computation for the marginally stable case *K*<sup>3</sup> ¼ �1, for which the solution form has already been obtained as: *A*Cosð Þþ *kπ=*3 *B*Sinð Þ *kπ=*3 , and hence plugging in the ICs, we have the solution for the βs as:

$$\beta\_k \equiv -\text{Cov}(k\pi/3) - \left(\mathbf{1}/\sqrt{3}\right)\text{Sim}(k\pi/3) \equiv \left(2/\sqrt{3}\right)\text{Cov}(k\pi/3 - \pi/6), k \ge 0 \quad \text{(50)}$$

And we can see from the above that the βs oscillate in value, from �1 to 0 to +1 infinitely often, i.e., the sequence { … 0, �1, �1, 0, +1,+1, 0 … ..} repeats infinitely often. And hence the series P<sup>∞</sup> *<sup>k</sup>*¼<sup>0</sup>*β*<sup>2</sup> *<sup>k</sup>* diverges to infinity.

Hence in this case the limiting inventory variance is *not bounded* and *not finite*. It can similarly be shown that the limiting inventory variance is not bounded for unstable solutions also.

Thus, the limiting inventory variance will be bounded only for stable cases.

#### *5.1.3 The complete solution*

We can also note from the above discussion and work up that we have also obtained the stochastic part of the solution, as: *<sup>x</sup>*3ð Þ*<sup>k</sup> stoc* � <sup>P</sup>*k*�<sup>1</sup> *<sup>l</sup>*¼<sup>0</sup> *<sup>β</sup>lε*ð Þ *<sup>k</sup>* � *<sup>l</sup>* , where the βs are as has been obtained above. Thus, the complete solution can be written as:

$$\boldsymbol{\infty}\_{3}(\boldsymbol{k}) \equiv \boldsymbol{\infty}\_{3}(\boldsymbol{k})^{\det} + \boldsymbol{\infty}\_{3}(\boldsymbol{k})^{\text{tot}} \equiv \boldsymbol{\infty}\_{3}(\boldsymbol{k})^{\det} + \sum\_{l=0}^{k-1} \beta\_{l} \boldsymbol{\varepsilon}(\boldsymbol{k} - l) \tag{51}$$

where *<sup>x</sup>*3ð Þ*<sup>k</sup>* det is the mean response derived from the solution of the deterministic part of the LDE, and the βs in the stochastic part are as obtained above by solution of the stochastic part of the system equation, the SDE.

The above representation proves useful for simulation purposes.

We next take up the P(ID) control. We discuss only the deterministic system LDE hereafter, since the stochastic part of the solution and the limiting inventory can be obtained by methods similar to that discussed above in all cases to follow.

#### **5.2 P(ID) control under zero lag**

In this type of control an additional demand-triggered component is also added to the control thereby making it more proactive. The control initiates corrective replenishment action no sooner than a demand deviation is observed. It does not wait for an inventory deviation to take place before initiating replenishment action though it does have an inventory-triggered component also.

The replenishment control flow is given by:

$$q\_3(k+1) \equiv K\_3 \varkappa\_3(k-1) + K\_0^3 r\_3(k-1) \tag{52}$$

where the first term is the inventory-triggered component and the second the demand-triggered component. Substituting for the control flow into the system equation yields:

$$\varkappa\_3(k+1) \equiv \varkappa\_3(k) + K\_3\varkappa\_3(k-1) + K\_0^3 r\_3(k-1) - r\_3(k+1) \tag{53}$$

which can be written as

$$\mathbf{x}\_3(k+1) - \mathbf{x}\_3(k) - K\_3 \mathbf{x}\_3(k-1) \equiv K\_0^3 r\_3(k-1) - r\_3(k+1) \tag{54}$$

We can note that the addition of the demand-triggered component has left the LHS of the LDE unaltered. Thus, the LHS Operator of the LDE remains the same and is unaffected by addition of demand-triggered components to the control. The system eqn. can hence be written in Operator form as:

$$\mathbb{E}\left[E^2 - E - K\_3\right] \mathbb{1}\_3(k) \equiv K\_0^3 r\_3(k) - r\_3(k+2) \text{ valid in } k \ge 0 \tag{55}$$

Since *r*3ð Þ� *k* 1, ∀*k*≥ 1, the system LDE can be written as:

*Control Systems in Engineering and Optimization Techniques*

$$[E^2 - E - K\_3] \varkappa\_3(k) \equiv K\_0^3 - 1 \text{ valid in } k \ge 1,\tag{56}$$

$$\text{with the ICs now as}: \{ \mathfrak{x}\_{\mathfrak{I}}(\mathfrak{1}) = -\mathfrak{1}, \mathfrak{x}\_{\mathfrak{I}}(\mathfrak{2}) = -\mathfrak{2} \}. \tag{57}$$

The ICs for the LDE have been obtained from the system Eq. (55) above using the standard system ICs; f g ð Þ� *x*3ð Þ*k* ,*r*3ð Þ*k* ð Þ 0, 0 , ∀*k*≤0 applied to the Eq. (55) above.

Since the LHS Operator is the same as for the earlier P(I) control, the stability analysis remains the same as earlier, as also the roots of the LHS Operator for various values of the inventory-trigger parameter discussed earlier, i.e.,f g *K*<sup>3</sup> ¼ �1*=*4, �1*=*2, �1 .

The solutions are the same as given earlier in Eqs. (25) and (26).

And substituting the solution back into the O-NHE yields the value of the extra constant D as

$$D = \frac{K\_0^3 - 1}{K\_3},\tag{58}$$

which hence yields the solutions for the two cases as:

$$\varkappa\_3(k) \equiv \left(K\_0^3 - 1\right) / K\_3 + C\_1 \lambda\_1^k + C\_2 \lambda\_2^k \text{ for the case of distinct roots} \tag{59}$$

*<sup>x</sup>*3ð Þ� *<sup>k</sup> <sup>K</sup>*<sup>3</sup> <sup>0</sup> � <sup>1</sup> � �*=K*<sup>3</sup> <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>0</sup> <sup>þ</sup> *<sup>C</sup>*1*<sup>k</sup> <sup>λ</sup><sup>k</sup>* for the case of repeated roots <sup>ð</sup>*K*<sup>3</sup> ¼ �1*=*4<sup>Þ</sup> (60)

where the offset term is now given by *K*<sup>3</sup> <sup>0</sup> � <sup>1</sup> � �*=K*<sup>3</sup> for both cases.

The important point to note in the above solution is that the offset can be made zero by choice of the demand-trigger parameter as *K*<sup>3</sup> <sup>0</sup> ¼ 1.

And hence we can observe the enhanced response of the P(ID) control over the earlier P(I) control, in that the Offset can now be controlled by us by choice of *K*<sup>3</sup> 0.

In fact, we can also achieve a (+)ve value of the offset by choosing *K*<sup>3</sup> <sup>0</sup> ≥ 1.

We can now obtain the full solutions for the three cases above, using the LDE ICs f g *<sup>x</sup>*3ð Þ¼� <sup>1</sup> 1, *<sup>x</sup>*3ð Þ¼� <sup>2</sup> <sup>2</sup> . We take *<sup>K</sup>*<sup>3</sup> <sup>0</sup> ¼ 1 to be able to obtain zero offset. Hence, we have:

$$\text{For } K\_3 = -1/4 \text{ (the repeated roots case)}: \varkappa\_3(k) \equiv (4 - 6k)(1/2)^k, k \ge 1 \quad \text{(61)}$$

$$\text{For } K\_3 = -\mathbf{1}/2: \varkappa\_3(k) \equiv \left( 2\sqrt{5} \right) \text{Cos}(k\pi/4 - \phi) \left( \mathbf{1}/\sqrt{2} \right)^k, \tan\phi = 2, k \ge \mathbf{1} \quad \text{(62)}$$

For *<sup>K</sup>*<sup>3</sup> ¼ �1 the marginal stability case � � : *<sup>x</sup>*3ð Þ� *<sup>k</sup>* 2Cosð Þ ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>π</sup>=*<sup>3</sup> , *<sup>k</sup>*≥<sup>1</sup> (63)

The solution curves are plotted in **Figure 3**, from which we can see that the response in all cases has zero offset.

We can similarly extend the modeling and analysis to PI(I), PID(I), and MA(ID) controls.

We could also have different types of input disturbances as indicated earlier in Section 3.1. Additionally, the third dimension of our analysis could be to have nonzero lags, i.e., lags of one, two periods, and so on. Cases with non-zero and higher lags will result in higher order system LDEs, and the LHS Operator would be of a higher order.

Further details of the above can be obtained in [1–9].

*Control of Supply Chains DOI: http://dx.doi.org/10.5772/intechopen.100523*

#### **6. Controls for multi-stage supply chains**

We look at the serial supply chain system as given in **Figure 1**.

We can see from **Figure 1** that the immediately succeeding downstream stage in a supply chain will provide the "demand perturbation" for the immediately preceding stage. Thus, the demand perturbation at the warehouse at the downstream end will successively be felt up the chain. And the single-stage analysis described above can be used in turn for each stage of the chain.

For non-serial supply chains, the arguments are similar and single-stage analysis can be used as described above.

Details of some of these analyses can be found in [1–9].

#### **7. Conclusion**

This chapter has presented the application of control concepts to the control of supply chains. The state variables have been taken to be the inventory levels, while the control variables are the replenishment flows into the various stages of the system. The conventional P, PI, PID controls have been discussed, as also some newer forms of control which are especially applicable to supply chains and warehouses. The performance of P(I) and P(ID) controls have been derived in detail, and their performance analyzed.

A significant feature of this chapter is that the conventional block diagrams and transfer functions of conventional control theory have not been used. Rather direct Operator Methods have been used to good advantage to solve the system equations. *Control Systems in Engineering and Optimization Techniques*

#### **Author details**

Kannan Nilakantan Mathematical Modelling Group, Operations Group, Institute of Management Technology, Nagpur, India

\*Address all correspondence to: nilakanthan@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Control of Supply Chains DOI: http://dx.doi.org/10.5772/intechopen.100523*

#### **References**

[1] Nilakantan, Kannan; Enhancing Supply Chain performance with improved Order-control Policies, *International Journal of System Science* Sep2010, Vol 41-9, p 1099-1113, Taylor and Francis.

[2] Nilakantan, Kannan; Replenishment Policies for Warehouse Systems under Cyclic demand, *International Journal of Business Performance and Supply Chain Modelling*, 2013,Inderscience, Vol 5-2, pp 148-176.

[3] Nilakantan, Kannan; Supply Chains subject to Demand Shocks, *International Journal of Logistics Systems and Management,* 2015, Inderscience, Vol 21-2, pp133-159.

[4] Nilakantan, Kannan; Supply Chains under Demands with Trend Components, and Robust Controls, *Annals of Management Science*, International Centre for Business and Management Excellence, Franklin, Tennessee, USA.Vol 3-2, Dec 2014, pp 27-64.

[5] Nilakantan, Kannan; Warehouse Control Systems for General Forms of Demand, *International Journal of Operations and Quantitative Management*, 2014, INFOMS, USA, Vol 20–4, pp 273–299,Dec 2014.

[6] Nilakantan, Kannan; Replenishment systems for responsive supply chains under dynamic and sudden lead-time disturbances, *International Journal of Systems Science-Operations and Logistics*, 2019, Elsevier, Vol 6-4, pp 320-345.

[7] Nilakantan, Kannan; Supply Chain Resilience to Sudden and Simultaneous Lead-time and Demand Disturbances *International Journal of Supply Chain and Operations Resilience,* 2019, Inderscience (USA), Vol 3 - 4, pp. 292-329.

[8] Nilakantan Kannan, Responsiveness and Recovery Performance Analysis,

and Replenishment System Design in Supply Chains under Large Replenishment Lead-times and Uncertain Demand, I*nternational Journal of Supply Chain Management 2021,(UK),* Vol 10-3, pp 1-17.

[9] Nilakantan Kannan The Ripple Effect in a Supply Chain: A Sudden Demand Increase and with a Sinusoidal Component, International Journal of Operations and Quantitative Management March *2021*, Vol 27-1, pp. 1-37.

[10] Kelley, W. G., & Peterson, A. C., *Difference Equations*, 2/e, Academic Press, Elsevier, USA.

[11] Gourieroux, G, & Monfort, A., Time Series and Dynamic Models, Cambridge University Press,(1990), Cambridge, UK.

#### **Chapter 4**

## The Fundamental of TCP Techniques

*Pritee Nivrutti Hulule*

#### **Abstract**

Strategies for prioritizing test cases plan test cases to reduce the cost of retrospective testing and to enhance a specific objective function. Test cases are prioritized as those most important test cases under certain conditions are made before the re-examination process. There are many strategies available in the literature that focus on achieving various pre-test testing objectives and thus reduce their cost. In addition, inspectors often select a few well-known strategies for prioritizing trial cases. The main reason behind the lack of guidelines for the selection of TCP strategies. Therefore, this part of the study introduces the novel approach to TCP strategic planning using the ambiguous concept to support the effective selection of experimental strategies to prioritize experimental cases. This function is an extension of the already selected selection schemes for the prioritization of probation cases.

**Keywords:** test case prioritization (TCP), final total effort (FTE), average effort (AE)

#### **1. Introduction**

In testing, part of Regression testing in the maintenance phase is the process of retesting the updated software to ensure that new errors have not been introduced into earlier validated code. In addition, the regression tests should take as little time as possible to perform a few test cases as possible. Due to its costly nature, several techniques in the literature focus on costs.

These are:


This document focuses on the techniques of prioritization of the test case. Testers may now want to increase code coverage in test software at a faster pace, increase or improve their reliability in software reliability in less time, or increase the speed at which test suites detect failures at that moment. System during the regression tests. The main problems with code-based prioritization techniques are that they focus only on the number of errors detected and therefore treat all failures in the same way [1–4].

#### **2. Test cases prioritization**

Testing software or applications is the most important part of the "Software Development Life Cycle" (SDLC). It plays a very important role in the quality and performance of the software and ensures that the final product is as per the client's requirement. Placement Priority is the expansion of software testing, which is used to determine the "critical test cases". Software testing is done to detect bugs and errors in its operation, depending on how the performance and quality of the software are continuously improved.

These preliminary test cases are determined by a variety of factors depending on the need for the software, which is provided for test cases to perform other processes. By prioritizing test cases, (testers and developers can reduce the time and cost of the software testing phase and ensure that the product delivered is of different quality) [3–6].

#### **2.1 What is test prioritization?**

The term "test prioritization" refers to the accountable and difficult part of testing that allows assessors to "manage risk, plan tests, estimate cost, and analyze" which tests will work in the context of a particular project. This process is known as Test Case Prioritization, which is the process of 'prioritizing and planning' test cases. These methods are used to run test cases that are very important to reduce time, cost, and effort during the software testing phase.

In addition, the prioritization of the test case helps in regression testing and improves its effectiveness Developers of forensic trial software can get help to fix bugs earlier than possible otherwise. In addition, to determine the prioritization of test cases, different factors are determined according to the need of the software.

In this way, inspectors can easily run test cases, which have a very high value and provide errors with previous defects. Also, the improved error detection rate during the test phase allows for faster response of the test system.

The major issues of code-based prioritization techniques are that they focus only on the number of faults detected and hence treats all faults equally. In practice, all faults cannot be treated the same. Therefore, there may be situations where the presence of an error is less important but its coverage of the requirements is important. The prioritization of needs-based assessment addresses those issues by providing the most relevant case-based assessment of service-based services. In addition, Testing does not guarantee error-free software is a process of verifying software compared to user descriptions and their requirements major problem with the specifics of Test Case Prioritization is based on specificity and that there is no effective way to measure the performance of selected Test Suits [3].

Strategies for the promotion of probation cases make probationary cases subject to certain conditions. Prioritization of test cases can serve a variety of purposes.

*The Fundamental of TCP Techniques DOI: http://dx.doi.org/10.5772/intechopen.100850*

The purpose of these priorities increases the likelihood that they will meet a particular goal more closely than they would otherwise have done at random. The Test Case Prioritization problem was officially described by Rothaermel and she learned nine TCP techniques. Among them, four relied on coding and two relied on early detection of errors. This was compared with no priorities and random prioritization strategies. The right category can only happen in books that are likely to 'benefit. Tests show an early detection of error with the greedy algorithm and additional greed in the code detection process. To measure the purpose of the experiment Rotermel defined metrics, the APFD rated an average of% of errors found over a fraction of the test suit (%) made. Its values are between 0 and 100 and the higher the value the better the error detection [3].

#### **2.2 Intelligence techniques**

#### *2.2.1 Uncertainty and metaheuristics*

The essence of this study is that it can be used in both the White-box test areas or the black box test environment. This approach aims to redesign test cases according to the extent of the alleged violation from the source code. The process of prioritizing probation cases using the absurd concept to improve the performance of a given assessment suit in violation of the evidence and further prioritization of probation cases. Anwar et al. conducted a study comparing various experimental precautionary measures and used the ambiguous notion of making good use of an experimental suit which is why the testing process backfired. Risk assessment of software needs is a major factor in improving software quality. Propose a new risk assessment approach using a sophisticated professional program to improve the effectiveness of TCP in the review process. All of these studies demonstrate the development of strategic priorities for testing. As time changes, the nature of these subjects changes. The proposed approach is a systematic study that minimizes theoretical aspects and drives research into a practical context. Although many strategies have been proposed many strategies are limited to code-based methods and focus on detecting a high number of errors. According to a study conducted by Catal (2013) on Test Case Prioritization Techniques, the highest number of strategies proposed so far is Coverage (code) based (40%) and minimal value is given to known costs (2%) and distribution-based (2%) strategies.

#### **2.3 Benefits of proposed methodology**

The major issues of code-based prioritization techniques are that they focus only on the number of faults detected and hence treats all faults equally. That is removed from the system.

There may be some cases where the existence of fault is not so important but its requirement coverage is that is performed in the proposed technique.

This part of the study is an extension of the selected selection schema and provides a Model of Priority Testing Priority Strategies based on three factors:


In selecting the schema identification of project features/features that need to be done to identify TCP strategies that include high project attributes and therefore requirements. Selection Schema assumes that the required tools are the same. Once the strategy has been identified the next step is to differentiate those strategies based on high integration and a small experimental effort again, with difficulty. To calculate the effort this study uses the same basis as used by Krishnamoorthi. Therefore, the experimental effort represents the average number of test cases required for a specific functional evaluation process. Methodology plays a role in the evaluation of experimental efforts. According to research, the difficulty can be taken from a scale of 1–10 which is usually defined by the engineer and analyst. Therefore, this part of the study achieves high coverage with minimal difficulty and experimental efforts [2, 7].

#### **3. Phases of TCP**

Stage-1 uses the priority matrix proposed in 'the Selection Schema'.

Stage-2 experiences the difficulty of a particular method compared to the availability of its requirements.

TCP strategy tension is measured on a scale of (1–10) based on research. Calculating the effort of a particular method above number 2 is used. According to the formulas, the effort will come on a scale of (0–1). In this way, we will get used to multiplying by 10 so that we consider the same amount as coverage and weight. In the final stage, the classification is done with the Fuzzy Rule-based system.

The output of this program is sorted into the following upcoming sets:" Select, Medium and, Discard". Input variables, as well as output variables, can take values between (1–10). In this case study, triangular membership functions are used for mapping random and flexible input sets during fuzzification as well as for making dynamic output and complex sets during defuzzification. The input variables are written in three non-linear sets each: Low, Middle, and Top various purposes. The priority of the test case can be explained.

Given: In the test, suit provided S of the given system (X; XS, set of S) permissions;

f, function from XS to real numbers. Problem: Get S'ϵ XS to (6S ") (S"ϵXS) (S<sup>0</sup> s's ') [f (S') ≥ f (S ")].

In this definition, 'XS' is a collection of existing combinations to prioritize test cases for test 'S', and f is an objective function. For example, testers may wish to increase code coverage in software under test at a faster rate, increase or improve their confidence in software reliability in the short term, or increase the rate at which test suits find errors in that system during deferred testing.

The Test case Prioritization problem was officially described by Rothermel and he learned the nine TCP techniques discussed. Four of them were based on coding and two were based on the early detection rate. This was compared with no priorities and random prioritization strategies. The appropriate category is only possible in the text that is almost impossible to achieve. Tests show an early detection of error with the greedy algorithm and additional greed in the code detection process. To measure the purpose of the experiment Rothermel explained the metric, the APFD measuring an average of % of errors found over a fraction of the test suit (%) made. Its values are between 0 and 100 and the higher the value the better the error detection.

*The Fundamental of TCP Techniques DOI: http://dx.doi.org/10.5772/intechopen.100850*

Think of a test suit T with several test cases; F is a set.

of m faults detected with test T T. TFi is the first test case in T'(one of T's orders) indicating error i. Thereafter T "s APFD is defined by the following equation [1, 4, 5, 8, 9]:

$$\text{Average Percentage of Fault detected} = 1 - \left(\frac{\text{TF1} + \text{TF2} + \dots \text{TFm}}{nm}\right) + \left(\frac{1}{2n}\right) \tag{1}$$

#### **4. Methods**

This proposed law-based program has a total of "17 rules" and is reviewed frequently based on expert knowledge. These rules are based on the following experts.

#### **4.1 Motivation and contribution**

In previous research, the researcher focuses on factors that I use, but they are not focused on the most important part 'time'. I work on that, and also how many testers are using the system, that tester priority is" low, medium or high" that things also captured and it generates the graph on basis of that.

#### **4.2 Research findings**


#### **4.3 Method analysis**

There Are Four Factors:


#### *4.3.1 Input (three inputs)*

1.Relevance of selected TCP Techniques based on maximum requirement coverage.

#### *Control Systems in Engineering and Optimization Techniques*


#### **Table 1.**

*Rule base for fuzzy based selection of TCP techniques.*

2.The complexity of selected TCP techniques


**Output:** Final class: TCP Techniques [2].

#### *4.3.2 Begin*


#### *4.3.3 Architecture*

relevant project 'attributes/features' are done to identify TCP techniques covering maximum project attributes consequently requirements.

The various stages of the proposed approach are as follows:

Stage-1 Identifying project features in terms of relevance and hence the coverage of requirements.

Stage-2 Identify the complexity of testing techniques.

Stage-3 calculating testing effort.

Stage-4 classifies TCP techniques using fuzzy inference.

Stage 5. Time to execute each technique. Selection of any technique most important factor is time to perform an execution [2, 11].

#### **4.4 Technical feasibility**

Technical feasibility deals with the study of performance and numerous constraints like availableness of "resources, technology", the risk concerning development that might affect the capability to attain an adequate system. It identifies if the technology used is companionable or not with the recent system.

Following are some technical issues are:


#### **4.5 Economic feasibility**

Economic analysis is the generally used methodology for estimating the efficiency of a recent system. The procedure of cost analysis is to establish the advantages and savings that are looking ahead from a system and evaluate them with their costs.

**Time-Based:** On a single click management can create any report.

**Cost-Based:** There is no need for any type of training to use the software or tools. For managing this tool there is no need for any investment. The software is freeware so there is a minimal cost [2].

### **4.6 Operational feasibility**

Operational feasibility is the activity of however well a projected system solves the issues. It receives the benefits of the opportunities known throughout the scope definition and the way it satisfies the needs known within the requirements analysis part of system implementation. If the user is aware of the all technicalities used for the system and the user having in detail knowledge about the system, then there are no difficulties at the time of implementing the system. Therefore, it's assumed that the user will not face any downside once handling the system implantation. Getting acceptance from users is the major difficulty for any developer at the time of developing any software tool. This system does not have any major problems. The small problem is that some time system gets slow due to the server and makes us wait for results. But it will automatically generate results fast. This is not a permanent problem of this system.

Case Studies for Development and Implementation:


#### **5. Mathematical model**

A] Mapping Diagram.

```
Where,
P1, P2 … . Pn = Tester.
S = System.
B] Set Theory.
S = {s, e, X, Y, ϕ}.
Where,
s = Start of the program.
```
#### 1.Authentication

L = Login, UN = User name, PWD = Password.

To access the facilities of system, TCP Classification.

X = Input of the program.

2. Identify INPUT as

Input should be 4 factors.

X = {S1, S2 … … Sn}.

Where,

S1, S2 … … Sn = No. of Factored selected by Patients.

3. Identify Process P as

P = {DF}.

Where,

DP = TCP Classification rules

4. Identify Output Y as

Y = {Bc1 … ...Bcn}. Where, Bc1 … ...Bcn = Select, discard, moderate class.

According to condition selected factor parameter TC will classify in one Class φ = Success or failure condition of the system [4].

### **6. Failures**

1.A huge database can lead to more time-consuming to get the information.

2.Hardware failure.

3. Software failure.

#### **7. Success**


#### **8. Our project is NP-complete**

Our project goes into NP-Finish because at some point it will give the result. With the resolution problem, so that it will provide a solution to the problem during the polynomial period. A collection of all decision-making problems the solution to which can be provided in the polynomial period. Functional requirement.

#### **8.1 User**

	- i. Requirement coverage,
	- ii. Efforts
	- iii. Complexity
	- iv. Time. View own TCP.

#### **8.2 Admin**


#### **9. Software quality attributes**

#### **9.1 Capacity**

The capacity of a project according to data is very less.

#### **9.2 Availability**

All functionality will work properly.

#### **9.3 Reliability**

The system is reliable for classifying large data.

#### **9.4 Security**

User when login to the system that time users Mail Id and password match accurately.

### **9.5 DFD diagram**

Fig-Data Flow Diagram.

**Fig-Use Case Diagram.**

#### **Class Diagram:**

**Fig-Class Diagram.**

#### **10. Architecture Modeling**

## **11. The design process for quality software**

#### **11.1 Implementation approach**

Describe the overall test method that will be used to evaluate the product of the project.

There are many ways such as:


Here we have used the "Black Box test" method. In Black Box Testing we simply provide input to the system and test its output regardless of how the system processes it.

#### **11.2 Passport test or test failure terms**

Where the actual results and expectations are the same the test will be passed. When the actual results and expectations are different the test will fail.

Entry method: as soon as we have a need, we can start testing.

Exit method: when the disturbance level falls below a certain level, we can stop the test.

#### **11.3 Implementation with screen output**

Present the system in this research we have proposed a novel-based technique for the classification of TCP techniques using Fuzzy Logic. "This work is an extension of the already proposed selection schema for test case prioritization techniques". with the help of tester login and that tester priority of use the factor, it can generate the graph and also, it how much time required for the execution of the test case it calculates. Model for selection of test case prioritization technique based on 4f factors:


#### **12. Future scope**

"The system will work for other projects testing like ERP.TCP techniques will enhance performance by using another solution to prioritize test cases".

*The Fundamental of TCP Techniques DOI: http://dx.doi.org/10.5772/intechopen.100850*

### **Author details**

Pritee Nivrutti Hulule Computing Engineering, Bharati Vidyapeeth (Deemed to be University) College of Engineering, Pune, India

\*Address all correspondence to: hululepritee9@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Duggal G, Suri B. Understanding Regression Testing Techniques. COIT; 2008

[2] Sujata KK, Purohit GN. A schema support for selection of test case prioritization techniques. In: Fifth International Conference on Advanced Computing & Communication Technologies (ACCT '15). 2015. pp. 547-551

[3] Vegas S, Basili V. A characterization schema for software testing techniques. Empirical Software Engineering. 2005; **10**(4):437-466

[4] Yoo S, Harman M. Regression testing minimization, selection, and prioritization: a survey. Software Testing, Verification, and Reliability. 2012;**22**:67-120

[5] Tyagi M, Malhotra S. An approach for test case prioritization based on three factors. International Journal of Information Technology and Computer Science. 2015;**4**:79-86

[6] Kumar V, Sujata K, Kumar M. Test case prioritization using fault severity. International Journal of Computer Science and Technology (IJCST). 2010; **1**(1):67-71

[7] Chen GY-H, Wang P-Q. Test case prioritization in specification-based environment. Journal of Software. 2014; **9**(8):205-2064

[8] Elbaum S, Malishevsky A, Rothermel G. Test case prioritization: a family of empirical studies. IEEE Transactions on Software Engineering. 2002;**28**:159-182

[9] Miranda B, Cruciani E. 2018 Copyright held by the owner/ author(s), FAST approaches to scalable similarity-based test case prioritization. 2018

[10] Silva D, Rabelo R, Campanhã M, Neto PS, Oliveira PA, Britto R. A hybrid approach for test case prioritization and selection. In: IEEE Congress on Evolutionary Computation (CEC). 2016. pp. 4508-4515

[11] Alakeel AM. Using fuzzy logic in test case prioritization for regression testing programs with assertions. The Scientific World Journal. 2014;**2014**: Article ID-316014

Section 2
