Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial Trade Credits to Credit-Risk Customers by Discounted Cash Flow Analysis

Nirmal Kumar Duari, Sorforaj Nowaj and Jobin George Varghese

## Abstract

Getting loans from banks are almost impossible after 2008 global financial crisis. As a result, about 80% of companies in the United Kingdom and the United States offer their products on various short terms, free-interest loans to customers. To compute the interest earned and charged during the credit period but not to the revenue and other costs which are considerably larger than the interest earned and charged, numerous researchers and academicians apply merely the discounted cash flow (DCF) analysis. In addition, some products deteriorate continuously and cannot sell after expiration date. However, a little number of researchers have considered the product lifetime expectance into their models. In this chapter, a supplierretailer-customer chain model is developed. The supplier provides an upstream full trade credit to the retailer, and the credit-risk customer gets a downstream partial trade credit from the retailer. The non-decreasing deterioration rate is 100% near particularly close to its expiration date. To compute all relevant costs, DCF analysis is applied. The retailer's optimal replenishment cycle time is not only exists but also unique that demonstrated in this proposal and that has been shown by the numerical examples.

Keywords: supply chain management, deterioration, expiration dates, trade credit, discount cash flow, credit-risk customer

## 1. Introduction

In traditional business transactions, it was implicitly assumed that the buyer must pay the procurement cost when products are received. However, in today's competitive markets, most companies offer buyers various credit terms to impel sales and hence reduce inventory. In the United Kingdom, "estimates suggest" that more than 80% of the business transactions are which made on credit, while about

80% of the United States' firms offer their products on trade credit, has been stipulated by Seifert et al. [1]. Conversely, trade credit decreases the inventory holding cost, therefore affecting order quantity of buyer.

study the effects of inflation and time value of money. Weibull non-decreasing deterioration rate is considered mostly as a special case of the proposed generalized deterioration rate. The retailer's objective function is formulated under different possible alternatives, in this study. We derived an algorithm to get optimal solution to each alternative using the existing theorem on concave functions. Finally, two

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial…

The remaining of the paper is follows as: Section 2 defines notations and makes necessary assumptions. Section 3 gives the mathematical model; Section 4 derives the present value of the retailer's annual total profit under each alternative. Section 5 provides the required algorithm which simplifies the search for the optimal solution. Section 6 presents numerical study. Finally, the conclusions and the future

numerical examples are solved in order to illustrate the problem.

The following notation and assumptions are used in this model.

α fraction of the purchasing cost must be paid at the time of placing an order, 0 ˂ α ˂ 1

Q(t) order quantity level at time t θð Þt deterioration rate at time t I(t) inventory level at time t D(t) demand rate, D ¼ ð Þ a þ bt PTP(T) present value of annual total profit

research direction are provided in Section 7.

DOI: http://dx.doi.org/10.5772/intechopen.90689

The following notations are used in this model:

2. Notation and assumption

C cost per unit D demand

I interest per

T time

Functions:

109

h holding cost per unit

OC ordering cost per order p selling price per unit

Ic interest charged by the retailer Ie interest earned by the supplier T replenishment cycle time

M expiration time or maximum lifetime

R downstream credit period in years by the retailer S upstream credit period in years by the supplier

2.1 Notations

In literature for inventory models with trade credit financing, Goyal [2] derived the retailer's optimal EOQ when the supplier provides a permissible delay in payment. Aggarwal and Jaggi [3] elongated the EOQ model for nondeteriorating items to deteriorating items. Jamal et al. [4] elaborate the model considering shortages. Teng [5] modified the previous models by using sales revenue to compute the interest earned from sales. Huang [6] then explored the problem to a supply chain system in an upstream and downstream trade credit environment. Liao [7] further generalized Huang's model with an unlimited replenishment rate with a limited replenishment rate for deteriorating items. Min et al. [8] proposed an EPQ model with both upstream and downstream trade credits when the demand is stock dependent. Teng et al. [9] extended the constant demand rate to product's life cycle dependent demand pattern. Under different credit terms, Chern et al. [10, 11] discussed Nash two-player equilibrium solutions between the supplier and the retailer. Chen et al. [12, 13] contributed the retailer's optimal model. Liao et al. [14] and Wu et al. [15] discussed optimal strategy for deteriorating items with capacity constraints.

The products like volatile liquids, blood banks, fruits, fashion merchandises, vegetables, and high-tech products deteriorate continuously due to evaporation, spoilage, and obsolescence, among other reasons. An exponentially decaying inventory model is built by Ghare and Schrader [16]. The constant deterioration rate is extended to Weibull failure rate by Covert and Philip [17]. Dave and Patel [18] proposed linear time functional demand. For shortages, Sachan [19] further generalized the EOQ model. The demand is log-concave with time dependent derived by Hariga [20]. Teng et al. [21] further expanded allowing shortages and continuous type demand pattern. Teng et al. [22] have permitted partial backlogging. For deteriorating products, Dye [23] investigated the effect of technology investment on refrigeration. No one of the above cited papers took the expiration date into consideration before Chen and Teng [24], Sarkar [25], Wang et al. [26], Wu et al. [27], and Sarkar et al. [28].

DCF is an important tool in inventory management. Researchers such as Hill and Pakkala [29], Chung and Liao [30], Dye et al. [31], Chang et al. [32], Mousavi et al. [33] work related to DCF analysis. Recently, Chen and Teng [34] and Duari and Chakraborti [35] applied the DCF analysis to obtain the optimal lot size and credit period in a supply chain with upstream and downstream trade credit financing.

A credit-worthy retailer generally gains a permissible delay on the entire purchasing quantity, in reality. However, a retailer often asks for credit-risk customers to cover a fraction of the purchasing cost at the time of placing an order and then provides a permissible delay (downstream credit). To reduce default risks with credit-risk customers, they use downstream partial trade credit as a strategy that has received relatively little attention by the researchers. Additionally, the majority of the recent studies consider merely the opportunity loss of trade credit. Most of the time, they ignore to take the opportunity loss of the other different costs in their study in order to take the effect of inflation and time value of money. For an exquisite and sharp analysis, the DCF analysis must be used on all relevant revenue and costs. Due to this fact, here a supplier-retailer-customer supply chain model is proposed. The supplier provides an upstream full credit period of S years to the retailer and the retailer gives to the customer a downstream partial credit period of R years. The deteriorating rate is constant or increasing and closer to 100% near expiration date. With time-dependent demand, the DCF analysis is applied to

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial… DOI: http://dx.doi.org/10.5772/intechopen.90689

study the effects of inflation and time value of money. Weibull non-decreasing deterioration rate is considered mostly as a special case of the proposed generalized deterioration rate. The retailer's objective function is formulated under different possible alternatives, in this study. We derived an algorithm to get optimal solution to each alternative using the existing theorem on concave functions. Finally, two numerical examples are solved in order to illustrate the problem.

The remaining of the paper is follows as: Section 2 defines notations and makes necessary assumptions. Section 3 gives the mathematical model; Section 4 derives the present value of the retailer's annual total profit under each alternative. Section 5 provides the required algorithm which simplifies the search for the optimal solution. Section 6 presents numerical study. Finally, the conclusions and the future research direction are provided in Section 7.

## 2. Notation and assumption

The following notation and assumptions are used in this model.

### 2.1 Notations

80% of the United States' firms offer their products on trade credit, has been stipulated by Seifert et al. [1]. Conversely, trade credit decreases the inventory

In literature for inventory models with trade credit financing, Goyal [2] derived the retailer's optimal EOQ when the supplier provides a permissible delay in payment. Aggarwal and Jaggi [3] elongated the EOQ model for nondeteriorating items to deteriorating items. Jamal et al. [4] elaborate the model considering shortages. Teng [5] modified the previous models by using sales revenue to compute the interest earned from sales. Huang [6] then explored the problem to a supply chain system in an upstream and downstream trade credit environment. Liao [7] further generalized Huang's model with an unlimited replenishment rate with a limited replenishment rate for deteriorating items. Min et al. [8] proposed an EPQ model with both upstream and downstream trade credits when the demand is stock dependent. Teng et al. [9] extended the constant demand rate to product's life cycle dependent demand pattern. Under different credit terms, Chern et al. [10, 11] discussed Nash two-player equilibrium solutions between the supplier and the retailer. Chen et al. [12, 13] contributed the retailer's optimal model. Liao et al. [14] and Wu et al. [15] discussed optimal strategy for deteriorating items with capacity

The products like volatile liquids, blood banks, fruits, fashion merchandises, vegetables, and high-tech products deteriorate continuously due to evaporation, spoilage, and obsolescence, among other reasons. An exponentially decaying inventory model is built by Ghare and Schrader [16]. The constant deterioration rate is extended to Weibull failure rate by Covert and Philip [17]. Dave and Patel [18] proposed linear time functional demand. For shortages, Sachan [19] further generalized the EOQ model. The demand is log-concave with time dependent derived by Hariga [20]. Teng et al. [21] further expanded allowing shortages and continuous type demand pattern. Teng et al. [22] have permitted partial backlogging. For deteriorating products, Dye [23] investigated the effect of technology investment on refrigeration. No one of the above cited papers took the expiration date into consideration before Chen and Teng [24], Sarkar [25], Wang et al. [26],

DCF is an important tool in inventory management. Researchers such as Hill and Pakkala [29], Chung and Liao [30], Dye et al. [31], Chang et al. [32], Mousavi et al. [33] work related to DCF analysis. Recently, Chen and Teng [34] and Duari and Chakraborti [35] applied the DCF analysis to obtain the optimal lot size and credit period in a supply chain with upstream and downstream trade credit financing. A credit-worthy retailer generally gains a permissible delay on the entire purchasing quantity, in reality. However, a retailer often asks for credit-risk customers to cover a fraction of the purchasing cost at the time of placing an order and then provides a permissible delay (downstream credit). To reduce default risks with credit-risk customers, they use downstream partial trade credit as a strategy that has received relatively little attention by the researchers. Additionally, the majority of the recent studies consider merely the opportunity loss of trade credit. Most of the time, they ignore to take the opportunity loss of the other different costs in their study in order to take the effect of inflation and time value of money. For an exquisite and sharp analysis, the DCF analysis must be used on all relevant revenue and costs. Due to this fact, here a supplier-retailer-customer supply chain model is proposed. The supplier provides an upstream full credit period of S years to the retailer and the retailer gives to the customer a downstream partial credit period of R years. The deteriorating rate is constant or increasing and closer to 100% near expiration date. With time-dependent demand, the DCF analysis is applied to

holding cost, therefore affecting order quantity of buyer.

Application of Decision Science in Business and Management

constraints.

108

Wu et al. [27], and Sarkar et al. [28].

The following notations are used in this model:


#### Functions:


For convenience, the asterisk symbol on a variable is denoted the optimal solution of the variable. For instance, T\* is the optimal solution of T.

## 2.2 Assumptions

The following assumptions are made to build the mathematical inventory model:

3. Mathematical model

DOI: http://dx.doi.org/10.5772/intechopen.90689

I ∣

I tðÞ¼ e

3.1 Sales revenue

SR ¼ p α

¼ p

111

cycle time T of the retailer is

ð T

2 4

ð Þ a þ bt e

0

e

paper. Solving the differential Eq. (4), we get

�δð Þt ð T

t e

� �

�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

�i Rð Þ <sup>þ</sup><sup>T</sup> <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biR � <sup>e</sup>

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>m</sup> � <sup>t</sup>

1 þ m

equation (Figure 1):

Figure 1.

Graphically shows the model.

The inventory level is depleted by demand and deterioration, during the replenishment cycle [0, T], and hence governed by the following differential

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial…

ðÞ¼� t D � θð Þt I tð Þ, 0≤t ≤T, D ¼ ð Þ a þ bt (4)

ðt

θð Þ u du

0

With the boundary condition I(T) = 0. Note that the prime symbol on a variable

<sup>δ</sup>ð Þ <sup>u</sup> ð Þ <sup>a</sup> <sup>þ</sup> bt du, where <sup>δ</sup>ðÞ¼ <sup>t</sup>

The customers get a downstream credit period of R years from the retailer. Thus, the retailer receives the cash payment for the time 0 to T. Also the retailer receives the credit payment from R to T + R. Hence, the present value of sales revenue per

ð Þ a þ bt e

iTð Þþ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biR biT � �ð Þ �<sup>1</sup> <sup>þ</sup> <sup>α</sup>

þ b þ ai � e

�itdt

�iTð Þ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biT � �<sup>α</sup>

3

5 (6)

(7)

T ð þR

R

i 2

ð Þ 1 þ m ðb tð Þ � T

þð Þ a þ b þ bm ðLog 1½ �� þ m � t Log 1½ � þ m � T ÞÞ (5)

is denoted the first order derivative with respect to the variable throughout the


$$0 \le \theta(t) \le 1, \theta^{\parallel}(t) \ge 0 \text{ and } \theta(m) = \mathbf{1}.\tag{1}$$

We assume the deterioration as

$$\theta(t) = \frac{1}{(1+m-t)}, \ 0 \le t \le m \tag{2}$$

a special case of (1):

e. It is assumed without loss of generality that both upstream and downstream credit periods R and S and the replenishment cycle time T are less than or equal to the expiration date m, since the deterioration rate reaches 100% after expiration date

$$\mathbf{R} \le \mathbf{m}, \mathbf{S} \le \mathbf{m}, \text{and } \mathbf{T} \le \mathbf{m} \tag{3}$$


Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial… DOI: http://dx.doi.org/10.5772/intechopen.90689

## 3. Mathematical model

For convenience, the asterisk symbol on a variable is denoted the optimal

The following assumptions are made to build the mathematical inventory model:

a. In a supplier-retailer-customer supply chain system, the retailer obtains a full upstream credit period of S years from his/her supplier and in turn gives a partial downstream trade credit to his/her credit-risk customers who must cover α portion of the purchasing cost at the time of placing an order and then get a credit period of R years on the outstanding quantity. For reliable customers, just set α ¼ 0, the retailer may provide a full trade credit.

b. The retailer deposits the sales revenue into an interest-bearing account after time R, if S ≥ R. When S ≥ (T + R), the retailer pays off the entire purchasing cost and collects all sales revenue at time S. Both, the credit payment sold by SR and the cash payment, are obtained from time 0 to S for S < T + R. To use the other activities and begin paying for the interest charges on the items sold, the retailer pays the supplier and retains the profit after

c. The retailer gets cash payments from customers and immediately deposits those into an interest-bearing account until time S, if S ≤ R. As to credit payments, the retailer must finance 1ð Þ � α c að Þ þ bt T at time S and then pay

d. A deteriorating item deteriorates continuously and cannot be sold after its maximum lifetime or expiration date. Therefore, its deterioration rate is percent near to its expiration date. As a result, it is assumed without loss of generality that the deterioration rate θð Þt at time t, 0 ≤t≤ m, satisfies the

1

e. It is assumed without loss of generality that both upstream and downstream credit periods R and S and the replenishment cycle time T are less than or equal to the expiration date m, since the deterioration rate reaches 100% after

ð Þt ≥0 and θð Þ¼ m 1: (1)

ð Þ <sup>1</sup> <sup>þ</sup> <sup>m</sup> � <sup>t</sup> , 0≤<sup>t</sup> <sup>≤</sup> <sup>m</sup> (2)

R≤ m, S≤ m, and T≤ m (3)

solution of the variable. For instance, T\* is the optimal solution of T.

Application of Decision Science in Business and Management

2.2 Assumptions

(S � R).

off the loan from time R to time T + R.

<sup>0</sup><sup>≤</sup> <sup>θ</sup>ð Þ<sup>t</sup> <sup>≤</sup>1, <sup>θ</sup><sup>∣</sup>

θðÞ¼ t

following conditions:

a special case of (1):

expiration date

f. No shortages allowed.

110

g. Replenishment rate is instantaneous.

We assume the deterioration as

The inventory level is depleted by demand and deterioration, during the replenishment cycle [0, T], and hence governed by the following differential equation (Figure 1):

Figure 1. Graphically shows the model.

$$I^\dagger(t) = -D - \theta(t)I(t), \mathbf{0} \le t \le T,\\ D = (a + bt) \tag{4}$$

With the boundary condition I(T) = 0. Note that the prime symbol on a variable is denoted the first order derivative with respect to the variable throughout the paper. Solving the differential Eq. (4), we get

$$I(t) = e^{-\delta(t)} \int\_{t}^{T} e^{\delta(u)}(a+bt) \, du, \text{where } \delta(t) = \int\_{0}^{t} \theta(u) \, du$$

$$= \left(1 + m - \frac{t}{1+m}\right)(1+m)(b(t-T) \, \text{ }$$

$$+ (a+b+bm)(\text{Log}[1+m-t] - \text{Log}[1+m-T])) \tag{5}$$

#### 3.1 Sales revenue

The customers get a downstream credit period of R years from the retailer. Thus, the retailer receives the cash payment for the time 0 to T. Also the retailer receives the credit payment from R to T + R. Hence, the present value of sales revenue per cycle time T of the retailer is

$$\begin{split} \text{SIR} &= p \left[ a \int\_{0}^{T} (a+bt)e^{-it}dt + (\mathbf{1}-\boldsymbol{a}) \int\_{\bar{R}}^{T+R} (a+bt)e^{-it}dt \right] \\ &= p \left[ \frac{e^{-i(\boldsymbol{R}+\boldsymbol{T})} \left(b+ai+bi\mathbf{R}-e^{i\boldsymbol{T}}(b+ai+bi\mathbf{R}) + b\dot{\boldsymbol{u}}\boldsymbol{T}\right)(-\mathbf{1}+\boldsymbol{a})}{i^{2}} \right] \end{split} \tag{6}$$

## 3.2 Different costs

## 3.2.1 Ordering cost

At time 0, the retailer orders deteriorating items. Hence, the present value of the retailer's ordering cost per cycle time T is

$$\textsf{OC} \tag{8}$$

3.2.4 Case I: when R ≤ S

DOI: http://dx.doi.org/10.5772/intechopen.90689

ii. T ≤ S ≤ T + R.

investigated accordingly.

3.2.4.1 Sub-case 1a: S ≤ T

IE ¼ pIe α

charged per cycle is given by

2 4 ð T

ð Þ T � t ð Þ a þ bt e

S

IC ¼ cIc α

113

ð S

2 4

t að Þ þ bt e

<sup>2</sup><sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> <sup>e</sup>�iSð Þ �aið Þ� <sup>1</sup> <sup>þ</sup> iS <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iSð Þ <sup>2</sup> <sup>þ</sup> iS � �<sup>α</sup> i 3

<sup>e</sup>�i Rð Þ <sup>þ</sup><sup>S</sup> eiSðai <sup>þ</sup> <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iR Þ þ <sup>e</sup>iR aið Þ �<sup>1</sup> <sup>þ</sup> i Rð Þ � <sup>S</sup>

0

iii. T + R ≤ S.

applicable.

cycle is

¼ pie

þ

(iii) T + R ≤ S.

i. S ≤ T.

Based on the values of S, T, and T + R, three sub-cases can occur:

Notice that for both cases (i) S ≤ T and (ii) T ≤ S ≤ T + R, the case S = T is

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial…

The interest earned and the interest charged for the above three cases are

In this sub-case, the retailer gets revenue and receives interest from the two

credit payment for the time R to S. So, the present value of the interest earned per

�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

ð S

ð Þ t � R ð Þ a þ bt e

#

�itdt

3 5

(11)

R

<sup>þ</sup>bð Þ �<sup>2</sup> <sup>þ</sup> i Rð Þ <sup>þ</sup> iRS � <sup>S</sup>ð Þ <sup>2</sup> <sup>þ</sup> iS � � � � ð Þ <sup>1</sup> � <sup>α</sup>

i 3

The retailer provides his customers a credit period of R years and gets customers'

credit payments from time R through (T + R), on the other hand. The retailer obtains αcð Þ a þ bt S dollars from cash payment and (1-α)cð Þ a þ bt (S-R) dollars from credit payment, at time S, and therefore pays his/her supplier

[αcð Þ a þ bt S+(1-αÞcð Þ a þ bt (S-R)] dollars. Consequently, the retailer must finance all items sold after time S for the cash payment and (S – R) for the credit payment at an interest charged Ic per dollar per year. So, the present amount of the interest

�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

T ð þR

S

ð Þ T þ R � t ð Þ a þ bt e

�itdt

3 5

possible sources: (a) the cash payment for the time 0 to S and (b) the

Similarly, the condition S = T + R is applicable for cases (ii) T ≤ S ≤ T + R and

## 3.2.2 Purchasing cost

Since the upstream trade credit is S years, the retailer must pay the supplier the whole purchasing cost cQ at time S. As a result, the present value of the retailer's purchasing cost per cycle time T is

$$PC = ce^{-iS}Q = ce^{-iS}I(\mathbf{0}) = c \int\_0^T (a+bt)e^{\delta(t)-iS}dt$$

$$= ce^{-iS}(\mathbf{1}+m)^2(-bT+(a+b+bm)(\text{Log}[\mathbf{1}+m]-\text{Log}[\mathbf{1}+m-T]))\tag{9}$$

3.2.3 Holding cost

The present value of the retailer's holding cost per cycle time T is

$$HC = h \left\{ e^{-\hat{\boldsymbol{\mu}}t} I(t) dt = h \prod\_{i=1}^{T} (a+bt) e^{\delta(u) - \delta(t) - \hat{\boldsymbol{\mu}}t} \text{d}u dt$$

$$= h \left( -\frac{b(1+m)^2 \left(-1 + e^{-iT} + iT\right)}{i^2} + \frac{b e^{-iT} \left(2 + iT + e^{iT} (-2 + iT)\right)}{i^3} \right)$$

$$e^{-i(1+m)} (1+m)^2 (a+b+bm) \begin{pmatrix} -\text{ExpltagralEi}[i(1+m)] \\ +\text{ExpltagralEi}[i(1+m-T)] \\ +e^{i(1+m)} \text{Log} \left[\frac{1+m}{1+m-T}\right] \end{pmatrix}$$

$$e^{-i(1+m)} (a+b+bm) (e^{i(1+m)} - e^{i(1+m-T)} - (1+i+m) \text{ExpltagralEi}[i(1+m)] )}$$

$$+ (1+i+m) \text{ExpltagralEi}[i(1+m-T)] + e^{i(1+m)} \text{Log} \left[\frac{1+m}{1+m-T}\right] \right)$$

$$i^2 \tag{10}$$

According to the values of R and S, there are two potential cases:

Case I: R ≤ S. Case II: R ≥ S.

�

Both cases are discussed separately.

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial… DOI: http://dx.doi.org/10.5772/intechopen.90689

3.2.4 Case I: when R ≤ S

Based on the values of S, T, and T + R, three sub-cases can occur:

i. S ≤ T.

3.2 Different costs

3.2.1 Ordering cost

3.2.2 Purchasing cost

retailer's ordering cost per cycle time T is

Application of Decision Science in Business and Management

purchasing cost per cycle time T is

2

HC ¼ h

<sup>e</sup>�ið Þ <sup>1</sup>þ<sup>m</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>m</sup> <sup>2</sup>

Both cases are discussed separately.

<sup>¼</sup> <sup>h</sup> � <sup>b</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>m</sup>

�ið Þ <sup>1</sup>þ<sup>m</sup> ð Þð <sup>a</sup> <sup>þ</sup> <sup>b</sup> <sup>þ</sup> bm <sup>e</sup>

þ

Case I: R ≤ S. Case II: R ≥ S.

�

112

e

ð T

0 e

i

ð T

ð Þ a þ bt e

The present value of the retailer's holding cost per cycle time T is

�itI tð Þdt <sup>¼</sup> <sup>h</sup>

<sup>2</sup> �<sup>1</sup> <sup>þ</sup> <sup>e</sup>�iT <sup>þ</sup> iT � �

ð Þ a þ b þ bm

þð Þ 1 þ i þ im ExpIntegralEi½ið Þ 1 þ m � T � þ e

According to the values of R and S, there are two potential cases:

<sup>i</sup>ð Þ <sup>1</sup>þ<sup>m</sup> � <sup>e</sup>

<sup>2</sup> þ

0

PC <sup>¼</sup> ce�iSQ <sup>¼</sup> ce�iSIð Þ¼ <sup>0</sup> <sup>c</sup>

<sup>¼</sup> ce�iSð Þ <sup>1</sup> <sup>þ</sup> <sup>m</sup>

3.2.3 Holding cost

At time 0, the retailer orders deteriorating items. Hence, the present value of the

Since the upstream trade credit is S years, the retailer must pay the supplier the whole purchasing cost cQ at time S. As a result, the present value of the retailer's

<sup>δ</sup>ð Þ�<sup>t</sup> iSdt

ð T ð T

ð Þ a þ bt e

þe

<sup>δ</sup>ð Þ�<sup>u</sup> <sup>δ</sup>ð Þ�<sup>t</sup> itdudt

be�iT <sup>2</sup> <sup>þ</sup> iT <sup>þ</sup> <sup>e</sup>iTð Þ �<sup>2</sup> <sup>þ</sup> iT � � i 3

�ExpIntegralEi½ � ið Þ 1 þ m þExpIntegralEi½ � ið Þ 1 þ m � T

<sup>i</sup>ð Þ <sup>1</sup>þ<sup>m</sup> Log <sup>1</sup> <sup>þ</sup> <sup>m</sup>

<sup>i</sup>ð Þ <sup>1</sup>þm�<sup>T</sup> � ð Þ <sup>1</sup> <sup>þ</sup> <sup>i</sup> <sup>þ</sup> im ExpIntegralEi½ � <sup>i</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>m</sup>

1 þ m � T � �

<sup>i</sup>ð Þ <sup>1</sup>þ<sup>m</sup> Log <sup>1</sup> <sup>þ</sup> <sup>m</sup>

<sup>2</sup> Þ

1

CCCCA

1 þ m � T � �!

(10)

t

0

0

BBBB@

i

i

ð Þ �bT þ ð Þ a þ b þ bm ð Þ Log 1½ �� þ m Log 1½ � þ m � T (9)

OC (8)

ii. T ≤ S ≤ T + R.

iii. T + R ≤ S.

Notice that for both cases (i) S ≤ T and (ii) T ≤ S ≤ T + R, the case S = T is applicable.

Similarly, the condition S = T + R is applicable for cases (ii) T ≤ S ≤ T + R and (iii) T + R ≤ S.

The interest earned and the interest charged for the above three cases are investigated accordingly.

## 3.2.4.1 Sub-case 1a: S ≤ T

In this sub-case, the retailer gets revenue and receives interest from the two possible sources: (a) the cash payment for the time 0 to S and (b) the credit payment for the time R to S. So, the present value of the interest earned per cycle is

$$IE = pI\_{\epsilon} \left[ a \int\_{0}^{S} (a+bt)e^{-it}dt + (1-a) \int\_{R}^{S} (t-R)(a+bt)e^{-it}dt \right]$$

$$= p i\_{\epsilon} \left[ \frac{e^{-i(R+S)} \left( e^{iS}(ai+b(2+iR)) + e^{iR} \left( \begin{matrix} ai(-1+i(R-S)) \\ + b(-2+i(R+iRS-S(2+iS)) \end{matrix} \right) \right) (1-a)}{i^3} \right]$$

$$+ \frac{(2b+ai+e^{-iS}(-ai(1+iS)-b(2+iS(2+iS))))a}{i^3} \tag{11}$$

The retailer provides his customers a credit period of R years and gets customers' credit payments from time R through (T + R), on the other hand. The retailer obtains αcð Þ a þ bt S dollars from cash payment and (1-α)cð Þ a þ bt (S-R) dollars from credit payment, at time S, and therefore pays his/her supplier [αcð Þ a þ bt S+(1-αÞcð Þ a þ bt (S-R)] dollars. Consequently, the retailer must finance all items sold after time S for the cash payment and (S – R) for the credit payment at an interest charged Ic per dollar per year. So, the present amount of the interest charged per cycle is given by

$$IC = cI\_{\mathcal{L}} \left[ a \int\_{S}^{T} (T - t)(a + bt)e^{-it}dt + (\mathbf{1} - a) \int\_{S}^{T+R} (T + R - t)(a + bt)e^{-it}dt \right]$$

$$\begin{aligned} \dot{\epsilon}\_{i} &= \dot{\epsilon}\_{i} \left[ \frac{\left( e^{-i(\mathbf{R}+T)} (a\dot{\mathbf{i}} + b(2 + i(\mathbf{R}+T))) + (1-a)e^{-iS} \begin{pmatrix} a(-1 + i(\mathbf{R}-S+T)) \\ + b \left( \frac{-2 + i(\mathbf{R}-2S+T)}{+i^{2}S(\mathbf{R}-S+T)} \right) \end{pmatrix} \right)}{i^{3}} \right] \\\\ &+ \left( \frac{e^{-iT} (ai + b(2 + iT))}{+b(-2 + i(T + S(-2 + i(-S + T))))} \right) \dot{\mathbf{a}} \\ &+ \frac{\left( \frac{-2 + i(\mathbf{R}+2S(-2 + i(-S + T)))}{+b} \right)}{i^{3}} \end{aligned} \tag{12}$$

¼ pie

þ

¼

¼ 1 T

115

( p α ð T

� h ð T

2 4 2 4

0

2 4

þ pIe α

annual total relevant profit is

ð Þ a þ bt e

ð Þ a þ bt e

t að Þ þ bt e

interest earned per cycle is given by

ð T

0

4.2.1 Sub-case 1c: T + R ≤ S

0

ð T

t

0 @

<sup>e</sup>�i Rð Þ <sup>þ</sup><sup>S</sup> <sup>e</sup>iSðai <sup>þ</sup> <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iR Þ þ eiR

DOI: http://dx.doi.org/10.5772/intechopen.90689

<sup>þ</sup> <sup>e</sup>�i Sð Þ <sup>þ</sup><sup>T</sup> �eiTð Þ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biS <sup>T</sup> <sup>þ</sup> <sup>e</sup>iST bð Þ <sup>þ</sup> ai <sup>þ</sup> biT � � i 2

<sup>2</sup><sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> <sup>e</sup>�iTð Þ �aið Þ� <sup>1</sup> <sup>þ</sup> iT <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iTð Þ <sup>2</sup> <sup>þ</sup> iT i 3

IC ¼ ð Þ 1 � α

þ b �2 þ i Rð Þþ � 2S þ T i

4.2 Profit of the second sub-case of the model

PTP2ð Þ¼ <sup>T</sup> <sup>1</sup>

�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

�itdt <sup>þ</sup>

aið Þ �1 þ i Rð Þ � S <sup>þ</sup>bð Þ �<sup>2</sup> <sup>þ</sup> i Rð Þ <sup>þ</sup> iRS � <sup>S</sup>ð Þ <sup>2</sup> <sup>þ</sup> iS ! ! ð Þ <sup>1</sup> � <sup>α</sup>

> � α #

> > (14)

i 3

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial…

By time T (≤ S), the retailer receives all cash payments so that there is no interest charged for the cash payment. However, the retailer must pay up all items

<sup>ð</sup>e�i Rð Þ <sup>þ</sup><sup>T</sup> <sup>ð</sup>ai <sup>þ</sup> <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> i Rð Þ <sup>þ</sup> <sup>T</sup> Þ þ <sup>e</sup>�iSðaið Þ �<sup>1</sup> <sup>þ</sup> i Rð Þ � <sup>S</sup> <sup>þ</sup> <sup>T</sup>

2 S Rð Þ � <sup>S</sup> <sup>þ</sup> <sup>T</sup> � �ÞÞð Þ <sup>1</sup> � <sup>α</sup> i

From (7)–(10), (14), and (15), it is known that the present value of the retailer's

ð Þ a þ bt e

�itdt

<sup>T</sup> ð Þ SR � PC � HC � OC � IC <sup>þ</sup> IE

�itdt

T ð þR

S

A þ ð Þ 1 � α

1

3 5 � c ð T

0

ð S

R

ð Þ a þ bt e

ð Þ T þ R � t ð Þ a þ bt e

<sup>δ</sup>ð Þ�<sup>t</sup> iSdt

ð Þ t � R ð Þ a þ bt e

�itdt

3 5

�itdt

(16)

3 5 )

ð Þ T þ R � t ð Þ a þ bt e

�itdt�

<sup>3</sup> (15)

sold during the interval [S–R, T]. Therefore, the annual interest charged is

T ð þR

S

T ð þR

R

T að Þ þ bt e

The retailer receives all the cash and credit payments before the supplier's upstream credit period S and no interest charged. However, the present value of the

<sup>δ</sup>ð Þ� <sup>u</sup> <sup>δ</sup>ð Þ�<sup>t</sup> itdudt � OC � ð Þ <sup>1</sup> � <sup>α</sup>

ð S

T

## 4. Total profit of the model

#### 4.1 Profit of the first sub-case of the model

As a result, the present value of the retailer's annual total profit by using (7)–(12) is

PTP1ð Þ¼ <sup>T</sup> <sup>1</sup> <sup>T</sup> ð Þ SR � PC � HC � OC � IC <sup>þ</sup> IE ¼ 1 <sup>T</sup> <sup>p</sup> <sup>α</sup> ð T 0 ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup> T ð þR R ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt 2 4 3 5 8 < : � c ð T 0 ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup><sup>δ</sup>ð Þ�<sup>t</sup> iSdt � <sup>h</sup> ð T 0 ð T t ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup><sup>δ</sup>ð Þ� <sup>u</sup> <sup>δ</sup>ð Þ�<sup>t</sup> itdudt � <sup>O</sup> � cIc α ð T S ð Þ <sup>T</sup> � <sup>t</sup> ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup> T ð þR S ð Þ <sup>T</sup> <sup>þ</sup> <sup>R</sup> � <sup>t</sup> ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt 2 4 3 5 þ pIe α ð S 0 t að Þ <sup>þ</sup> bt <sup>e</sup>�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup> ð S R ð Þ <sup>t</sup> � <sup>R</sup> ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt 2 4 3 5 9 = ; (13)

#### 4.1.1 Sub-case 1b: T ≤ S ≤ T+R

The retailer accumulates revenue and obtains interest from two sources: (a) the cash payment starting from time 0 to S and (b) the credit payment starting from time R to S. So, the present value of the interest earned per cycle is

$$\text{sATE} = pI\_{\varepsilon} \left[ a \left( \int\_{0}^{T} (a+bt)e^{-\dot{t}t}dt + \int\_{T}^{S} T(a+bt)e^{-\dot{t}t}dt \right) + (\mathbf{1}-a) \int\_{R}^{S} (t-R)(a+bt)e^{-\dot{t}t}dt \right]$$

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial… DOI: http://dx.doi.org/10.5772/intechopen.90689

$$\begin{aligned} \dot{\epsilon} &= \text{pi}\_{\epsilon} \left[ \frac{e^{-i(R+S)} \left( e^{iS} (ai + b(2+iR)) + e^{iR} \left( \frac{ai(-1+i(R-S))}{+b(-2+i(R+iRS-S(2+iS)))} \right) \right) (1-a)}{i^3} \right] \\ &+ \left( \frac{e^{-i(S+T)} \left( -e^{iT} (b+ai+biS)T + e^{iS}T(b+ai+biT) \right)}{i^2} \right) \\ &+ \frac{2b+ai+e^{-iT} \left( -ai(\mathbf{1}+iT) - b(2+iT(2+iT)) \right)}{i^3} \Big| a \right] \end{aligned} \tag{14}$$

By time T (≤ S), the retailer receives all cash payments so that there is no interest charged for the cash payment. However, the retailer must pay up all items sold during the interval [S–R, T]. Therefore, the annual interest charged is

$$IC = (\mathbf{1} - a) \int\_{S}^{T+R} (T + R - t)(a + bt)e^{-it}dt$$

$$= \frac{(e^{-i(\mathbf{R} + T)}(ai + b(2 + i(\mathbf{R} + T))) + e^{-iS}(ai(-\mathbf{1} + i(\mathbf{R} - S + T)))}{i^3}$$

$$= \frac{-b\left(-2 + i(\mathbf{R} - 2S + T) + i^2S(\mathbf{R} - S + T)\right))(\mathbf{1} - a)}{i^3} \tag{15}$$

#### 4.2 Profit of the second sub-case of the model

From (7)–(10), (14), and (15), it is known that the present value of the retailer's annual total relevant profit is

$$\textbf{PTP}\_{2}(T) = \frac{\textbf{1}}{T}(\textbf{SR} - \textbf{PC} - \textbf{OC} - \textbf{OC} + \textbf{IE})$$

$$= \frac{1}{T} \left\{ p \left[ a \int\_{0}^{T} (a + bt)e^{-\hat{u}} dt + (1 - a) \int\_{R}^{T+R} (a + bt)e^{-\hat{u}} dt \right] - c \int\_{0}^{T} (a + bt)e^{\delta(t) - iS} dt \right.$$

$$\left. - \left[ h \int\_{0}^{T} (a + bt)e^{\delta(u) - \delta(t) - it} du dt - OC - (1 - a) \int\_{S}^{\top} (T + R - t)(a + bt)e^{-\hat{u}} dt \right] \right.$$

$$\left. + pI\_{\varepsilon} \left[ a \left( \int\_{0}^{T} (a + bt)e^{-\hat{u}} dt + \int\_{T}^{S} (a + bt)e^{-\hat{u}} dt \right) + (1 - a) \int\_{R}^{S} (t - R)(a + bt)e^{-\hat{u}} dt \right] \right\}$$

#### 4.2.1 Sub-case 1c: T + R ≤ S

The retailer receives all the cash and credit payments before the supplier's upstream credit period S and no interest charged. However, the present value of the interest earned per cycle is given by

¼ cic

þ

(7)–(12) is

PTP1ð Þ¼ <sup>T</sup> <sup>1</sup>

¼ 1 <sup>T</sup> <sup>p</sup> <sup>α</sup>

> � c ð T

> > 0

2 4

> 2 4

� cIc α

þ pIe α

IE ¼ pIe α

114

0

0

BB@

BB@

<sup>e</sup>�i Rð Þ <sup>þ</sup><sup>T</sup> <sup>ð</sup>ai <sup>þ</sup> <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> i Rð Þ <sup>þ</sup> <sup>T</sup> Þ þ ð Þ <sup>1</sup> � <sup>α</sup> <sup>e</sup>�iS

Application of Decision Science in Business and Management

<sup>e</sup>�iTð Þ ai <sup>þ</sup> <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iT

i 3

<sup>T</sup> ð Þ SR � PC � HC � OC � IC <sup>þ</sup> IE

ð T ð T

t

time R to S. So, the present value of the interest earned per cycle is

ð S

T að Þ þ bt e

T

0

ð Þ <sup>T</sup> � <sup>t</sup> ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

t að Þ <sup>þ</sup> bt <sup>e</sup>�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

�itdt <sup>þ</sup>

ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

þbð Þ �2 þ i Tð Þ þ Sð Þ �2 þ ið Þ �S þ T

As a result, the present value of the retailer's annual total profit by using

T ð þR

R

ð S

R

The retailer accumulates revenue and obtains interest from two sources: (a) the cash payment starting from time 0 to S and (b) the credit payment starting from

�itdt

1

A þ ð Þ 1 � α

ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt

ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup><sup>δ</sup>ð Þ� <sup>u</sup> <sup>δ</sup>ð Þ�<sup>t</sup> itdudt � <sup>O</sup>

T ð þR

S

ð Þ <sup>t</sup> � <sup>R</sup> ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt

3 5

ð Þ <sup>T</sup> <sup>þ</sup> <sup>R</sup> � <sup>t</sup> ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup>�itdt

3 5 9 = ;

> ð S

ð Þ t � R ð Þ a þ bt e

R

3 5 (13)

�itdt

3 5

!

<sup>þ</sup>e�iS :

4.1 Profit of the first sub-case of the model

4. Total profit of the model

ð T

2 4

8 < :

0

ð T

S

ð S

0

4.1.1 Sub-case 1b: T ≤ S ≤ T+R

ð T

0 @

2 4

t að Þ þ bt e

0

ð Þ <sup>a</sup> <sup>þ</sup> bt <sup>e</sup><sup>δ</sup>ð Þ�<sup>t</sup> iSdt � <sup>h</sup>

aið Þ �1 þ i Rð Þ � S þ T

þi 2

�2 þ i Rð Þ � 2S þ T

!

1

1

CCA

(12)

CCA

S Rð Þ � S þ T

þ b

1

CCAα

0

BB@

i 3

$$\begin{split} I\mathbf{E} &= pI\_{\epsilon} \left( a \left( \int\_{0}^{T} t(a+bt)e^{-it}dt + \int\_{T}^{S} Te^{-it}dt \right) \right) \\ &+ (1-a) \left( \int\_{T+R}^{S} T(a+bt)e^{-it}dt \int\_{R}^{T+R} (t-R)(a+bt)e^{-it}dt \right) \Bigg|\_{} \\ &= pi\_{\epsilon} \left[ \left( -\frac{e^{-i(R+T)}\left( ai(1-\epsilon^{T}+iT) + b\left(2-\epsilon^{T}(2+iR) + i^{2}T(R+T) + i(R+2T)\right) \right)}{i^{3}} \right) \right. \\ &\left. + \frac{\epsilon^{-i(R+S+T)}\left( -\epsilon^{i(R+T)}(b+ai+biS)T + \epsilon^{S}T(ai+b(1+i(R+T))) \right)}{i^{2}} \right) (1-a) \\ &+ \left( \frac{\epsilon^{-i(S+T)}\left( -\epsilon^{iT}(b+ai+biS)T + \epsilon^{S}T(b+ai+biT) \right)}{i^{2}} \right. \\ &\left. + \frac{2b+ai+e^{-iT}\left( -ai(1+iT) - b(2+iT(2+iT)) \right)}{i^{3}} \right) a \right] \end{split} \tag{17}$$

4.3.1 Case II: when R ≥ S

DOI: http://dx.doi.org/10.5772/intechopen.90689

4.3.1.1 Sub-case 2a: S ≤ T

IC ¼ cIc α

þ T ð þR

¼ cic

117

"

R

ð T

0 @

2 4

S

The following sub-cases may occur based on values of S and T: S ≤ T, and S ≥ T.

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial…

As R ≥ S, there is no interest earned from the credit payment. However, the

ð S

ð Þ <sup>a</sup> <sup>þ</sup> bt te�itdt

�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

3 5

�itdt 9 = ;

<sup>þ</sup> <sup>e</sup>�i Rð Þ <sup>þ</sup><sup>T</sup> ai <sup>þ</sup> <sup>b</sup>ð<sup>2</sup> <sup>þ</sup> i Rð Þ <sup>þ</sup> <sup>T</sup> Þ þ <sup>e</sup>iTð Þ aið Þþ �<sup>1</sup> <sup>þ</sup> iT <sup>b</sup>ð Þ �<sup>2</sup> <sup>þ</sup> i Tð Þ <sup>þ</sup> <sup>R</sup>ð Þ �<sup>1</sup> <sup>þ</sup> iT � � i 3

Consequently, the present value of the retailer's annual total profit is

�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

ð T ð T

t

0

<sup>þ</sup> <sup>e</sup>�iTðai <sup>þ</sup> <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iT Þ þ <sup>e</sup>�iSð Þ aið�<sup>1</sup> <sup>þ</sup> <sup>i</sup>ð Þ �<sup>S</sup> <sup>þ</sup> <sup>T</sup> Þ þ <sup>b</sup>ð Þ �<sup>2</sup> <sup>þ</sup> i Tð Þ <sup>þ</sup> <sup>S</sup>ð Þ �<sup>2</sup> <sup>þ</sup> <sup>i</sup>ð Þ �<sup>S</sup> <sup>þ</sup> <sup>T</sup> � �

i 3

<sup>T</sup> ð Þ SR � PC � HC � OC � IC <sup>þ</sup> IE

T ð þR

R

ð Þ a þ bt e

ð Þ a þ bt e

�itdt

<sup>δ</sup>ð Þ�<sup>u</sup> <sup>δ</sup>ð Þ�<sup>t</sup> itdudt � OC

3 5

<sup>3</sup> (21)

T að Þ þ bt e

�itdt

!

ð Þ 1 � α

α

(22)

� :

ð R

8 < :

S

0

<sup>¼</sup> <sup>p</sup> <sup>2</sup><sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> <sup>e</sup>�iSð Þ �aið Þ� <sup>1</sup> <sup>þ</sup> iS <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iSð Þ <sup>2</sup> <sup>þ</sup> iS � �αie i

At time S, the retailer should finance (1-α)cð Þ a þ bt T for the credit payment and αcð Þ a þ bt (T–S) for the cash payment, respectively. Then, the retailer renders the loan for the cash payment at time T and pays off the loan for the credit payment at

present value of the annual interest earned from the cash payment is

IE ¼ αpIe

t = T + R. So, the present amount of the interest charged per cycle is

ð Þ T � t ð Þ a þ bt e

ð Þ T þ R � t ð Þ a þ bt e

4.4 Profit of the first sub-case of the second case of the model

PTP4ð Þ¼ <sup>T</sup> <sup>1</sup>

ð T

ð Þ a þ bt e

<sup>δ</sup>ð Þ�<sup>t</sup> iSdt � <sup>h</sup>

0

ð Þ a þ bt e

¼ 1

� c ð T

<sup>T</sup> <sup>p</sup> <sup>α</sup>

0

8 < : 2 4

<sup>e</sup>�i Rð Þ <sup>þ</sup><sup>S</sup> �eiSð Þ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biR <sup>T</sup> <sup>þ</sup> <sup>e</sup>iRð Þ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biS <sup>T</sup> � � i 2

#### 4.3 Profit of the third sub-case of the model

So the present value of the retailer's annual total profit is

$$\begin{aligned} \textbf{PTP}\_3(T) &= \frac{1}{T}(\textbf{SR} - \textbf{PC} - \textbf{OC} + \textbf{IE}) \\ = \frac{1}{T} \left\{ p \left[ a \int\_0^T (a+bt)e^{-\hat{u}t}dt + (1-a) \int\_0^{T+R} (a+bt)e^{-\hat{u}t}dt \right] \right. \\ &\left. - c \int\_0^T (a+bt)e^{\delta(t) - \hat{u}t}dt - h \int\_0^T (a+bt)e^{\delta(u) - \delta(t) - \hat{u}t}dudt - OC \\ &+ pI\_\varepsilon \left[ a \left( \int\_0^T (a+bt)e^{-\hat{u}t}dt + \int\_T^T \mathbf{C}e^{-\hat{u}t}dt \right) \right. \\ &+ (1-a) \left( \int\_{T+R}^S T(a+bt)e^{-\hat{u}t}dt + \int\_R^{T+R} (t-R)(a+bt)e^{-\hat{u}t}dt \right) \right] \right\} \end{aligned} \tag{18}$$

Combining (13), (16), and (18), the present value of the retailer's annual total profit is given as

$$\begin{aligned} \text{PTP}(T) &= \text{PTP}\_1(T) \text{ if } S \le T\\ \text{PTP}(T) &= \text{PTP}\_2(T) \text{ if } S - R \le T \le S \end{aligned}$$

$$\text{PTP}(T) = \text{PTP}\_3(T) \text{ if } T \le S - R \tag{19}$$

It is clear from (13), (16), and (18) that

$$PTP\_1(\mathcal{S}) = PTP\_2(\mathcal{S}), and \, PTP\_2(\mathcal{S} - R) = PTP\_3(\mathcal{S} - R) \tag{20}$$

This implies that PTP(T) is continuous in T ≥ 0.

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial… DOI: http://dx.doi.org/10.5772/intechopen.90689

#### 4.3.1 Case II: when R ≥ S

IE ¼ pIe α

"

þ

¼ 1

� c ð T

<sup>T</sup> <sup>p</sup> <sup>α</sup>

0

2 4

þ pIe α

þ ð Þ 1 � α

profit is given as

116

8 < : 2 4 ð T

0

ð Þ a þ bt e

0 @

ð T

0

0 @ ð S

TþR

It is clear from (13), (16), and (18) that

This implies that PTP(T) is continuous in T ≥ 0.

ð T

0 @

2 4

þ ð Þ 1 � α

t að Þ þ bt e

ð S

TþR

<sup>¼</sup> pie � <sup>e</sup>�i Rð Þ <sup>þ</sup><sup>T</sup> ai <sup>1</sup> � <sup>e</sup>iT <sup>þ</sup> iT � � <sup>þ</sup> <sup>b</sup> <sup>2</sup> � <sup>e</sup>iTð Þþ <sup>2</sup> <sup>þ</sup> iR <sup>i</sup>

<sup>þ</sup> <sup>e</sup>�i Sð Þ <sup>þ</sup><sup>T</sup> �eiTð Þ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biS <sup>T</sup> <sup>þ</sup> <sup>e</sup>iST bð Þ <sup>þ</sup> ai <sup>þ</sup> biT � � i 2

<sup>2</sup><sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> <sup>e</sup>�iTð Þ �aið Þ� <sup>1</sup> <sup>þ</sup> iT <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iTð Þ <sup>2</sup> <sup>þ</sup> iT i 3

So the present value of the retailer's annual total profit is

�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

ð T ð T

t

ð S

T

�itdt <sup>þ</sup>

PTP Tð Þ¼ PTP1ð Þ T if S≤ T

0

�itdt <sup>þ</sup>

PTP3ð Þ¼ <sup>T</sup> <sup>1</sup>

<sup>δ</sup>ð Þ�<sup>t</sup> iSdt � <sup>h</sup>

T að Þ þ bt e

t að Þ þ bt e

4.3 Profit of the third sub-case of the model

ð Þ a þ bt e

�itdt <sup>þ</sup>

T að Þ þ bt e

<sup>þ</sup> <sup>e</sup>�i Rð Þ <sup>þ</sup>Sþ<sup>T</sup> �ei Rð Þ <sup>þ</sup><sup>T</sup> ð Þ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biS <sup>T</sup> <sup>þ</sup> <sup>e</sup>iST ai ð Þ <sup>þ</sup> <sup>b</sup>ð Þ <sup>1</sup> <sup>þ</sup> i Rð Þ <sup>þ</sup> <sup>T</sup> � � i 2

ð S

Te�itdt

R

i 3

T Rð Þþ <sup>þ</sup> <sup>T</sup> i Rð Þ <sup>þ</sup> <sup>2</sup><sup>T</sup> � � � �

� α �

<sup>T</sup> ð Þ SR � PC � HC � OC <sup>þ</sup> IE

ð Þ a þ bt e

�itdt

<sup>δ</sup>ð Þ� <sup>u</sup> <sup>δ</sup>ð Þ�<sup>t</sup> itdudt � OC

ð Þ t � R ð Þ a þ bt e

PTP Tð Þ¼ PTP3ð Þ T if T ≤S � R (19)

PTP1ð Þ¼ S PTP2ð ÞS , and PTP2ð Þ¼ S � R PTP3ð Þ S � R (20)

3 5

�itdt

1 A 3 5 9 = ;

T ð þR

R

ð Þ a þ bt e

Te�itdt

T ð þR

R

Combining (13), (16), and (18), the present value of the retailer's annual total

PTP Tð Þ¼ PTP2ð Þ T if S � R≤T ≤S

1 A 1 A

ð Þ t � R ð Þ a þ bt e

2

�itdt

!

ð Þ 1 � α

(17)

(18)

T

�itdt T ð þR

0

0 @

Application of Decision Science in Business and Management

The following sub-cases may occur based on values of S and T: S ≤ T, and S ≥ T.

## 4.3.1.1 Sub-case 2a: S ≤ T

As R ≥ S, there is no interest earned from the credit payment. However, the present value of the annual interest earned from the cash payment is

$$IE = apI\_{\epsilon} \begin{vmatrix} \\ \\ \\ 0 \end{vmatrix} (a+bt)t e^{-it} dt$$

$$= \frac{p\left(2b+ai+e^{-iS}(-ai(1+iS)-b(2+iS(2+iS)))\right)ai\_{\epsilon}}{i^3} \tag{21}$$

At time S, the retailer should finance (1-α)cð Þ a þ bt T for the credit payment and αcð Þ a þ bt (T–S) for the cash payment, respectively. Then, the retailer renders the loan for the cash payment at time T and pays off the loan for the credit payment at t = T + R. So, the present amount of the interest charged per cycle is

$$\begin{split} IC &= cI\_{\varepsilon} \left[ a \left( \int\_{S}^{T} (T-t)(a+bt)e^{-\hat{a}t}dt + (1-a) \left\{ \int\_{S}^{R} T(a+bt)e^{-\hat{a}t}dt \right. \right. \\ &\left. + \int\_{R}^{T+R} (T+R-t)(a+bt)e^{-\hat{a}t}dt \right) \right] \\ &= cI\_{\varepsilon} \left[ \left( \frac{e^{-i(R+S)} \left( -\epsilon^{\mathcal{S}}(b+ai+b\dot{a}R)T + \epsilon^{\mathcal{S}}(b+ai+b\dot{a}S)T \right)}{i^{2}} \right. \\ &\left. + \frac{e^{-i(R+T)} \left( ai+b(2+i(R+T)) + \epsilon^{\mathcal{T}}(a\dot{a}(-1+iT) + b(-2+i(T+R(-1+iT)))) \right)}{i^{3}} \right) (1-a) \\ &+ \underbrace{\left( \epsilon^{-\mathcal{T}}(ai+b(2+iT)) + \epsilon^{-\mathcal{S}}(ai(-1+i(-S+T)) + b(-2+i(T+S(-2+i(-S+T)))))) \right)}\_{i^{3}} \end{split} \tag{22}$$

### 4.4 Profit of the first sub-case of the second case of the model

Consequently, the present value of the retailer's annual total profit is

$$\begin{aligned} \textbf{PTP}\_4(T) &= \frac{\textbf{1}}{T}(\textbf{SR} - \textbf{PC} - \textbf{HC} - \textbf{OC} - \textbf{IC} + \textbf{IE}) \\ &= \frac{\textbf{1}}{T} \left\{ p \left[ a \int\_0^T (a + bt)e^{-it} dt + (1 - a) \int\_R^{T+R} (a + bt)e^{-it} dt \right] \right. \\ &\left. \begin{aligned} ^T & \int\_0^T (a + bt)e^{\delta(t) - iS} dt - h \int\_0^T (a + bt)e^{\delta(u) - \delta(t) - it} du dt - OC \\ 0 &= 0 \end{aligned} \right\} $$

$$\begin{aligned} &1 - cI\_{\varepsilon} \left[ a \left( \int\_{S}^{T} (T - t)(a + bt)e^{-it} dt + (1 - a) \left\{ \int\_{S}^{R} T(a + bt)e^{-it} dt \right. \right. \\ &+ \int\_{R}^{T + R} (T + R - t)(a + bt)e^{-it} dt \right] \Bigg) \\ &+ \int\_{R}^{T + R} (T + R - t)(a + bt)e^{-it} dt \Bigg] \Bigg( \begin{aligned} (23) \end{aligned} \tag{23}$$

¼ 1

� c ð T

<sup>T</sup> <sup>p</sup> <sup>α</sup>

0

þ pIe α

total relevant profit is

5. Algorithm

to Step 5.

by (16).

T∗

T∗

T∗

119

� ð Þ 1 � α cIc

2 4

8 < : 2 4 ð T

ð Þ a þ bt e

DOI: http://dx.doi.org/10.5772/intechopen.90689

ð R

8 < :

ð T

8 < :

0

problem. The algorithm is as follows:

Step 3: Compute all PTPj T<sup>∗</sup>

<sup>1</sup> <sup>¼</sup> <sup>S</sup>. Calculate PTP<sup>1</sup> <sup>T</sup> <sup>∗</sup>

If <sup>S</sup> � <sup>R</sup> <sup>≤</sup>T<sup>2</sup> <sup>≤</sup>S; we set <sup>T</sup><sup>∗</sup>

Step 5: Compute PTPj T<sup>∗</sup>

<sup>4</sup> <sup>¼</sup> <sup>S</sup>. Calculate PTP<sup>4</sup> <sup>T</sup><sup>∗</sup>

<sup>5</sup> <sup>¼</sup> <sup>S</sup>. Calculate PTP<sup>5</sup> <sup>T</sup> <sup>∗</sup>

Otherwise we set T<sup>∗</sup>

Step 1: Input all the parameters values.

j

1 � � by (13).

Step 4: Find the maximum among PTPj T <sup>∗</sup>

solution {<sup>T</sup> <sup>∗</sup> , PTP T<sup>∗</sup> ð Þ} accordingly, and then stop.

4 � � by (23). Step 5.2: Find the unique root T<sup>5</sup> in (34). If S≥T<sup>5</sup> we set T<sup>∗</sup>

> 5 � � by (26).

j

Step 3.1: Find the unique root T<sup>1</sup> in (30). If S≤T<sup>1</sup> we set T<sup>∗</sup>

Step 3.2: Find the unique root <sup>T</sup><sup>2</sup> in (31). If <sup>T</sup><sup>2</sup> <sup>≤</sup> <sup>S</sup> � <sup>R</sup>; we set <sup>T</sup><sup>∗</sup>

Step 3.3: Find the unique root <sup>T</sup><sup>3</sup> in (32). If <sup>T</sup><sup>3</sup> <sup>≤</sup><sup>S</sup> � <sup>R</sup>; we set <sup>T</sup><sup>∗</sup>

� �, for j = 4 and 5.

Step 5.1: Find the unique root T<sup>4</sup> in (33). If S≤T<sup>4</sup> we set T<sup>∗</sup>

<sup>3</sup> <sup>¼</sup> <sup>S</sup> � <sup>R</sup>. Calculate PTP<sup>3</sup> <sup>T</sup><sup>∗</sup>

S

t að Þ þ bt e

<sup>δ</sup>ð Þ�<sup>t</sup> iSdt � <sup>h</sup>

T að Þ þ bt e

�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

ð T ð T

t

�itdt <sup>þ</sup>

ð S

T

0

�itdt <sup>þ</sup>

T ð þR

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial…

R

ð Þ a þ bt e

T ð þR

R

T að Þ þ bt e

Combining (23) and (26), we know that the present value of the retailer's annual

PTP Tð Þ¼ PTP4ð Þ T , if S≤ T

It is clear that PTP(T) is continuous in T and has the following properties:

We developed the following algorithm to find the optimal solution of the

Step 2: Assimilate the values of R and S. If R ≤ S then go to Step 3, otherwise go

� �, for j = 1, 2, and 3.

<sup>2</sup> <sup>¼</sup> <sup>T</sup>2. If <sup>T</sup><sup>2</sup> <sup>≥</sup> <sup>S</sup>, ; we set <sup>T</sup> <sup>∗</sup>

3 � � by (18).

j

ð Þ a þ bt e

�itdt

<sup>δ</sup>ð Þ� <sup>u</sup> <sup>δ</sup>ð Þ�<sup>t</sup> itdudt � <sup>O</sup>

ð Þ T þ R � t ð Þ a þ bt e

9 = ;

3 5 9 = ;

PTP Tð Þ¼ PTP5ð Þ T , if S≤T (27)

PTP4ð Þ¼ S PTP5ð ÞS (28)

�itdt

�itdt

9 = ;

3 5

<sup>1</sup> ¼ T<sup>1</sup> else, we set

<sup>2</sup> ≤S � R.

<sup>3</sup> ≤T3.

<sup>4</sup> ¼ T<sup>4</sup> else we set

<sup>5</sup> ¼ T<sup>5</sup> else we set

2 � �

<sup>2</sup> <sup>¼</sup> <sup>S</sup>. Calculate PTP<sup>2</sup> <sup>T</sup><sup>∗</sup>

� � for j = 1, 2, and 3, set the optimal

(26)

3 5

0

ð Þ a þ bt e

We then discuss the last sub-case in which R ≥ S ≥ T.

## 4.4.1 Sub-case 2b: S ≥ T

Similarly, the present value of the interest earned from the cash payment per cycle is

$$\begin{split} IE &= pI\_{\epsilon} \left[ a \left\{ \int\_{0}^{T} t(a+bt)e^{-it}dt + \int\_{T}^{S} T(a+bt)e^{-it}dt \right\} \right] \\ &= p \left( \frac{e^{-i(S+T)} \left( -e^{iT}(b+ai+biS)T + e^{iS}T(b+ai+biT) \right)}{i^{2}} \\ &+ \frac{2b+ai+e^{-iT}(-ai(1+iT)-b(2+iT(2+iT)))}{i^{3}} \right) a i\_{\epsilon} \end{split} \tag{24}$$

For the cash payment, there is no interest to charge. On the other hand, the retailer must finance (1–α)cð Þ a þ bt T for the credit payment at time S and start paying off the loan from time R to T + R.

Hence, the present amount of the interest charged per cycle is

$$IC = (1 - a)cI\_{\epsilon} \left[ \left\{ \int\_{S}^{R} T(a + bt)e^{-it}dt + \int\_{R}^{T+R} (T + R - t)(a + bt)e^{-it}dt \right\} \right]$$

$$= c i\_{\epsilon} \left[ \left( \frac{e^{-i(R+S)} \left( -e^{iS}(b + ai + biR)T + e^{iR}(b + ai + biS)T \right)}{i^2} \right.$$

$$+ \frac{e^{-i(R+T)} \left( ai + b(2 + i(R+T)) + e^{iT}(ai(-1 + iT) + b(-2 + i(T + R(-1 + iT)))) \right)}{i^3} \right) (1 - a) \left. \tag{21}$$

$$+ \frac{(e^{-iT}(ai + b(2 + iT)) + e^{-iS}(ai(-1 + i(-S + T)) + b(-2 + i(T + S(-2 + i(-S + T))))))a}{i^3} \right]$$

#### 4.5 Profit of the second sub-case of the second case of the model

Consequently, the present value of the retailer's annual total profit is

$$\text{PTPP}\_5(T) = \frac{1}{T}(\text{SR} - \text{PC} - \text{HC} - \text{OC} - \text{IC} + \text{IE})^{\frac{1}{2}}$$

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial… DOI: http://dx.doi.org/10.5772/intechopen.90689

$$\begin{split} \mathcal{I} &= \frac{1}{T} \left\{ p \left[ a \int\_{0}^{T} (a+bt)e^{-it}dt + (1-a) \int\_{R}^{T+R} (a+bt)e^{-it}dt \right] \right. \\ &\left. \left. \begin{aligned} & \int\_{0}^{T} (a+bt)e^{\delta(t)-iS}dt - h \left[ \int\_{0}^{T} (a+bt)e^{\delta(u)-\delta(t)-it}dt dt - O \\ & \int\_{0}^{T} (a+bt)e^{-it}dt + \int\_{R}^{T+R} (T+R-t)(a+bt)e^{-it}dt \right] \right) \\ & \quad + pI\_{\varepsilon} \left\{ a \left[ \int\_{0}^{T} (t(a+bt)e^{-it}dt + \int\_{T}^{S} T(a+bt)e^{-it}dt \right] \right) \right\} \end{split} \tag{26}$$

Combining (23) and (26), we know that the present value of the retailer's annual total relevant profit is

$$\text{PTP}(T) = \text{PTP}\_4(T), \text{if } \mathbb{S} \le T$$

$$\text{PTP}(T) = \text{PTP}\_5(T), \text{if } \mathbb{S} \le T \tag{27}$$

It is clear that PTP(T) is continuous in T and has the following properties:

$$PTP\_4(\mathcal{S}) = PTP\_5(\mathcal{S}) \tag{28}$$

## 5. Algorithm

� cIc α

R

4.4.1 Sub-case 2b: S ≥ T

¼ p

þ

IC ¼ ð Þ 1 � α cIc

¼ cic

118

"

cycle is

þ T ð þR

2 4 ð T

0 @

ð Þ T � t ð Þ a þ bt e

Application of Decision Science in Business and Management

We then discuss the last sub-case in which R ≥ S ≥ T.

ð T

8 < :

2 4

t að Þ þ bt e

0

ð Þ T þ R � t ð Þ a þ bt e

IE ¼ pIe α

paying off the loan from time R to T + R.

ð R

8 < :

PTP5ð Þ¼ <sup>T</sup> <sup>1</sup>

2 4

S

�itdt <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup>

1 A 3

Similarly, the present value of the interest earned from the cash payment per

�itdt <sup>þ</sup>

<sup>e</sup>�i Sð Þ <sup>þ</sup><sup>T</sup> �eiTð Þ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biS <sup>T</sup> <sup>þ</sup> <sup>e</sup>iST bð Þ <sup>þ</sup> ai <sup>þ</sup> biT � � i 2

For the cash payment, there is no interest to charge. On the other hand, the retailer must finance (1–α)cð Þ a þ bt T for the credit payment at time S and start

�itdt <sup>þ</sup>

<sup>þ</sup> <sup>e</sup>�i Rð Þ <sup>þ</sup><sup>T</sup> ai <sup>þ</sup> <sup>b</sup>ð<sup>2</sup> <sup>þ</sup> i Rð Þ <sup>þ</sup> <sup>T</sup> Þ þ eiTð Þ aið Þþ �<sup>1</sup> <sup>þ</sup> iT <sup>b</sup>ð Þ �<sup>2</sup> <sup>þ</sup> i Tð Þ <sup>þ</sup> <sup>R</sup>ð Þ �<sup>1</sup> <sup>þ</sup> iT � � i 3

4.5 Profit of the second sub-case of the second case of the model

Consequently, the present value of the retailer's annual total profit is

<sup>þ</sup> <sup>e</sup>�iTðai <sup>þ</sup> <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iT Þ þ <sup>e</sup>�iSð Þ aið�<sup>1</sup> <sup>þ</sup> <sup>i</sup>ð Þ �<sup>S</sup> <sup>þ</sup> <sup>T</sup> Þ þ <sup>b</sup>ð Þ �<sup>2</sup> <sup>þ</sup> i Tð Þ <sup>þ</sup> <sup>S</sup>ð Þ �<sup>2</sup> <sup>þ</sup> <sup>i</sup>ð Þ �<sup>S</sup> <sup>þ</sup> <sup>T</sup> � �

i 3

<sup>T</sup> ð Þ SR � PC � HC � OC � IC <sup>þ</sup> IE

T ð þR

R

<sup>e</sup>�i Rð Þ <sup>þ</sup><sup>S</sup> �eiSð Þ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biR <sup>T</sup> <sup>þ</sup> eiRð Þ <sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> biS <sup>T</sup> � � i 2

<sup>2</sup><sup>b</sup> <sup>þ</sup> ai <sup>þ</sup> <sup>e</sup>�iTð Þ �aið Þ� <sup>1</sup> <sup>þ</sup> iT <sup>b</sup>ð Þ <sup>2</sup> <sup>þ</sup> iTð Þ <sup>2</sup> <sup>þ</sup> iT i 3

Hence, the present amount of the interest charged per cycle is

T að Þ þ bt e

ð S

T að Þ þ bt e

�itdt

� αie

ð Þ T þ R � t ð Þ a þ bt e

�itdt

9 = ;

3 5

!

ð Þ 1 � α

α

(25)

#

9 = ;

3 5

T

5 þ αpIe

�itdt 9 = ; ð R

8 < :

T að Þ þ bt e

ð Þ <sup>a</sup> <sup>þ</sup> bt te�itdt

�itdt

(23)

(24)

S

ð S

0

S

We developed the following algorithm to find the optimal solution of the problem. The algorithm is as follows:

Step 1: Input all the parameters values.

Step 2: Assimilate the values of R and S. If R ≤ S then go to Step 3, otherwise go to Step 5.

Step 3: Compute all PTPj T<sup>∗</sup> j � �, for j = 1, 2, and 3.

Step 3.1: Find the unique root T<sup>1</sup> in (30). If S≤T<sup>1</sup> we set T<sup>∗</sup> <sup>1</sup> ¼ T<sup>1</sup> else, we set T∗ <sup>1</sup> <sup>¼</sup> <sup>S</sup>. Calculate PTP<sup>1</sup> <sup>T</sup> <sup>∗</sup> 1 � � by (13).

Step 3.2: Find the unique root <sup>T</sup><sup>2</sup> in (31). If <sup>T</sup><sup>2</sup> <sup>≤</sup> <sup>S</sup> � <sup>R</sup>; we set <sup>T</sup><sup>∗</sup> <sup>2</sup> ≤S � R. If <sup>S</sup> � <sup>R</sup> <sup>≤</sup>T<sup>2</sup> <sup>≤</sup>S; we set <sup>T</sup><sup>∗</sup> <sup>2</sup> <sup>¼</sup> <sup>T</sup>2. If <sup>T</sup><sup>2</sup> <sup>≥</sup> <sup>S</sup>, ; we set <sup>T</sup> <sup>∗</sup> <sup>2</sup> <sup>¼</sup> <sup>S</sup>. Calculate PTP<sup>2</sup> <sup>T</sup><sup>∗</sup> 2 � � by (16).

Step 3.3: Find the unique root <sup>T</sup><sup>3</sup> in (32). If <sup>T</sup><sup>3</sup> <sup>≤</sup><sup>S</sup> � <sup>R</sup>; we set <sup>T</sup><sup>∗</sup> <sup>3</sup> ≤T3. Otherwise we set T<sup>∗</sup> <sup>3</sup> <sup>¼</sup> <sup>S</sup> � <sup>R</sup>. Calculate PTP<sup>3</sup> <sup>T</sup><sup>∗</sup> 3 � � by (18).

Step 4: Find the maximum among PTPj T <sup>∗</sup> j � � for j = 1, 2, and 3, set the optimal solution {<sup>T</sup> <sup>∗</sup> , PTP T<sup>∗</sup> ð Þ} accordingly, and then stop.

Step 5: Compute PTPj T<sup>∗</sup> j � �, for j = 4 and 5.

Step 5.1: Find the unique root T<sup>4</sup> in (33). If S≤T<sup>4</sup> we set T<sup>∗</sup> <sup>4</sup> ¼ T<sup>4</sup> else we set T∗ <sup>4</sup> <sup>¼</sup> <sup>S</sup>. Calculate PTP<sup>4</sup> <sup>T</sup><sup>∗</sup> 4 � � by (23).

Step 5.2: Find the unique root T<sup>5</sup> in (34). If S≥T<sup>5</sup> we set T<sup>∗</sup> <sup>5</sup> ¼ T<sup>5</sup> else we set T∗ <sup>5</sup> <sup>¼</sup> <sup>S</sup>. Calculate PTP<sup>5</sup> <sup>T</sup> <sup>∗</sup> 5 � � by (26).

Step 6: Find the maximum among PTPj T<sup>∗</sup> j for j = 4 and 5, Set the optimal solution {<sup>T</sup> <sup>∗</sup> , PTP T<sup>∗</sup> ð Þ} accordingly, and then stop.

By using the proposed algorithm, we obtain the optimal solutions of each case shown in Table 1. As a result, for S = 0.74, then the optimal solution to the problem

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial…

Example 2: Using the same data as those in Example 1 except R = 0.45 years again using the proposed algorithm, we obtain the optimal solutions for S = 0.18 and 0.44, respectively, each case shown in Table 2. Optimal solutions when θðÞ¼ t

<sup>22</sup>:21 units, T<sup>∗</sup> <sup>¼</sup> <sup>0</sup>:18 yrs <sup>¼</sup> 45 days and Total Profit <sup>∗</sup> <sup>¼</sup> <sup>31892</sup>:34. Considering all the sub-cases, we can conclude that the second sub-case of the first case is more

is <sup>Q</sup> <sup>∗</sup> <sup>¼</sup> <sup>21</sup>:<sup>91</sup> units, <sup>T</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:18, years <sup>¼</sup> <sup>25</sup> days, and PTP<sup>∗</sup> <sup>¼</sup> <sup>43119</sup>:31.

As a result, at S = 0.44, the optimal solution to the problem is Q <sup>∗</sup> <sup>¼</sup>

1

Figure 3.

Figure 4.

121

Graphically the results of sub-cases of the second case.

Graphical comparison of profit functions of both cases I and II.

ð Þ <sup>1</sup>þm�<sup>t</sup> and <sup>S</sup>≤<sup>R</sup> (Figure 3).

DOI: http://dx.doi.org/10.5772/intechopen.90689

## 6. Numerical examples

Example 1: Let us assume that <sup>θ</sup>ðÞ¼ <sup>t</sup> <sup>1</sup> ð Þ <sup>1</sup>þm�<sup>t</sup> , m = 1, <sup>α</sup> = 0:20, c = \$10 per unit, a = 50 units per year, b = 30 units per year, h = \$15 per unit per year = 0.4, Ie ¼ 0:4,Ic ¼ 0:5, p = \$10 per unit, R = 0.25, and T = 0.18. Optimal solutions when <sup>θ</sup>ðÞ¼ <sup>t</sup> <sup>1</sup> ð Þ <sup>1</sup>þm�<sup>t</sup> and <sup>S</sup>≥<sup>R</sup> (Figure 2).

Figure 2.

Graphically the results of sub-cases of the first case.


#### Table 1.

The optimal solution to the three sub-cases of case I.


#### Table 2.

The optimal solution to the two sub-cases of case II.

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial… DOI: http://dx.doi.org/10.5772/intechopen.90689

By using the proposed algorithm, we obtain the optimal solutions of each case shown in Table 1. As a result, for S = 0.74, then the optimal solution to the problem is <sup>Q</sup> <sup>∗</sup> <sup>¼</sup> <sup>21</sup>:<sup>91</sup> units, <sup>T</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:18, years <sup>¼</sup> <sup>25</sup> days, and PTP<sup>∗</sup> <sup>¼</sup> <sup>43119</sup>:31.

Example 2: Using the same data as those in Example 1 except R = 0.45 years again using the proposed algorithm, we obtain the optimal solutions for S = 0.18 and 0.44, respectively, each case shown in Table 2. Optimal solutions when θðÞ¼ t 1 ð Þ <sup>1</sup>þm�<sup>t</sup> and <sup>S</sup>≤<sup>R</sup> (Figure 3).

As a result, at S = 0.44, the optimal solution to the problem is Q <sup>∗</sup> <sup>¼</sup> <sup>22</sup>:21 units, T<sup>∗</sup> <sup>¼</sup> <sup>0</sup>:18 yrs <sup>¼</sup> 45 days and Total Profit <sup>∗</sup> <sup>¼</sup> <sup>31892</sup>:34. Considering all the sub-cases, we can conclude that the second sub-case of the first case is more

Figure 3. Graphically the results of sub-cases of the second case.

Figure 4. Graphical comparison of profit functions of both cases I and II.

Step 6: Find the maximum among PTPj T<sup>∗</sup>

Application of Decision Science in Business and Management

solution {<sup>T</sup> <sup>∗</sup> , PTP T<sup>∗</sup> ð Þ} accordingly, and then stop.

Example 1: Let us assume that <sup>θ</sup>ðÞ¼ <sup>t</sup> <sup>1</sup>

ð Þ <sup>1</sup>þm�<sup>t</sup> and <sup>S</sup>≥<sup>R</sup> (Figure 2).

Graphically the results of sub-cases of the first case.

The optimal solution to the three sub-cases of case I.

The optimal solution to the two sub-cases of case II.

Case II Q <sup>∗</sup>

Case I Q <sup>∗</sup>

6. Numerical examples

<sup>θ</sup>ðÞ¼ <sup>t</sup> <sup>1</sup>

Figure 2.

Table 1.

Table 2.

120

j 

a = 50 units per year, b = 30 units per year, h = \$15 per unit per year = 0.4, Ie ¼ 0:4,Ic ¼ 0:5, p = \$10 per unit, R = 0.25, and T = 0.18. Optimal solutions when

<sup>i</sup> T <sup>∗</sup>

1a(1) 21.91 0.18 42861.95 1b(2) 21.91 0.18 43119.31 1c(3) 21.91 0.18 35794.41

<sup>i</sup> T <sup>∗</sup>

2a(4) 22.21 0.18 30994.26 2b(5) 22.21 0.18 31892.34

for j = 4 and 5, Set the optimal

ð Þ <sup>1</sup>þm�<sup>t</sup> , m = 1, <sup>α</sup> = 0:20, c = \$10 per unit,

<sup>i</sup> PTP<sup>∗</sup>

<sup>i</sup> PTP<sup>∗</sup>

i

i

profitable for the retailer. Consequently, the background and the conditions to take the best policy of that model are better than irrespective of all other cases (Figure 4).

References

245-256

36(4):335-338

48(8):826-833

852-861

3273-3285

123

[1] Seifert D, Seifert RW, Protopappa-Sieke M. A review of trade credit literature: Opportunity for research in operations. European Journal of Operational Research. 2013;231(2):

DOI: http://dx.doi.org/10.5772/intechopen.90689

[9] Teng JT, Min J, Pan Q. Economic order quantity model with trade credit financing for non-decreasing demand.

[10] Chern MS, Chan YL, Teng JT, Goyal SK. Nash equilibrium solution in a vendor-buyer supply chain model with

permissible delay in payments.

[11] Chern MS, Pan Q, Teng JT,

payments. International Journal of Production Economics. 2013;144(1):

[12] Chen SC, Cárdenas-Barrón LE, Teng JT. Retailer's economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity. International Journal of Production Economics. 2014a;155:284-291

[13] Chen SC, Teng JT, Skouri K. Economic production quantity models for deteriorating items with up-stream full trade credit and down-stream partial trade credits. International Journal of Production Economics.

[14] Liao JJ, Huang KN, Ting PS. Optimal strategy of deteriorating items with capacity constraints under two-levels of

Mathematics and Computation. 2014;

[15] Wu J, Al-khateeb FB, Teng J-T, Cárdenas-Barrón LE. Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis. International Journal of Production Economics. 2016a;171:105-115

2014b;155:302-309

233:647-658

trade credit policy. Applied

with permissible delay in

2014;70(1):116-123

397-404

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial…

Computers and Industrial Engineering.

Chan YL, Chen SC. Stackelberg solution in a vendor-buyer supply chain model

Omega. 2012;40(3):328-335

[2] Goyal SK. Economic order quantity under conditions of permissible delay in

[3] Aggarwal SP, Jaggi CK. Ordering policies of deteriorating items under permissible delay in payments. The Journal of the Operational Research

[4] Jamal AMM, Sarker BR, Wang S. An ordering policy for deteriorating items with allowable shortage and permissible delay in payment. The Journal of the Operational Research Society. 1997;

[5] Teng JT. On the economic order quantity under conditions of permissible delay in payments. The Journal of the Operational Research

Society. 2002;53(8):915-918

[6] Huang YF. Optimal retailer's ordering policies in the EOQ model under trade credit financing. The Journal of the Operational Research Society. 2003;54(9):1011-1015

[7] Liao JJ. An EOQ model with noninstantaneous receipt and exponentially deteriorating items under two-level trade credit. International Journal of Production Economics. 2008;113(2):

[8] Min J, Zhou YW, Zhao J. An

inventory model for deteriorating items under stock-dependent demand and two-level trade credit. Applied

Mathematical Modelling. 2010;34(11):

payments. The Journal of the Operational Research Society. 1985;

Society. 1995;46(5):658-662

## 7. Conclusions and future research

Usually, the retailer uses a downstream partial trade credit as a strategy to reduced fault risks with credit-risk customers. The deterioration rate is time dependent and near cent percent near expiration date. We have investigated an EOQ model for deteriorating items in a general framework that Goyal [2], Teng [5, 36], Huang [6], Teng and Goyal [37], Chen and Teng [24], Wu and Chan [38], and Wu et al. [15] did special cases. In addition, to reflect the effects of inflation and time value of money, a discounted cash flow analysis has been adopted to obtain the present value of the total profit for time-dependent demand. By applying the existing theoretical results in concave functions, we have demonstrated the proper algorithm to find the optimal solution for possible alternatives. Then we have used the two most commonly used deterioration rates to run several numerical simulations. With increase in the purchasing cost, the holding cost, or the interest rate reduces the order quantity, the cycle time and the annual total profit is sensitive. In contrast, along product maximum life-span elevates the order quantity, the cycle time, and annual total profit is also sensitive. In addition, the total profit is very sensitive to the selling price. Consequently, to prolong product life-span and increase profit, retailers must negotiate with suppliers for low purchase cost and invest in preservation technology (such as refrigeration).

If time approaches to the expiration date, it may be profitable to have a closeout sale at a markdown price. One may extend the model from zero-ending inventory to nonzero-ending inventory. Furthermore, a researcher might consider allowing for shortages, allow partial backlogging, and allow for failure, scrap, and rework. Finally, the proposed model with a single player can be extended to an integrated cooperative model for both the retailer and the customer.

## Author details

Nirmal Kumar Duari\*, Sorforaj Nowaj and Jobin George Varghese Techno India University, Kolkata, West Bengal, India

\*Address all correspondence to: abnu1985@gmail.com

© 2020 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.

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial… DOI: http://dx.doi.org/10.5772/intechopen.90689

## References

profitable for the retailer. Consequently, the background and the conditions to take the best policy of that model are better than irrespective of all other cases

Usually, the retailer uses a downstream partial trade credit as a strategy to reduced fault risks with credit-risk customers. The deterioration rate is time dependent and near cent percent near expiration date. We have investigated an EOQ model for deteriorating items in a general framework that Goyal [2], Teng [5, 36], Huang [6], Teng and Goyal [37], Chen and Teng [24], Wu and Chan [38], and Wu et al. [15] did special cases. In addition, to reflect the effects of inflation and time value of money, a discounted cash flow analysis has been adopted to obtain the present value of the total profit for time-dependent demand. By applying the existing theoretical results in concave functions, we have demonstrated the proper algorithm to find the optimal solution for possible alternatives. Then we have used the two most commonly used deterioration rates to run several numerical simulations. With increase in the purchasing cost, the holding cost, or the interest rate reduces the order quantity, the cycle time and the annual total profit is sensitive. In contrast, along product maximum life-span elevates the order quantity, the cycle time, and annual total profit is also sensitive. In addition, the total profit is very sensitive to the selling price. Consequently, to prolong product life-span and increase profit, retailers must negotiate with suppliers for low purchase cost and

If time approaches to the expiration date, it may be profitable to have a closeout sale at a markdown price. One may extend the model from zero-ending inventory to nonzero-ending inventory. Furthermore, a researcher might consider allowing for shortages, allow partial backlogging, and allow for failure, scrap, and rework. Finally, the proposed model with a single player can be extended to an integrated

(Figure 4).

Author details

122

7. Conclusions and future research

Application of Decision Science in Business and Management

invest in preservation technology (such as refrigeration).

cooperative model for both the retailer and the customer.

Nirmal Kumar Duari\*, Sorforaj Nowaj and Jobin George Varghese

© 2020 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,

Techno India University, Kolkata, West Bengal, India

\*Address all correspondence to: abnu1985@gmail.com

provided the original work is properly cited.

[1] Seifert D, Seifert RW, Protopappa-Sieke M. A review of trade credit literature: Opportunity for research in operations. European Journal of Operational Research. 2013;231(2): 245-256

[2] Goyal SK. Economic order quantity under conditions of permissible delay in payments. The Journal of the Operational Research Society. 1985; 36(4):335-338

[3] Aggarwal SP, Jaggi CK. Ordering policies of deteriorating items under permissible delay in payments. The Journal of the Operational Research Society. 1995;46(5):658-662

[4] Jamal AMM, Sarker BR, Wang S. An ordering policy for deteriorating items with allowable shortage and permissible delay in payment. The Journal of the Operational Research Society. 1997; 48(8):826-833

[5] Teng JT. On the economic order quantity under conditions of permissible delay in payments. The Journal of the Operational Research Society. 2002;53(8):915-918

[6] Huang YF. Optimal retailer's ordering policies in the EOQ model under trade credit financing. The Journal of the Operational Research Society. 2003;54(9):1011-1015

[7] Liao JJ. An EOQ model with noninstantaneous receipt and exponentially deteriorating items under two-level trade credit. International Journal of Production Economics. 2008;113(2): 852-861

[8] Min J, Zhou YW, Zhao J. An inventory model for deteriorating items under stock-dependent demand and two-level trade credit. Applied Mathematical Modelling. 2010;34(11): 3273-3285

[9] Teng JT, Min J, Pan Q. Economic order quantity model with trade credit financing for non-decreasing demand. Omega. 2012;40(3):328-335

[10] Chern MS, Chan YL, Teng JT, Goyal SK. Nash equilibrium solution in a vendor-buyer supply chain model with permissible delay in payments. Computers and Industrial Engineering. 2014;70(1):116-123

[11] Chern MS, Pan Q, Teng JT, Chan YL, Chen SC. Stackelberg solution in a vendor-buyer supply chain model with permissible delay in payments. International Journal of Production Economics. 2013;144(1): 397-404

[12] Chen SC, Cárdenas-Barrón LE, Teng JT. Retailer's economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity. International Journal of Production Economics. 2014a;155:284-291

[13] Chen SC, Teng JT, Skouri K. Economic production quantity models for deteriorating items with up-stream full trade credit and down-stream partial trade credits. International Journal of Production Economics. 2014b;155:302-309

[14] Liao JJ, Huang KN, Ting PS. Optimal strategy of deteriorating items with capacity constraints under two-levels of trade credit policy. Applied Mathematics and Computation. 2014; 233:647-658

[15] Wu J, Al-khateeb FB, Teng J-T, Cárdenas-Barrón LE. Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis. International Journal of Production Economics. 2016a;171:105-115

[16] Ghare PM, Schrader GP. A model for an exponentially decaying inventory. Journal of Industrial Engineering. 1963;14(5):238-243

[17] Covert RB, Philip GS. An EOQ model with Weibull distribution deterioration. AIIE Transactions. 1973; 5(4):323-326

[18] Dave U, Patel LK. (T, Si) policy inventory model for deteriorating items with time proportional demand. The Journal of the Operational Research Society. 1981;32(2):137-142

[19] Sachan RS. On (T, Si) policy inventory model for deteriorating items with time proportional demand. The Journal of the Operational Research Society. 1984;35(11):1013-1019

[20] Hariga MA. Optimal EOQ models for deteriorating items with timevarying demand. The Journal of the Operational Research Society. 1996; 47(10):1228-1246

[21] Teng JT, Chern MS, Yang HL, Wang YJ. Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand. Operations Research Letters. 1999;24 (1–2):65-72

[22] Teng JT, Chang HJ, Dye CY, Hung CH. An optimal replenishment policy for deteriorating items with timevarying demand and partial backlogging. Operations Research Letters. 2002;30(6):387-393

[23] Dye CY. The effect of preservation technology investment on a noninstantaneous deteriorating inventory model. Omega. 2013;41(5):872-880

[24] Chen SC, Teng JT. Retailer's optimal ordering policy for deteriorating items with maximum life time under supplier's trade credit financing. Applied Mathematical Modelling. 2014; 38(15–16):4049-4061

[25] Sarkar B. An EOQ model with delay in payments and time varying deterioration rate. Mathematical and Computer Modelling. 2012;55(3–4): 367-377

Computers and Industrial Engineering.

DOI: http://dx.doi.org/10.5772/intechopen.90689

Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial…

Niaki STA, Alikar N. Optimizing multiitem multi-period inventory control system with discounted cash flow and inflation: Two calibrated metaheuristic algorithms. Applied

Mathematical Modelling. 2013;37(4):

[34] Chen SC, Teng JT. Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis. European Journal of Operational Research. 2015;

[35] Duari NK, Chakraborti T. A cash

[36] Teng JT. Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers. International Journal of Production Economics. 2009;119(2):

[37] Teng JT, Goyal SK. Optimal ordering policies for a retailer in a supply chain with up-stream and downstream trade credits. The Journal of the Operational Research Society. 2007;

[38] Wu J, Chan YL. Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to creditrisk customers. International Journal of Production Economics. 2014;155:

flow-oriented EOQ model of deteriorating items with increasing demand rate and shortages under permissible delay in payment. International Journal of Operational Research. 2015;24(2):145-160

2010;59(4):770-777

2241-2256

243(2):566-575

415-423

58(9):1252-1255

292-301

125

[33] Mousavi SM, Hajipour V,

[26] Wang WC, Teng JT, Lou KR. Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum life time. European Journal of Operational Research. 2014; 232(2):315-321

[27] Wu J, Ouyang LY, Cárdenas-Barrón LE, Goyal SK. Optimal credit period and lot size for deteriorating items with expiration dates under twolevel trade credit financing. European Journal of Operational Research. 2014; 237(3):898-908

[28] Sarkar B, Saren S, Cárdenas-Barrón LE. An inventory model with trade-credit policy and variable deterioration for fixed life time products. Annals of Operations Research. 2015;229(1):677-702

[29] Hill RM, Pakkala TPM. A discounted cash flow approach to the base stock inventory model. International Journal of Production Economics. 2005;93-94:439-445

[30] Chung KJ, Liao JJ. The optimal ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity. International Journal of Production Economics. 2006;100(1):116-130

[31] Dye CY, Ouyang LH, Hsieh TP. Inventory and pricing strategies for deteriorating items with shortages: A discounted cash flow approach. Computers and Industrial Engineering. 2007;52(1):29-40

[32] Chang CT, Ouyang LY, Teng JT, Cheng MC. Optimal ordering policies for deteriorating items using a discounted cash-flow analysis when a trade credit is linked to order quantity. Inventory Policies for Deteriorating Items with Maximum Lifetime under Downstream Partial… DOI: http://dx.doi.org/10.5772/intechopen.90689

Computers and Industrial Engineering. 2010;59(4):770-777

[16] Ghare PM, Schrader GP. A model

Application of Decision Science in Business and Management

[25] Sarkar B. An EOQ model with delay

[26] Wang WC, Teng JT, Lou KR. Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum life time. European Journal of Operational Research. 2014;

[27] Wu J, Ouyang LY, Cárdenas-Barrón LE, Goyal SK. Optimal credit period and lot size for deteriorating items with expiration dates under twolevel trade credit financing. European Journal of Operational Research. 2014;

[28] Sarkar B, Saren S, Cárdenas-Barrón LE. An inventory model with trade-credit policy and variable deterioration for fixed life time products. Annals of Operations Research. 2015;229(1):677-702

[29] Hill RM, Pakkala TPM. A

base stock inventory model. International Journal of Production Economics. 2005;93-94:439-445

discounted cash flow approach to the

[30] Chung KJ, Liao JJ. The optimal ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity. International Journal of Production Economics. 2006;100(1):116-130

[31] Dye CY, Ouyang LH, Hsieh TP. Inventory and pricing strategies for deteriorating items with shortages: A discounted cash flow approach.

Computers and Industrial Engineering.

[32] Chang CT, Ouyang LY, Teng JT, Cheng MC. Optimal ordering policies for deteriorating items using a discounted cash-flow analysis when a trade credit is linked to order quantity.

2007;52(1):29-40

in payments and time varying deterioration rate. Mathematical and Computer Modelling. 2012;55(3–4):

367-377

232(2):315-321

237(3):898-908

[17] Covert RB, Philip GS. An EOQ model with Weibull distribution deterioration. AIIE Transactions. 1973;

[18] Dave U, Patel LK. (T, Si) policy inventory model for deteriorating items with time proportional demand. The Journal of the Operational Research

Society. 1981;32(2):137-142

47(10):1228-1246

(1–2):65-72

[19] Sachan RS. On (T, Si) policy inventory model for deteriorating items with time proportional demand. The Journal of the Operational Research Society. 1984;35(11):1013-1019

[20] Hariga MA. Optimal EOQ models for deteriorating items with timevarying demand. The Journal of the Operational Research Society. 1996;

[21] Teng JT, Chern MS, Yang HL, Wang YJ. Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand. Operations Research Letters. 1999;24

[22] Teng JT, Chang HJ, Dye CY, Hung CH. An optimal replenishment policy for deteriorating items with time-

[23] Dye CY. The effect of preservation technology investment on a noninstantaneous deteriorating inventory model. Omega. 2013;41(5):872-880

[24] Chen SC, Teng JT. Retailer's optimal ordering policy for deteriorating items

Applied Mathematical Modelling. 2014;

with maximum life time under supplier's trade credit financing.

38(15–16):4049-4061

124

varying demand and partial backlogging. Operations Research Letters. 2002;30(6):387-393

for an exponentially decaying inventory. Journal of Industrial Engineering. 1963;14(5):238-243

5(4):323-326

[33] Mousavi SM, Hajipour V, Niaki STA, Alikar N. Optimizing multiitem multi-period inventory control system with discounted cash flow and inflation: Two calibrated metaheuristic algorithms. Applied Mathematical Modelling. 2013;37(4): 2241-2256

[34] Chen SC, Teng JT. Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis. European Journal of Operational Research. 2015; 243(2):566-575

[35] Duari NK, Chakraborti T. A cash flow-oriented EOQ model of deteriorating items with increasing demand rate and shortages under permissible delay in payment. International Journal of Operational Research. 2015;24(2):145-160

[36] Teng JT. Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers. International Journal of Production Economics. 2009;119(2): 415-423

[37] Teng JT, Goyal SK. Optimal ordering policies for a retailer in a supply chain with up-stream and downstream trade credits. The Journal of the Operational Research Society. 2007; 58(9):1252-1255

[38] Wu J, Chan YL. Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to creditrisk customers. International Journal of Production Economics. 2014;155: 292-301

Chapter 8

Analysis

Andrea Lippi

Abstract

differently.

1. Introduction

127

private investors, retail investors

The Role of Wealth in Gain and

People with significantly different initial starting capitals may perceive gains and losses differently. In order to test this hypothesis, we consider and compare two samples of investors: retail investors as those with a maximum of €500,000 worth of assets under management (AUM) and private investors as those with more than €500,000 AUM. Based on the answers obtained from specifically devised questionnaires, we test the differences in gain and loss perception and check the level of satisfaction/dissatisfaction in situations of gain and loss. The results obtained demonstrate that private and retail investors perceive gains and losses

A plethora of experiments (e.g., [1, 2]) demonstrate that decisions made in an economic and financial setting are influenced by subjective perceptions. The framing effect [3] is a perceptual phenomenon implying that different presentations of the same information may lead to different choices. Chen et al. [4], Del Vecchio et al. [5], Gourville [6], Levin et al. [3], McKechnie et al. [7], Sinha and Smith [8], Tombu and Mandel [9] and Tversky and Kahneman [10] have investigated how the framing effect could influence the decision-making process. DelVecchio [11], DelVecchio et al. [5], Gourville [6], Kahneman [12, 13], McKechnie et al. [7] and Mellers [14] have examined the incoherence of judgement when faced with similar or indifferent situations. Kahneman and Tversky [15], Kühberger [16] and Olsen [17, 18] demonstrate that the framing effect can influence the decision-making process so as to cause a shift from 'risk-adverse' to 'risk-seeking' and vice versa, the so-called risky-choice framing effect. The framing effect is predicted by Kahneman and Tversky [15] in their prospect theory. According to this theory, individuals' choices are always made considering the gains and losses compared with an initial starting capital (reference point). Kahneman and Tversky [19] argue that investors decide by mentally referring to their status quo (i.e. the current level of wellbeing). In any situation in which it risks being altered, the decision-making procedure is adjusted [10, 20] in order to preserve it as far as possible [15]. According to

Kahneman and Tversky [15, 19], the absolute value perceived of losses appears to be

Keywords: perception of gain, perception of loss, decision-making,

Loss Perception: An Empirical

## Chapter 8
