**6. Phase "control"**

Due to intense competition, variable economic and environmental conditions, changing wage rates and fluctuating oil prices, the control phase of the DMAIC methodology will be focused on establishing the changes and standardizing the results given in the previous phases. Consequently, sensitivity analysis is chosen to assess the effect of changes in demand and variations in subcontracting and transportation costs on the supply chain cost. Three parameters are examined in this sensitivity analysis: demand, transportation cost and subcontracting cost to assess their impact on planning decisions and supply chain performance.

During our experimentation, fifteen scenarios are considered by varying (1) the demand, (2) the cost of transport and (3) the cost of subcontracting between 50% to +50% of their current values. By considering the three scenarios (α = 100%, monthly fixed α, monthly varying α), the cost of the supply chain is calculated for each case.

#### **6.1 Sensitivity analysis of demand**

A 50% increase in demand leads to an increase in supply chain costs, as explained in **Table 4**. Nevertheless, if we consider a RPC, a decrease in this total cost is recorded. For all considered scenarios, the best cost is obtained when a monthly variable RPC is considered.


**Table 4.** *Cost variation according to demand*

*Enhancement of Textile Supply Chain Performance through Optimal Capacity Planning DOI: http://dx.doi.org/10.5772/intechopen.96292*

These results confirm the importance of our approach and encourage the idea of using RPC to reduce supply chain costs. We note a saving of 4% compared to current practice when demand is reduced by half. This proves that the use of a monthly variable RPC yields better results. If demand is increased by half, the gain is 11%. The proposed approach becomes essential when demand is relatively high. If demand is low, the internal capacity will accommodate the demand without any additional costs. As a result, the RPC becomes less important and will avoid situations of under capacity due to urgent orders arriving at the operational level.

#### **6.2 Sensitivity analysis of transportation and subcontracting costs**

Identical trends are observed in transport and subcontracting costs savings are also obtained when we consider a RPC at the tactical planning level (**Figure 10**). It is worth noting that the greatest savings are achieved when considering a monthly variable RPC. Moreover, savings become more important with increases in these two costs. Lower transportation costs lead to the outsourcing of some internal production to overseas subcontractor's manufacturing unit, as the latter offers very competitive prices, especially for most basic products. Subsequently, at the tactical planning level, some internal production capacity is unused; therefore, enforcing a RPC is meaningless. For this reason, the lowest savings are observed when transport costs are halved. Nevertheless, outsourcing abroad is no longer the preferred option when transport costs increase. This promotes the use of a RPC to avoid the use of full production capacity at the tactical level. Internal production (regular and overtime) and locally subcontracting are the adequate options to cover capacity requirements.

#### **Figure 10.**

*Transportation cost variation.*

When the cost of subcontracting is reduced by half, this activity is more profitable than the internal production. In this case, part of the internal production is manufactured at local subcontractors. In addition, the under-utilization cost prevents the full transfer of quantities to local subcontractors' plants. Internal production capacity is currently under-utilized; however, this situation results in lower supply chain costs due to lower production costs. In this case, the consideration of a RPC is no longer significant.

#### *Lean Manufacturing*

Nevertheless, increased supply chain costs are noted when the cost of subcontracting increases, especially for α = 100% (current situation). In practice, subcontracting is prevented until internal production capacity is completely used; thus leading to local subcontracting at higher costs (**Figure 11**). In these situations, reserve capacity appears to be a substantial consideration in ensuring the use of foreign subcontracting (which is cheaper than local subcontracting) at the tactical level. If the cost of subcontracting is increased by half, taking into account a RPC that varies on a monthly basis makes it possible to achieve a cost savings of 11% regards current practice.

#### **Figure 11.**

*Subcontracting cost variation.*

The sensitivity analysis confirms the interest of our approach taking into account a RPC. Indeed, when demand or transportation or subcontracting costs increase, our approach allows us to adequately place urgent and unpredictable orders that arrive at the operational level at the lowest cost.

This approach provides a decision tool for textile and apparel manufacturers who are constantly faced with two types of orders: long lead time orders dedicated for the next season and urgent and unpredictable orders that are related to the current season. Moreover, this approach is applied to any type of industry where there are two types of customers: - premium customers with short lead time orders, classic customers with long lead time orders. Indeed, through flexibility and responsiveness to needs, our approach will be able to place orders in the right location and at the lowest cost, taking into consideration a RPC and a rolling horizon. This will guarantee customer satisfaction that will gain in competitiveness, on both cost and lead time aspects, in today's highly competitive environment.

Furthermore, taking into account the type of information introduced, the performance of the supply chain can be improved. The more reliable this information is, the better the performance. Indeed, it is important for the customer to share his sales information with the producer so that the latter can prepare in advance, using adequate forecasting methods, to best accommodate these orders. In this case, the estimate of the RPC can be adjusted for more reliability and flexibility in order to forecast future orders that may come in. This is one of the perspectives and ideas to be explored for this work.

*Enhancement of Textile Supply Chain Performance through Optimal Capacity Planning DOI: http://dx.doi.org/10.5772/intechopen.96292*

## **7. Conclusions**

In this chapter, the DMAIC methodology was chosen and applied to perform a complicated problem of a textile company. Our aim is to satisfy customer needs at lower cost while ensuring prompt and punctual deliveries. To achieve this, a sequential approach integrating tactical and operational decisions for textile and apparel supply chain planning has been implemented with an emphasis on the flexibility provided by the consideration of RPC at the tactical level. As a result, newly arrived urgent orders, with short lead times, can be placed optimally at production sites, via the rolling horizon.

During the definition phase of the DMAIC methodology, we have defined the problem statement and presented the proposed planning approach. Then, we established the SIPOC diagram in order to identify the different steps of our approach which ensures flexibility of production and distribution activities' planning considering textile and apparel sector specificities: fashion effect, demand fluctuations.

In the first step, we have detailed our measurement system analysis by introducing the two mathematical models used to evaluate the performance of the current situation of the apparel company, taking into account the full available production capacity. Next, we presented our data collection plan by describing the experimental data that were collected. Finally, we outlined the desired situation taking into account additional flexibility at the tactical level.

In the "analysis" phase, we presented the obtained results when assessing the current situation by detailing production assignments over different locations. We also performed an extended analysis using the Ishikawa diagram and the 5P tool in order to underline the interest of our approach.

During the "improve" phase, we outlined the Improvements achieved in the current situation. To do so, we started by testing different RPC values in order to identify the optimal one to be taken into account at the tactical level. Then, we evaluated the performance of our approach by considering a fixed RPC then a monthly variable one. Finally, we evaluated the efficiency of our approach to optimally respond to urgent orders arriving at the operational level. Our approach is evaluated over a six-month planning horizon, but it remains applicable over longer planning horizons.

"Control" phase is devoted to sensitivity analysis while studying the effect of some parameters' variation on the cost of the supply chain. The three considered parameters are: demand, transport cost and subcontracting cost. The main focus of this section is to prove the interest of our approach to place, at the lowest cost, urgent orders that arrive at the operational level, even when demand and cost increase. For a better performance of the considered supply chain, the importance of cooperation between the manufacturer and the retailers, based on information sharing, was also emphasized.

#### **Appendix A. The tactical planning model**

In model formulation, we consider the following sets and indices, parameters, and decision variables.

Sets and indices:

K: set of manufacturing units k ∈K; K = U ∪V.

U: set of internal manufacturing units, k ∈ U.

V: set of subcontractors' manufacturing units, k ∈ V.

I: set of retailers, i∈ I.

J: set of warehouses, j ∈ J.

P: set of products, p ∈P.

L: set of transportation modes, L = {trucks, ships, aircraft}, l ∈L.

T: set of periods included in the planning horizon, t ∈ [1 .. |T|]. Parameters:

In the tactical model, the parameter (Dpit) expresses the need of retailer i, in product p, to serve during period t. During period t, the quantities to be delivered are manufactured in production sites k characterized by a limited capacity (Ukt). Production incurs variable and fixed production costs per product per period (Cpkt, Spkt) or subcontracting costs (Gpkt). A monthly cost of under-utilization of internal production capacity (CSUkt) is also considered to penalize the non-utilization of available internal resources. Each product is characterized by a production lead time (Tpp) and a product unit volume (Vp). The quantities manufactured are then transported to warehouses and incur inventory holding costs (KPpjt). The warehouses are characterized by a limited storage capacity (Wj). The delivery lead time is noted by (el). Each means of transport has limited capacity (Capl). Variable and fixed distribution costs from sites to warehouses (CTkjplt, CFkjplt) and from warehouses to retailers (CSjiplt, CFSjiplt) are also addressed.

Decision variables:

Z1kjplt: quantity of product p to deliver, via transportation mode l, from manufacturing unit k to warehouse j over period t,

Z2jiplt: quantity of product p to deliver, via transportation mode l, from warehouse j to retailer i over period t,

Xpkt: produced quantity, of product p, in manufacturing unit k over period t. SUkt: unused production capacity at internal manufacturing unit k over period t. Jpjt: inventory level of product p in warehouse j at the end of period t.

Ypkt =1 if product p is produced in manufacturing unit k over period t; 0 otherwise. N1kjlt: transported quantity from manufacturing unit k to warehouse j over period t by use of transportation mode l.

N2jilt: transported quantity from warehouse j to retailer i over period t by use of transportation mode l.

Model formulation (M1)

The tactical production–distribution planning model is formulated as an ILP that aims at minimizing the overall cost in the considered supply chain network.

$$\begin{split} & \text{Min} \left( \sum\_{\iota \in \mathcal{T}} \sum\_{\tau \in \mathcal{P}} \sum\_{\substack{\kappa \in U \\ k \in U}} \mathsf{C\_{pkl}} \sum\_{\iota \in \mathcal{U}} \sum\_{\tau \in \mathcal{T}} \sum\_{\substack{\kappa \in \mathcal{P}}} \sum\_{\substack{\lambda \in U}} \sum\_{k \in U} \mathsf{S\_{pkl}} Y\_{pk} \\ & + \sum\_{k \in U} \sum\_{\iota \in \mathcal{T}} \sum\_{\ell \in \mathcal{K}} \sum\_{k \in K} \sum\_{\substack{\eta \in \mathcal{I}}} \sum\_{p \in \mathcal{P}} \text{KP}\_{pj\bullet} \left( V\_{p\mathrm{j}l-1} + J\_{p\mathrm{j}l} \right) /\_{2} \\ & + \sum\_{\iota \in T} \sum\_{p \in \mathcal{D}} \sum\_{k \in K} \sum\_{\ell \in \mathcal{K}} \sum\_{\substack{\eta \in \mathcal{I}}} \sum\_{\substack{\eta \in \mathcal{I}}} \text{CF}\_{kj\mathrm{pl}} \ast \mathbf{V}\_{p} \ast \mathbf{Z} 1\_{\hat{k} \text{pl}} + \sum\_{\iota \in T} \sum\_{p} \sum\_{k \in K} \sum\_{\ell \in \mathcal{L}} \sum\_{\substack{\zeta \in \mathcal{I} \\ \eta \in \mathcal{I}}} \sum\_{\substack{\eta \in \mathcal{I}}} \sum\_{\substack{\eta \in \mathcal{I}}} \sum\_{\substack{\eta \in \mathcal{I}}} \sum\_{\substack{\eta \in \mathcal{I}}} \sum\_{\substack{\zeta \in \mathcal{I}}} \sum\_{\substack{\zeta \in \mathcal{I}}} \sum\_{\substack{\zeta \in \mathcal{I} \\ \zeta \in \mathcal{I}}} \sum\_{\substack{\eta \$$

Subject to

$$J\_{\rm pjt} = J\_{\rm pjt-1} + \sum\_{l \in L} \sum\_{k \in K} \mathbf{Z1}\_{k \rm pjlt-e\_l} - \sum\_{l \in L} \sum\_{k \in K} \mathbf{Z2}\_{k \rm pjlt}, \; j \in I; p \in P; t \in T \text{ and } t \ge e\_l \tag{1}$$

$$\sum\_{p \in P} J\_{p\not\ni} \le W\_j, \ j \in I \; and \; t \in T \tag{2}$$

$$\sum\_{p \in P} T p\_p \ast X\_{PKT} \le a\_{kt} \ast V\_{kt}, k \in K \text{ and } t \in T \tag{3}$$

*Enhancement of Textile Supply Chain Performance through Optimal Capacity Planning DOI: http://dx.doi.org/10.5772/intechopen.96292*

$$X\_{pkt} \le M \ast Y\_{pkt}, k \in K, p \in Pandt \in T \tag{4}$$

$$Y\_{pkt} \le X\_{pkt}, k \in V, p \in \text{Pand} t \in T \tag{5}$$

$$SV\_{kt} \ge a\_{kt} \* V\_{kt} - \sum\_{p \in P} T p\_p \* X\_{pkt}, k \in \text{Kand} t \in T \tag{6}$$

$$X\_{pkt} = \sum\_{j \in J} \sum\_{l \in L} Z \mathbf{1}\_{kjlt}, k \in Kandt \in T \tag{7}$$

$$D\_{\rm jet} = \sum\_{j \in I} \sum\_{l \in L} Z \mathbf{2}\_{j \rm jplt - e\_l}, i \in I, p \in P, t \in T, t \ge e\_l \tag{8}$$

$$\sum\_{p} V\_{p} \* Z2\_{jplt} \le \mathcal{N} \mathcal{Z}\_{jilt} \* \mathcal{C}ap\_{l}, j \in \mathcal{I}, i \in \mathcal{I}, l \in \mathcal{L}and t \in \mathcal{T} \tag{9}$$

$$\sum\_{p} V\_{p} \* Z \mathbf{1}\_{k \text{jilt}} \le N \mathbf{1}\_{k \text{jilt}} \* \mathbf{C} \mathbf{a} p\_{l}, j \in \mathcal{J}, k \in \mathcal{K}, l \in \mathcal{L} and t \in T \tag{10}$$

$$Y\_{pkt} \in \{0, 1\} \\ k \in K, p \in P and t \in T \tag{11}$$

$$\begin{aligned} Z\mathbf{1}\_{kjplt} \in \mathbb{N}, Z\mathbf{2}\_{jplt} \in \mathbb{N}, X\_{pkt} \in \mathbb{N}, J\_{pjt} \in \mathbb{N}, N\mathbf{1}\_{kjplt} \in \mathbb{N}, \\ N\mathbf{2}\_{jplt} \in \mathbb{N}, S\mathbf{V}\_{kt} \in \mathbb{N} \quad k \in K, j \in J, p \in P, t \in T, l \in \mathbf{L}and i \in I \end{aligned} \tag{12}$$

$$N2\_{j
\mid j
\mid t} \in \mathbb{N}, SV\_{kt} \in \mathbb{N} \quad k \in K, j \in J, p \in P, t \in T, l \in Land i \in I$$

The objective function aims at minimizing the total cost composed of set-up cost, variable production cost, subcontracting cost, internal capacity underutilization cost, inventory holding cost, transportation costs from manufacturing units to warehouses and transportation cost from warehouses to retailers. Transportation costs are composed of variable and fixed costs. The first, depends on quantity to deliver using transportation mode. While the second is proportional to the number of shipments.

The constraints (1) determine the stock level of product p in warehouse j at the end of period t. Constraints (2) guarantee that over each period, the total stored quantity is limited by the storage capacity. Constraints (3) ensure that the produced quantities do not exceed production capacity. Constraints (4) and (5) establish the relationship between binary and integer variables. Constraints (6) with the objective function identify the underutilized internal production capacity. Constraints (7) guarantee the delivery of all produced quantities to warehouses. Constraints (8) ensure that delivered products from warehouses to retailers meet on time demand. Constraints (9) and (10) guarantee the respect of transportation capacity. Constraints (11) and (12) are integrity constraints.

#### **B. The set of periods in the operational planning model**

The set of periods in the operational planning model used at week δ of month Ө is presented at the table below. For example, to construct an operational planning at the beginning of the second week (δ = 2) of month Ө, the periods involved are (Ө,2), (Ө,3), (Ө,4), (Ө + 1,1), (Ө + 1,2), (Ө + 1,3), (Ө + 1,4), (Ө + 2,1), (Ө + 2,2), (Ө + 2,3), and (Ө + 2,4) and they are listed in the third column of table below (TSӨ2).

Set of periods in the operational planning model used at week δ of month Ө.



### **C. The operational planning model**

The same predetermined parameters of the tactical model are maintained for the operational planning model except few adjustments. Since the tactical and operational models consider different periods, w has been added here to the parameters and decision variables to indicate that they are related to a one-week period. The operational planning model determines the weekly quantities to be produced, stored and delivered (t,s) ∈ TSӨδ. It is worth knowing that the production plans obtained from the tactical model, for month t such as (t,s) ∈ TSӨδ, represent inputs to be considered at the operational level and must be weekly detailed.

In addition to the notation introduced in the tactical planning model, we consider the following two parameters and two decision variables related to overtime: Parameters:

*UHw*: overtime production capacity in internal manufacturing unit k ∈ U at week s of month t with (t,s) in TSӨδ.

*CHwpkts*: overtime production cost in internal manufacturing unit k ∈ U at week s of month t with (t,s) in TSӨδ.

Decision variables

*XHwpkts*: quantity of product p produced during overtime in internal manufacturing unit k ∈ U at week s of month t with (t,s) in TSӨδ.

*YHwpkts* ¼ 1 if there is production of p during overtime in internal manufacturing unit k at week s of month t; 0 otherwise with (t,s) in TSӨδ.

Model formulation (M2)

The main objective is to minimize the overall cost composed of: -weekly production cost, � weekly set-up cost during regular working hours and overtime, � weekly subcontracting cost, � weekly internal production capacity underutilization cost, � weekly holding inventory cost, � weekly variable and fixed transportation costs from manufacturing units to warehouses and from warehouses to retailers.

Min X ð Þ *t;s* ∈*TSθδ* X *p*∈*P* X *k*∈*V CwpktsXwpkts* <sup>þ</sup> <sup>X</sup> ð Þ *t;s* ∈*TSθδ* X *p*∈*P* X *k*∈*V Swpkts Ywpkts* <sup>þ</sup> *YHwpkts* � � <sup>0</sup> @ <sup>þ</sup> <sup>X</sup> ð Þ *t;s* ∈*TSθδ* X *p*∈*P* X *k* ∈*V GwpktsXwpkts* <sup>þ</sup> <sup>X</sup> ð Þ *t;s* ∈*TSθδ* X *p*∈*P* X *k*∈*V CHwpktsXHpkts* <sup>þ</sup> <sup>X</sup> ð Þ *t;s* ∈*TSθδ* X *kϵV CSVwktsSVwkts* <sup>þ</sup><sup>X</sup> *j* ∈*J* X ð Þ *t;s* ∈*TSθδ* X *p*∈*P KPwpjts Jwpjts*�<sup>1</sup> <sup>þ</sup> *Jwpjts* � �*=*<sup>2</sup> <sup>þ</sup> <sup>X</sup> ð Þ *t;s* ∈*TSθδ* X *p*∈*P* X *k* ∈*K* X *l*∈*l* X *j* ∈*J CTwkjpltsVpZ*1*wkjplts* <sup>þ</sup> <sup>X</sup> ð Þ *t;s* ∈*TSθδ* X *p*∈*P* X *i* ∈*I* X *l*∈*l* X *j* ∈*J CSwjipltsVpZ*2*wjiplts* <sup>þ</sup> <sup>X</sup> ð Þ *t;s* ∈*TSθδ* X *p*∈*P* X *l* ∈*L* X *k* ∈*K* X *j* ∈*J CFwkjpltsN*1*wkjplts* <sup>þ</sup> <sup>X</sup> ð Þ *t;s* ∈*TSθδ* X *l* ∈*L* X *p* ∈*P* X *i* ∈*I* X *j* ∈*J CFSwkjpltsN*2*wkjplts* 1 A

#### *Enhancement of Textile Supply Chain Performance through Optimal Capacity Planning DOI: http://dx.doi.org/10.5772/intechopen.96292*

Constraints (1), (2), and (8)–(12) of the tactical model are also included at the operational level while introducing weekly parameters and decision variables. They ensure the balance of production flows, the respect of storage capacity, the satisfaction of retailer demand, the respect of transportation capacity [(9) and (10)], and guarantee the integrity of the decision variables [(11) and (12)]. Constraints (3)–(7) are changed to incorporate full production capacity and overtime as follows:

$$\sum\_{p \in P} T p\_p \ast X H w\_{p\&s} \le V H w\_{k\&s}, \mathfrak{a} \varepsilon \mathcal{U}; (\mathfrak{e}, \mathfrak{s}) \varepsilon \text{ TS}\_{\partial \mathbb{S}} \tag{13}$$

$$\sum\_{p \in P} T p\_p \ast X w\_{pkts} \le V w\_{kts}, \text{def } \mathcal{K}; (\mathfrak{t}, \mathfrak{s}) \varepsilon \text{ } T \mathbb{S}\_{\Theta \delta} \tag{14}$$

$$\text{LXH}w\_{\text{pkts}} \leq \text{M} \ast \left( \text{YH}w\_{\text{pkts}} + \text{Y}w\_{\text{pkts}} \right), \text{ \textit{et } \mathcal{U}; \mu \epsilon \mathcal{P}; (\text{\textit{t}}, \mathfrak{s}) \epsilon \text{ } \mathcal{T} \mathbb{S}\_{\Theta} \delta \tag{15}$$

$$\text{L\ $}\,\text{Xw}\_{\text{pkts}} \leq \text{M} \,\ast \left( \text{YH}w\_{\text{pkts}} + \text{Yw}\_{\text{pkts}} \right), \text{ \textit{4c} } \mathcal{K}; \text{\textit{2c}} \,\prime \mathcal{P}; \text{(\textit{4},\bullet} \,\text{)} \epsilon \text{ T\$ }\,\text{8} \tag{16}$$

$$\text{YH}w\_{\text{pkts}} + \text{Yw}\_{\text{pkts}} \le \mathbf{1}, \text{ és } \mathcal{U}; \text{\(\mu \gets \mathcal{P}; (\mathsf{t}, \mathsf{s})\) \in \mathsf{TS}\_{\Theta}\mathsf{S} \tag{17}$$

$$\text{YH}w\_{pkts} \le \text{XH}w\_{pkts}, \text{ 4e } \mathcal{U}; \text{4e } \mathcal{P}; (\mathfrak{t}, \mathfrak{s}) \text{e } \text{TS}\_{\Theta}\delta \tag{18}$$

$$\text{Yw}\_{pkts} \le \text{Xw}\_{pkts}, \text{ 4e } \mathbb{K}; \text{\(\mu\)e } \mathcal{P}; (\mathfrak{t}, \mathfrak{s}) \epsilon \text{ \(\mathcal{T}\)}\_{\mathcal{O}\mathcal{S}} \tag{19}$$

$$\text{SV}w\_{\text{kts}} \ge \text{V}w\_{\text{kts}} - \sum\_{p \in P} \text{Tp}\_p \ast \text{X}w\_{\text{pkts}}, \mathfrak{a} \varepsilon \mathcal{U}; \ (\mathfrak{e}, \mathfrak{s}) \varepsilon \text{ \textdegree \textdegree \textdegree \textdegree \dots} \tag{20}$$

$$\mathbf{X}\mathbf{H}\mathbf{w}\_{\text{pkts}} + \mathbf{X}\mathbf{w}\_{\text{pkts}} = \sum\_{l \in L} \sum\_{j \in J} \mathbf{Z} \mathbf{1} w\_{\text{kips}} \text{ lts}, \text{ } \text{\(\epsilon\)} \text{ } \text{\(\epsilon\)} \text{\(\epsilon\)} \text{\(\epsilon\)} \text{\(\epsilon\)} \text{\(\epsilon\)} \text{\(\epsilon\)}$$

Constraints (13) and (14) guarantee the respect of the production capacity in regular working hours and on overtime. Constraints (15), (17) and (18) ensure that the cost of overtime is only taken into account if the same products are not previously produced. Constraints (16) and (19) establish the relationship between binary and integer variables. Constraints (20) with the objective function set the underutilized internal production capacity. Constraints (21) ensure that all production quantities are delivered to the warehouses.

Constraints (22)–(26) are also considered at the operational model:

$$\sum\_{(\mathbf{t},\mathbf{t})\in T\mathbb{S}\_{\theta\delta}/\delta\geq\mathbf{1}} \mathbf{X}w\_{pk\mathbf{t}\mathbf{t}}=\mathbf{X}\_{pk\mathbf{t}}-\sum\_{\mathbf{t}=\mathbf{1}}^{\delta-1} \mathbf{X}w\_{pk\mathbf{t}\mathbf{t}},\ \text{de }\mathbb{K};\mathfrak{a}\ \mathbf{e}\ \mathcal{P};\mathfrak{a}=\Theta\tag{22}$$

$$\sum\_{(\mathfrak{t},\mathfrak{t}) \in T\mathbb{S}\_{\mathfrak{t}\mathfrak{t}}/\delta \ge 1} \mathcal{X}w\_{p\mathfrak{t}\mathfrak{t}} = \mathcal{X}\_{p\mathfrak{k}\mathfrak{t}+1}, \text{ \textit{a}\varepsilon}\,\mathcal{K}; \mathfrak{a}\varepsilon\,\mathcal{P}; \mathfrak{t} = \mathcal{O} + \mathbbm{1} \tag{23}$$

$$\sum\_{(\mathfrak{t},\mathfrak{s})\in T\mathbb{S}\_{\mathfrak{st}}/\delta\geq 1} \mathcal{X}w\_{p\mathsf{k}\mathsf{t}\mathsf{s}} = \mathcal{X}\_{p\mathsf{k}\theta+2}, \text{ \textit{a}\mathsf{c}} \ \mathcal{K}; \mathfrak{a}\mathsf{e}\ \mathcal{P}; \mathfrak{t} = \Theta + 2\tag{24}$$

$$\text{YH}w\_{pkts} \in \{0, 1\}, \text{ 4e } \mathcal{K}\_{\text{;}} \text{ } \mu \text{e } \mathcal{P}\_{\text{;}} \text{ (}\mathfrak{t}, \mathfrak{s}\text{)} \epsilon \text{ } \text{TS}\_{\Theta\\$} \tag{25}$$

$$\text{YH}w\_{pk\mathfrak{k}} \in \mathbb{N}, \text{ 4e } \mathbb{K}, \text{ } \mathfrak{a} \in \mathcal{P}; \text{ } (\mathfrak{t}, \mathfrak{s}) \text{ } \mathsf{T}\mathbb{S}\_{\Theta} \tag{26}$$

Constraints (22), (23) and (24) guarantee coherence with the tactical decisions made. Finally, constraints (25) and (26) ensure the integrity of the new decision variables.

*Lean Manufacturing*
