# Horizontal Cooperation in Subcontracting: A Case Study in Home Textile Industry

Authors: Derya Eren Akyol; Ozlem Sarısakal
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

Home textile is the most common business line in Denizli, Turkey. However, most companies are not integrated, i.e., they do not include the entire production processes in their facilities. Different stages of manufacturing process are outsourced to other companies located in the same geographic region. As a result of this, there is a considerable increase in transportation costs. In this paper, we consider collaboration between companies in a cluster to reduce their transportation costs. A game theoretic approach is proposed such that the subcontractors can be used jointly and their fabric production capacity can be split among the companies in the coalition in order to meet their demands. A case study of four home textile companies is provided to illustrate the approach. Coalition cost is allocated using different cost allocation methods, and compared to the current situation, we obtain significant cost reductions.


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## Body

## 1. Introduction

The most important source of livelihood in Denizli is home textile that includes products ranging from bed linens to towels. Although there are enterprises of all sorts of sizes operating in the region, most of them are not integrated companies, i.e., a majority of factories can perform only some parts of production processes. For instance, some companies are experts on weaving and they do only weaving process, and these companies outsource other manufacturing processes such as dyeing, confection, and packing processes to other companies according to their expertise in other processes. For this reason, regional concentration (clustering) is still keeping its importance in Denizli.Textile clusters are geographic concentrations of interconnected companies, specialized suppliers, and institutions in textile and apparel sector that are located near each other and provide productive advantage from their mutual connections. In other words, each company benefits from being concentrated in one area with companies functioning in the same industry. Another main strength of textile companies in Denizli is to be able to respond to demand easily using other companies’ (subcontractors’) capacity in peak seasons. Each company has its own subcontractors and outsources the required processes to these particular companies that are specialized in specific tasks. On the other hand, due to the increase in the number of tours required to perform outsourced processes, transportation costs increase enormously. However, the companies in the cluster do not cooperate on reducing their transportation costs which form the biggest portion of their total costs. They need to engage in a dialogue with other companies on obtaining joint solutions to their problems. One way to reduce transportation cost is through cooperation among the companies. The cooperation can be vertical, i.e., different branches of business can cooperate (e.g., a dye house and a weaving facility), or it can be horizontal, i.e., companies in the same business line can cooperate (e.g., between two weaving facilities).In this paper, we propose a game theory-based approach in a textile cluster where a coalition of the companies cooperates to jointly use their subcontractors. They still keep the same vehicle fleet in contrast to other studies in the literature which focus on the joint use of vehicles. In the proposed coalition case, the problem converts to a multidepot vehicle routing problem which is NP-hard. As we assume that the subcontractor’s capacity can be divided among the companies in the coalition, a clustering algorithm is proposed which takes into account the divisible capacities. In the proposed approach, the “cluster first, route second” method is used for the solution of the problem as long as the subcontractors have enough capacity to meet the demands of the companies. The subcontractors are assigned based on their distance to each company, and then the single-depot heterogeneous fleet vehicle routing problem (SDHFVRP) is solved for each company. When the customer demand is not met, some regulations are made about allocation of subcontractors and a subcontractor selection procedure is proposed so that some subcontractors are used jointly by the companies. The selection continues until the customers in the coalition meet their requirements. We then allocate the coalitional costs using different game theory-based methods such as the Shapley value, nucleolus, and weighted relative savings method (WRSM), and our numerical experiments confirm the potential of collaboration.The focus of this paper is horizontal cooperation among the weaving facilities. In the literature, there are many examples of horizontal cooperation in different sectors, such as grocery delivery [1], rural area distribution [2], furniture industry [3, 4], freight carriers [5], forest transportation [6], rail industry [7], and city distribution [8]. In most of the studies in the literature, the vehicles of the cooperating companies are used jointly. Different from other horizontal cooperation studies in the literature, we study cooperation between the textile companies, in an attempt to jointly use their subcontractors. Our objective is to develop a theoretical collaborative framework that determines the suitable subcontractors for the companies in textile clusters by ensuring that their demand requirements are met at the minimum cost. As far as we know, this study is the first collaboration work in home textile sector. Such an approach has not yet been applied in textile industry. Another notable side of this study is that the fabric production capacity of the subcontractors can be split among the companies in the coalition in order to meet their demands.The contribution of this paper is threefold. Firstly, we develop a model that may help to reduce transportation costs of the companies in the textile industry clustered in a particular field. We introduce a collaborative framework and illustrate that it may improve the current situation. Secondly, different from the existing collaboration works in the form of sharing vehicles, the proposed collaboration model considers that the companies can subcontract the providers that other companies work with. They keep the same vehicle fleet. In clustered regions, the suppliers (subcontractors) share similar characteristics. For example, in textile clusters, one firm’s fabric supplier is also capable of producing the other firm’s fabric. Therefore, transportation costs can easily be reduced in clustered sectors by outsourcing to closest suppliers from a pool of suppliers that can produce the same fabric and meet the demand requirements. For this reason, a selection mechanism and a heuristic are proposed to support this decision-making process and obtain the coalitional costs. The insights obtained from the model can help to come up with a solution that addresses the outsourcing policies of the companies in the region. Finally, a real-life case study is presented and it is seen that significant cost reductions can be achieved through cooperation, and the average savings from the proposed coalitional model are reported.A case study comprising four companies is presented to illustrate the implementation of the approach and the results. The individual transportation costs are determined by solving the single-depot heterogeneous fleet vehicle routing problem (SDHFVRP). After that, the coalitional costs are obtained via solving the multidepot heterogeneous fleet vehicle routing problem (MDHFVRP) by the proposed procedure. From the solution of the coalition models, we see that it is possible to obtain average cost savings of 33%.The remainder of the paper is organized as follows. The relevant literature is reviewed in Section2. Section 3 presents the individual and coalitional conditions within the proposed methodology. We introduce a case study in Section 4. The costs obtained from establishing the coalition are allocated among participants, and the numerical results of the individual and coalitional states are also compared in this section. Finally, results of the case study and conclusions are provided in Section 5.


## 2. Literature Review

In the last decade, due to the fierce competition in transport sector, there has been an extensive effort in improving the efficiency of distribution operations to meet growing customer satisfactions regarding the quality and level of service [9]. One of the strategies that companies follow is to collaborate with other companies. The common ways of cooperation are vertical cooperation and horizontal cooperation. In the vertical cooperation, firms of different stages of the value chain work together, whereas horizontal cooperation occurs at the same level of the value chain.In this paper, we introduce a game theory-based collaborative approach where a coalition of companies in textile industry cooperates to jointly use their subcontractors. This type of cooperation between the companies can be regarded as an example of horizontal cooperation. According to Cruijssen et al. [10], horizontal cooperation is determining and making use of mutually profitable situations between two or more companies at the same stage of the supply chain in order to enhance performance.Recently, collaborative practices have attracted the attention of companies and researchers, and several studies have been conducted in transportation to show the potential benefits of cooperation. However, research related to collaborative transportation is scarce. Cruijssen et al. [10] published a detailed overview of earlier studies. A survey was conducted by Cruijssen et al. [11] among the logistics service providers in Flanders to determine both the opportunities and limitations for horizontal cooperation. Cruijssen et al. [11] mentioned that finding an acceptable allocation mechanism for the attained benefits is a problem in cooperations. In an effort to fairly distribute costs arising from the coalition, in the literature, cooperative game theory methods have been used.Krajewska et al. [5] considered collaboration among freight carriers, and the Shapley value was used for profit sharing. They showed that a substantial decrease in transportation costs can be obtained. Cruijssen et al. [12] employed a procedure called insinking in which a logistics service provider determines the shippers from a group of shippers with a high synergy capability. The synergy benefits are allocated to the shippers through cooperative game theory. Ballot and Fontane [13] showed that collaboration provides not only cost savings and productivity but also reduces the negative environmental impacts of transportation.Fair cost allocation is an important issue, and Audy et al. [4] presented a new fair cost allocation which is called the equal profit method. Another interesting application was done by Frisk et al. [6] who focused on collaborative planning in forest transportation, and savings were distributed among the participants by different methods including the Shapley value, nucleolus, and equal profit method.Yılmaz and Savasaneril [14] studied the coalition among shippers in uncertain demand condition. Lozano et al. [15] proposed a mathematical model to calculate the savings that companies may obtain when they combine their transportation requirements. They used cooperative game theory to allocate savings. Vanovermeire and Sörensen [16] integrated a cost allocation method in the operational planning problem. Vanovermeire and Sörensen [17] proposed a cost allocation method for horizontal logistics alliances. Mixed-integer linear programming models were formulated by Guajardo and Rönnqvist [18] to determine coalition frameworks and cost allocation. Kimms and Kozeletskyi [19] worked on a multiobjective transportation problem and presented an allocation approach to enable the joint objectives of players. Liu and Cheng [20] proposed a cost allocation method and focused on horizontal cooperation among freight carriers. Guajardo and Ronnqvist [21] provided an extended survey on different methods of cost allocation in collaborative transportation. For a detailed review of recent literature, see [22, 23].Although several horizontal cooperation studies exist in the literature, the potential of horizontal cooperation in textile clusters has not yet been studied. The proposed type of collaboration has the potential to facilitate the development of a subcontractor selection pool which might assist the companies to work with suppliers that fulfill their demand requirements at the least cost. In this research, we address this gap by proposing a framework that enables companies to work with other companies’ subcontractors rather than their own. For this reason, we propose a subcontractor selection mechanism and a heuristic in order to calculate the coalitional costs of the companies in the coalition. We focus on horizontal cooperation in home textile sector. In Denizli, the presence of regional concentration obliges horizontal cooperation among companies. This study differs from the others with the following aspects: subcontractors are used jointly and their production capacity can be consumed by more than one company. A clustering solution procedure is proposed which takes into account the divisible capacities.


## 3. Methodology

The first step involves the data collection for the problem to be analyzed. The non-cooperative model is solved after data collection. In the non-cooperative case, each company works with its own subcontractors and the problem is SDHFVRP. In order to calculate each company’s individual transportation costs, SDHFVRPs are solved for each company. In the cooperative case, we assume that subcontractors can be used jointly, i.e., they can serve more than one company; therefore, the problem converts to an MDHFVRP where the companies pick their fabrics from the subcontractors back to their companies. The problem is NP-hard, and a procedure is proposed to solve the proposed coalition structure. As a final step, the coalitional costs (savings) are allocated using different methods from the literature including the Shapley value, nucleolus, and weighted relative savings method.


### 3.1. Individual Cost Calculation

Each company has several subcontractors andSi denotes the subcontractor i, where i = 1, …, n, and n is total number of subcontractors. The companies are denoted by Cj, where j = A, B, C, D. After the completion of the process outsourced, the company goes to the subcontractor’s factory to pick the fabrics and carries back all the fabrics to its own factory. Therefore, companies pick up the fabrics from subcontractors and put them in their own factories (like depot). For each company considered, SDHFVRP is solved and the individual transportation costs are obtained.Ji and Chen [24] considered postal delivery problem and formulated an integer linear programming model to collect mails from different post offices in Hong Kong. In this study, the problem can also be considered as a mail collection. The only difference is that the fabrics are collected instead of mails. So, the model proposed by Ji and Chen [24] can be applied to this work with minor changes. The mathematical model is given below.


#### 3.1.1. Parameters

i, j: subcontractors and company point indexv: company vehicle indexn: number of customersN: number of vehiclesdij: travelling distance of arc (i, j)kv: the capacity of vehicle vci: demand at point iav: the cost of vehicle v (TL/km)


#### 3.1.2. Decision Variables

xijv: 1 if vehicle k travels from i to j, and 0 otherwiseyij: 1 if some vehicles travel on arc (i, j), and 0 otherwise(1) Objective Function(1)MinimizeZ=∑v=1N∑i=0n∑j=0ndij∗xijv∗av.(2) Constraints(2)∑v=1Nxijv=yij∀i,j=1,…,n,(3)∑j=1nyij=1∀i=1…n,(4)∑i=1nyij=1∀j=1…n,(5)∑j=1ny0j≤N,(6)∑i=1nyi0≤N,(7)∑i=1n∑j=0ncixijv≤kv∀v=1,…,n,S⊂V0S≥2,(8)∑i∈Si=1n∑j∈Sj=1nxijv≤S−1∀v=1,…,n,(9)∑j=0,j≠inxijv=∑j=0,j≠inxjiv∀i=1,…,n;v=1,…,N,(10)xijv∈0,1∀i,j=0,…,n;v=1,…,N,(11)xijv∈0,1∀i,j=0,…,n.In the model proposed by Ji and Chen [24], the total distance travelled is tried to be minimized. However, in the model above, our objective is to minimize total transportation costs. Vehicle paths are defined by constraint (2). Constraint (3) implies that exactly one vehicle departs from a specified post office. Constraint (4) states that a particular post office is visited by a vehicle from the preceding one once. Constraint (5) enforces all vehicles to initiate from a single depot. Constraint (6) ensures that all the vehicles go back to the depot. Constraint (7) is the vehicle capacity limitation. Constraint (8) eliminates subtours. Constraint (9) ensures route continuity. Constraints (10) and (11) are binary constraints.


### 3.2. Coalitional Cost Calculation

In the proposed coalition case, the problem converts to an MDHFVRP which is NP-hard. Each vehicle originates from one depot, picks up from the subcontractors assigned to that depot, and comes back to the same depot. In other words, there are several companies (depots) that pick up their own fabrics from the subcontractors (customers) back to the companies. Since we assume that the subcontractor’s capacity can be divided among the companies in the coalition, a clustering algorithm is proposed which takes into account the divisible capacities. The “cluster first, route second” method is used for the solution of the problem as long as the subcontractors have enough capacity to meet the demands of the companies. The subcontractors are assigned based on their distance to each company, and then the single-depot heterogeneous fleet vehicle routing problem is solved for each company. When the customer demand is not met, some regulations are made about allocation of subcontractors and a subcontractor selection procedure is proposed so that some subcontractors are used jointly by the companies. The selection continues until the customers in the coalition meet their requirements. The summary of the proposed cluster first, route second algorithm is given in Figure1.Figure 1
The summary of the proposed cluster first, route second algorithm.


## 3.1. Individual Cost Calculation

Each company has several subcontractors andSi denotes the subcontractor i, where i = 1, …, n, and n is total number of subcontractors. The companies are denoted by Cj, where j = A, B, C, D. After the completion of the process outsourced, the company goes to the subcontractor’s factory to pick the fabrics and carries back all the fabrics to its own factory. Therefore, companies pick up the fabrics from subcontractors and put them in their own factories (like depot). For each company considered, SDHFVRP is solved and the individual transportation costs are obtained.Ji and Chen [24] considered postal delivery problem and formulated an integer linear programming model to collect mails from different post offices in Hong Kong. In this study, the problem can also be considered as a mail collection. The only difference is that the fabrics are collected instead of mails. So, the model proposed by Ji and Chen [24] can be applied to this work with minor changes. The mathematical model is given below.


### 3.1.1. Parameters

i, j: subcontractors and company point indexv: company vehicle indexn: number of customersN: number of vehiclesdij: travelling distance of arc (i, j)kv: the capacity of vehicle vci: demand at point iav: the cost of vehicle v (TL/km)


### 3.1.2. Decision Variables

xijv: 1 if vehicle k travels from i to j, and 0 otherwiseyij: 1 if some vehicles travel on arc (i, j), and 0 otherwise(1) Objective Function(1)MinimizeZ=∑v=1N∑i=0n∑j=0ndij∗xijv∗av.(2) Constraints(2)∑v=1Nxijv=yij∀i,j=1,…,n,(3)∑j=1nyij=1∀i=1…n,(4)∑i=1nyij=1∀j=1…n,(5)∑j=1ny0j≤N,(6)∑i=1nyi0≤N,(7)∑i=1n∑j=0ncixijv≤kv∀v=1,…,n,S⊂V0S≥2,(8)∑i∈Si=1n∑j∈Sj=1nxijv≤S−1∀v=1,…,n,(9)∑j=0,j≠inxijv=∑j=0,j≠inxjiv∀i=1,…,n;v=1,…,N,(10)xijv∈0,1∀i,j=0,…,n;v=1,…,N,(11)xijv∈0,1∀i,j=0,…,n.In the model proposed by Ji and Chen [24], the total distance travelled is tried to be minimized. However, in the model above, our objective is to minimize total transportation costs. Vehicle paths are defined by constraint (2). Constraint (3) implies that exactly one vehicle departs from a specified post office. Constraint (4) states that a particular post office is visited by a vehicle from the preceding one once. Constraint (5) enforces all vehicles to initiate from a single depot. Constraint (6) ensures that all the vehicles go back to the depot. Constraint (7) is the vehicle capacity limitation. Constraint (8) eliminates subtours. Constraint (9) ensures route continuity. Constraints (10) and (11) are binary constraints.


## 3.1.1. Parameters

i, j: subcontractors and company point indexv: company vehicle indexn: number of customersN: number of vehiclesdij: travelling distance of arc (i, j)kv: the capacity of vehicle vci: demand at point iav: the cost of vehicle v (TL/km)


## 3.1.2. Decision Variables

xijv: 1 if vehicle k travels from i to j, and 0 otherwiseyij: 1 if some vehicles travel on arc (i, j), and 0 otherwise(1) Objective Function(1)MinimizeZ=∑v=1N∑i=0n∑j=0ndij∗xijv∗av.(2) Constraints(2)∑v=1Nxijv=yij∀i,j=1,…,n,(3)∑j=1nyij=1∀i=1…n,(4)∑i=1nyij=1∀j=1…n,(5)∑j=1ny0j≤N,(6)∑i=1nyi0≤N,(7)∑i=1n∑j=0ncixijv≤kv∀v=1,…,n,S⊂V0S≥2,(8)∑i∈Si=1n∑j∈Sj=1nxijv≤S−1∀v=1,…,n,(9)∑j=0,j≠inxijv=∑j=0,j≠inxjiv∀i=1,…,n;v=1,…,N,(10)xijv∈0,1∀i,j=0,…,n;v=1,…,N,(11)xijv∈0,1∀i,j=0,…,n.In the model proposed by Ji and Chen [24], the total distance travelled is tried to be minimized. However, in the model above, our objective is to minimize total transportation costs. Vehicle paths are defined by constraint (2). Constraint (3) implies that exactly one vehicle departs from a specified post office. Constraint (4) states that a particular post office is visited by a vehicle from the preceding one once. Constraint (5) enforces all vehicles to initiate from a single depot. Constraint (6) ensures that all the vehicles go back to the depot. Constraint (7) is the vehicle capacity limitation. Constraint (8) eliminates subtours. Constraint (9) ensures route continuity. Constraints (10) and (11) are binary constraints.


## 3.2. Coalitional Cost Calculation

In the proposed coalition case, the problem converts to an MDHFVRP which is NP-hard. Each vehicle originates from one depot, picks up from the subcontractors assigned to that depot, and comes back to the same depot. In other words, there are several companies (depots) that pick up their own fabrics from the subcontractors (customers) back to the companies. Since we assume that the subcontractor’s capacity can be divided among the companies in the coalition, a clustering algorithm is proposed which takes into account the divisible capacities. The “cluster first, route second” method is used for the solution of the problem as long as the subcontractors have enough capacity to meet the demands of the companies. The subcontractors are assigned based on their distance to each company, and then the single-depot heterogeneous fleet vehicle routing problem is solved for each company. When the customer demand is not met, some regulations are made about allocation of subcontractors and a subcontractor selection procedure is proposed so that some subcontractors are used jointly by the companies. The selection continues until the customers in the coalition meet their requirements. The summary of the proposed cluster first, route second algorithm is given in Figure1.Figure 1
The summary of the proposed cluster first, route second algorithm.


## 4. Case Study

In this study, we consider a special branch of home textile sector, i.e., towel and bathrobe production. Four textile companies are chosen based on having similar contextual conditions. These companies operate in the same region, i.e., Denizli, have customers with similar purchasing power, and have similar marketing activities.In Denizli, most textile companies focus on their core businesses; for example, some companies perform only “sizing and weaving process” and some companies are only responsible for “dyeing process.” To produce a towel or a bathrobe, several processes such as warping and sizing, weaving, and dyeing are necessary. If a company cannot perform a process, it has to be outsourced to a company which is an expert in that process. After the completion of the process, semifinished goods are carried back to the company which results in high transportation costs.Especially during the peak season such as summer, generally, weaving capacity is insufficient to meet the requirements. In this situation, the company outsources some orders to subcontractors in order to fulfill the demand requirements. The task of the subcontractor is only “weaving,” i.e., the subcontractor does not need to take the responsibility of other processes. When the subcontractor weaves the fabric, the company goes to the subcontractor’s factory to pick the fabrics and carries back all the fabrics to its own factory. Therefore, the companies pick up the fabrics from the subcontractors and put them in their own factories (like depot).In this study, there are four main companies, and each of them has different subcontractors to get the fabrics weaved. As the transportation cost makes up the majority of the total cost of these companies, we predict that the transportation cost of the companies and indirectly the total cost will decrease if these companies merge. In most of the studies in the literature, the vehicles of the companies in cooperation are used jointly. However, in our situation, this type of collaboration is not suitable. This is because each company wants to keep its customer information or special design a secret. On the other hand, before dyeing and finishing processes, all the fabrics must be controlled, but all of the companies want to control the fabrics themselves. So, the common use of the vehicles is not a suitable cooperation approach. Instead, the companies informed us about an application that they use in real life, i.e., they work with the other companies’ subcontractors in case they need. Let us explain this situation with an example. As mentioned before, each company has its own subcontractors. Let us assume that there are two companies,A and B. Some of the subcontractors of company A are close to company B. Also, some of the subcontractors of company B are close to company A. If company A uses company B’s subcontractor, which is near to company A, instead of using its own farther subcontractor, the vehicles of company A travel less distance. The same situation holds for company B. Therefore, the total distance travelled reduces and the total transportation cost decreases.Motivated from the practical application that the companies have, in this work, we assume that the companies use the subcontractors jointly and they still keep the same vehicle fleet in contrast to other studies in the literature which focus on the joint use of vehicles. Then, we compare the costs for two situations: the individual cost that is incurred before merging and the allocated cost of the proposed coalition. If the allocated costs are less than the companies’ individual costs, they will be more willing to cooperate [25]. In the next sections, individual costs and the coalitional costs are figured out.


### 4.1. Case Data

There are four textile companies, namely,A, B, C, and D. Table 1 summarizes the descriptive statistics of these companies including the information about the number of subcontractors, the daily fabric requirements (demands of the companies), the numbers of vehicles, and the cost of each type of vehicle.Table 1
Descriptive statistics.
Company# of subcontractors# of vehiclesVehicleVehicle capacity (kg)Cost of each vehicle (t/km)A831, 2, 31500, 3000, 1500.48, 1.34, 0.24B1131, 2, 31250, 500, 5000.41, 0.30, 0.30C1231, 2, 32000, 1500, 10000.55, 0.48, 0.35D1031, 2, 31500, 1500, 12500.48, 0.48, 0.41The companies use their own vehicle fleets that are heterogeneous, i.e., the capacity of each vehicle differs from the other. The cost of each driver is fixed, i.e., the drivers are paid monthly and the salary is almost the same for all companies in the coalition. In the study, the transportation cost consists of fuel cost (t/km). The distances between companies are known. Each company records the information about their vehicles such as where and when they go and the distance the vehicle travels. The addresses of the companies are known, and “Google Maps” is used for obtaining the distance matrix. Each subcontractor’s daily production capacity is deterministic and is equal to the demand of the company. Semifinished goods can be transported at any time of the working hours. The objective is to minimize the transportation cost.


### 4.2. Individual Cost

The model is solved using LINGO version 9.0. The results before merging are shown in Table2.Table 2
Individual cost of each company.
Company# of subcontractors# of vehiclesVehicleTotal transportation cost (TL)A831, 249.88B1131, 2, 317.82C1231, 2, 333.81D1031, 2, 330.34Each company uses all its vehicles except companyA. Company A uses only two of its vehicles because the demand of company A is less than the total vehicle capacity of company A. Therefore, the use of two vehicles is sufficient to collect all of the fabrics of company A. In the next section, the coalitional costs will be calculated.


### 4.3. Coalitional Costs

In the coalition, we have more than one company (depot) and the companies carry their own fabrics back, so the problem type is multidepot heterogeneous fleet vehicle routing problem (MDHFVRP). Totally, there are 2N − 1 (24 − 1 = 15) vehicle routing problems in the coalition, including 4 SDHFVRPs and 11 MDHFVRPs. 4 SDHFVRPs were solved in the previous section. In this section, the solutions of the remaining 11 MDHFVRPs will be explained.The multidepot vehicle routing problem is NP-hard [26, 27]. Therefore, we solve it by a heuristic. The MDHFVRP can be solved as a clustering problem as the output becomes a set of vehicle routes clustered by depot.In our problem, two-player-coalition (e.g., companiesA and B use their subcontractors jointly) problems and three-player-coalition problems (e.g., companies A, B, and C use their subcontractors jointly) will be solved and finally the model of grand coalition (companies A, B, C, and D use their subcontractors jointly) will be solved. To solve two, three, and four-player-coalition models, we have to solve the MDHFVRP. The problem can be solved in two stages: firstly, subcontractors must be allocated to the companies (depots); then routes must be established that connect subcontractors assigned to the same company (depot). We solve the MDHFVRP by using the proposed “cluster first, route second” method.The proposed assignment algorithm allocates subcontractors to the companies taking into account that the subcontractor meets the demand of the company and the vehicle capacity is not exceeded. The proposed algorithm is explained with an example. Let us consider two companies (j = A, B): company A and company B. Assume that both companies have 3 subcontractors and the fabric production capacity of subcontractor i is denoted by pi. S1, S2, S3 belong to company A and S4, S5, S6 belong to company B. The fabric production capacities of company A’s subcontractors are 150 kg, 200 kg, and 150 kg, respectively. The fabric demand of company A is equal to sum of subcontractors’ capacities, i.e., 500 kg. The demand of company B is 1000 kg, and its subcontractors’ capacities are 450 kg, 300 kg, and 250 kg. To allocate the subcontractors to the companies in the coalition established by companies A and B, we follow the steps explained below:(1)
Firstly, the distances of all the subcontractors to each company in the coalition are found. The distances are given in Table3.For each subcontractor, the row minimum is found and each subcontractor (Si) is assigned to the closest company (see Table 4).(2)
After allocating all the subcontractors to the closest company, we check whether the assigned subcontractors meet the daily demand of each company. In other words, the sum of subcontractors’ capacity must be equal to or greater than the daily demand of company. If subcontractors meet the demand of the company (this condition is checked for each company in the coalition), clusters are obtained for the companies and vehicle routing problems can be solved. Otherwise, some of companies will have excess supply while others’ supply will be inadequate. In order to satisfy the requirements of all the partners in the coalition, some regulations are necessary about the allocation of the subcontractors. These details are explained in the next step. In Tables5 and 6, we have demand/capacity controls for company A and company B, respectively.(3)
The company that has insufficient supply chooses a new subcontractor from the company that has extra subcontractor. The subcontractors of the company which have extra supply become the candidates for the company which has inadequate supply. The problem is how to choose the subcontractors among the candidates. A candidate subcontractor may be close to the company, but its fabric production capacity may not be adequate. Meanwhile, there can be another candidate subcontractor that is farther, but its production capacity may be adequate. In this situation, a subcontractor selection criterion is needed. In this paper, a subcontractor selection criterion is defined by a ratio that is obtained by dividing the distance of the candidate subcontractor to the company, which has insufficient supply, to the fabric production capacity of the candidate subcontractor. This ratio corresponds to the distance to be visited for getting 1 kilogram from the subcontractor (kilometer per kilogram: km/kg). So, the less the rate is, the better it is. The subcontractor selection ratio (km/kg) is obtained for all the subcontractors of the company that needs extra subcontractors (see Table7).(4)
The subcontractor selection continues until the demands of all the companies in the coalition are satisfied. It is worth to note that as some subcontractors are used jointly by the companies, we assume that the capacities of some subcontractors can be split and they can serve more than one company.Table 3
Distances (km) to each company.
Distance toCADistance toCBS136S274S3210S459S583S672Table 4
Assignment of subcontractors.
Distance toCADistance toCBMinimum distanceAssigned companyS1363AS2744BS32102AS4595AS5833BS6722BTable 5
Demand/capacity control forCA.
Subcontractor ofCApi (kg)S1150S3150S4450Total capacity750The demand ofCA500Excess250Table 6
Demand/capacity control forCB.
Subcontractor ofCBpi (kg)S2200S5300S6250Total capacity750The demand ofCB1000Requirement250Table 7
Subcontractor selection amongCA’s subcontractors in order to meet the demand of CB.
Candidate subcontractor forCBDistance toCB for each subcontractor (X)Fabric production capacity (Y)Subcontractor selection criterion (Z = X/Y)S161500.040S3101500.066S494500.020Minimum rate0.020In the example, to meet the demand of companyB, S4 is selected among the candidate subcontractors. When subcontractor 4 produces for only company B, 250 kg of subcontractor 4’s capacity is consumed by company B and 200 kg becomes unused. On the other hand, company A needs 200 kg fabric if S4 works only for company B. Therefore, we propose to split the capacity of S4 among companies A and B in order to prevent the idle capacity and meet the demands of all companies. For this reason, S4 is used jointly.In the last situation, the subcontractors of companies are assigned as below.After obtaining the subcontractor clusters for each company (see Tables8 and 9) in the coalition, the vehicle routing problem is solved as SDHFVRP. The mathematical model given in the “Individual Cost Calculation” part is used in order to solve this problem. The following coalitional results are obtained (see Table 10).Table 8
Cluster of subcontractors forCA.
Subcontractor ofCApi (kg)S1150S3150(Jointly used)S4200Total capacity500The demand ofCA500Excess0Table 9
Cluster of subcontractors forCB.
Subcontractor ofCBpi (kg)S2200S5300S6250(Jointly used)S4250Total capacity1000The demand ofCB1000Requirement0Table 10
Coalitional costs.
Companies in the coalition# of subcontractorsTotal number of vehiclesCoalitional cost (TL)(A, B)19649.333(A, C)20659.753(A, D)18655.226(B, C)23657.161(B, D)21657.291(C, D)22653.483(A, B, C)31976.319(A, B, D)29982.055(A, C, D)30965.884(B, C, D)33963.124(A, B, C, D)411288.413Now, we have the individual and coalitional costs. In order to decide whether joining the coalition is beneficial or not, we need to know the shared cost of each company. So, the grand coalition cost must be allocated fairly among the companies. There are different methods to allocate coalitional costs. In the next section, the cost allocation methods used are explained and the numerical results are presented.


### 4.4. Cost Allocation Methods

In order to distribute the coalitional costs among the companies, three different methods are chosen: Shapley value, nucleolus, and weighted relative savings method. A coalition is denoted by “S” that is a subset of participants (companies) where “N” defines the grand coalition which includes all participants (companies). The cost of coalition S is shown by vS and the cost of grand coalition is vN. In cooperative game theory, vS is called the characteristic cost function and each participant (company) in the coalition is defined as a player.An allocation method that allocates the total cost,vN, among the companies (participants) j∈N is an efficient allocation method, if ∑j∈Nyj=vN, where yj is the allocated cost to company j. A cost allocation method should be individual rational if no player pays more than its individual cost. The individual rationality property is defined mathematically as yj ≤ vj.The core of the game is defined as those allocations,y, that satisfy the following conditions.(12)∑j∈Syj≤vS,S⊂N,∑j∈Nyj=vN,efficiency.A cost allocation in the core is said to be stable.The difference between the cost of coalitionS and the sum of the costs allocated to the companies in the coalition is defined as excess.Mathematically,eS,y=vS−∑j∈Syj. The meaning of at least one negative excess is that the cost allocation is not in the core.The cost allocation methods used in the paper are briefly explained below.


#### 4.4.1. Shapley Value

This method allocates to each playerj an average of the marginal costs it implies when entering coalitions [21]. The cost allocated to the company j is equal to(13)yj=∅jN,v=∑S⊂N:j∈SS−1!N−S!N!vS−vS−j,where ∙ denotes the number of companies in the considered coalition. The summation is over all coalitions that contain participant j. Shapley value gives a unique solution. It is based on four axioms formulated by Shapley [28]. According to these axioms, a cost allocation computed by this solution concept satisfies the properties of efficiency, symmetry, dummy property, and additivity [6]. Symmetry means that if two arbitrary participants, i and j, have the same marginal cost with respect to all coalitions not containing i and j, the costs allocated to these two participants must be equal. The dummy property expresses that if there is a dummy player which means that he neither contributes to nor harms any coalition he may join, then the allocated cost to him should be zero. Finally, additivity states that given three different characteristic cost functions v1, v2, and v1+v2, for each participant, the allocated cost based on v1+v2 must be equal to the sum of the allocated costs based on v1 and v2, respectively. However, the property of “stability” is not guaranteed. For exact formulation of these axioms, refer to Shapley [28].


#### 4.4.2. The Nucleolus

With this method, the worst inequity is tried to be minimized such that individual rationality is satisfied. This method was introduced by Schmeidler [29]. In this method, we ask each coalition S how dissatisfied it is with the proposed allocation y and we aim at minimizing the maximum dissatisfaction of any coalition. The dissatisfaction of a cost allocation y for a coalition S is expressed by the excess, which measures the amount by which coalition S falls short of its potential vS in the allocation y [6]. The nucleolus satisfies both the symmetry axiom and the dummy axiom. If the core is non-empty, the nucleolus is in the core, i.e., it represents a stable cost allocation. Mathematically, the nucleolus model can be expressed as(14)Minimizeεs.t.∑j∈Syj−vS≤ε,∀SS⊂N,S≠0,∑j∈Nyj=vN,yj≤vi.The solution of this method is also unique [30].


#### 4.4.3. Weighted Relative Savings Method (WRSM)

It minimizes the largest difference between relative savings among the companies [30]. The relative savings of company i can be expressed as yi/vi.Therefore, the difference in relative savings between companiesi and j can be calculated as yi/vi−yj/vj. The contribution of each company to the grand coalition is different. So, it is necessary to find a contribution ratio weight wi. Each company’s contribution to the coalition cost is expressed as [vS−vS−i]. To find the weight of each company’s contribution, we use(15)wi=1−∑i∈SvS−vS−i∑i∈N∑i∈SvS−vS−i.The weighted relative savings of companyi are equal to wiyi/vi. The difference in relative savings between two companies i and j is written as(16)wiyivi−wjyjvj.The model is expressed as(17)Minimizefsubjecttowiyivi−wjyjvj≤f∀i,j∈N,∑j∈Nyj≥vS∀S⊂N∑j∈Nyj=vN,where f represents the objective function to minimize the maximum difference. The first constraint set calculates the difference between weighted relative savings. The rest of the constraints guarantee stable allocations [30].


### 4.5. Allocated Costs

The allocated coalitional costs that are found by three different methods are given in Table11.Table 11
Allocated costs.
CompanyShapley valueThe nucleolusWRSMA30.34820.57936.128B18.65317.81911.250C19.69433.80921.627D19.71816.20619.408The nucleolus and the weighted relative savings methods are LP models and are solved by the LINGO software. In Table12, the allocated costs are compared with the individual costs of the companies.Table 12
Distribution of costs and savings.
CompanyIndividual costShapley valueCost reductionThe nucleolusCost reductionWRSMCost reductionA49.8830.34839%20.57959%36.12828%B17.8218.653−5%17.8190%11.25037%C33.8119.69442%33.8090%21.62736%D30.3419.71835%16.20647%19.40836%Sum131.8588.41388.41388.413StabilityNoYesYesTo our knowledge, no horizontal cooperation examples are available in the literature for the textile clusters, so no test instances are known for the collaboration problem studied in this paper. Therefore, the proposed framework is tested with real data obtained from the companies in textile clusters operating in Denizli.From Table12, we see that the total cost obtained after applying the proposed coalition framework decreases from 131.85 to 88.413. From Table 12, we also see that the cost value of all the companies improves when we use the WRSM method and decreases or stays the same in the grand coalition compared to individually determined costs when we use the nucleolus. The only exception is when the allocation is performed through the Shapley value. When the Shapley value is used, we see that the cost of city B increases by 5% which results in unstable solutions, and we can say that the Shapley values do not belong to the core. Guajardo and Ronnqvist [21] pointed out this and mentioned that Shapley values do not necessarily belong to the core. We see similar examples in [16, 18] where Shapley values do not generate stable solutions as in this paper.We see that the cost allocation computed according to Shapley satisfies efficiency property since the sum of the Shapley values of all companies is equal to value of the grand coalition.∑j∈Nyj = vN, i.e., (30.348 + 18.653 + 19.694 + 19.718) = v (A, B, C, D) = 88.413.According to symmetry property, if two players have equal contributions to every coalitionS, then their Shapley values must be the same, i.e., for all S which contains neither i nor j, vS∪i=vS∪j⟶∅c,Ni=∅c,Nj. In our example, we do not have interchangeable partners, so the Shapley values are different for each firm.According to dummy property, dummy firm entering or leaving the coalition brings neither loss nor profit. In this case, the Shapley value of that firm is 0. However, in our results, there is no company which does not contribute to the game, so we do not have a company with a Shapley value of 0. According to additivity, the sum of the Shapley values of the different games defined on a set of firms should be equal to the Shapley value of the sums of the games. But as we consider a single game, the proof of this axiom is out of the scope of this paper.The cost allocations computed according to Shapley value are not stable. Individual rationality is not satisfied. It is worth to note that according to Shapley values given in Table12, company B loses if it cooperates compared to the situation when it operates alone as 18.653 > 17.82. Therefore, we focus on the nucleolus and the WRSM as the allocations by these methods satisfy individual rationality and are stable. These allocations also satisfy the efficiency property; therefore, it can be said that the core is non-empty.The numerical results give rise to the following comments. First, horizontal collaboration through subcontracting sharing can hence produce large cost benefits to companies, and by the proposed collaborative approach, in the grand coalition where all the companies cooperate, we obtain significant cost reductions around 33% on average131.85−88.413/131.85×100. There are similar cooperative studies where significant cost reduction is possible through sharing of resources with collaborating partners, for example, Verdonck et al. [31] obtained an average of 17.9% decrease in transport costs by sharing distribution centers rather than vehicle capacity sharing mostly present in the literature. Second, we show that efficient cost allocation is possible by using the proposed framework. Although the individual savings by means of the nucleolus are 59% for company A and 47% for company D, there are no cost savings for company B and C. Besides its individual rationality property, some partners may receive a very small share of the gain which might give that partner a small incentive to collaborate. The individual savings by means of WRSM range from 28% to 37%, and as we see from Table 12, all other companies obtain cost savings. So, cost allocation through WRSM may be preferred by the cooperating companies.Existing studies on horizontal cooperation generally focus on collaboration opportunities within a transport context, especially when two or more partners collaborate through joint distribution [4, 14, 15, 32–35]. However, in this study, we present a new approach to textile firms operating in clusters: the sharing of subcontractors with collaborating companies. We propose that the companies cooperate with the other companies in the clustered region in order to share their subcontractors based on a distance and capacity related measure. In other words, there are predetermined subcontractors already that the companies work with, and by the proposed cooperation scheme, the companies select the appropriate subcontractors so that they gain cost advantages than their current non-cooperative scheme. In the literature, there are several subcontractor selection studies which consider various criteria in the decision-making process where multicriteria decision-making methods are employed. Many studies are carried out in different subjects [36–43], and only a few studies exist in textile industry although textile industry is one of the main industries in the world [44–47]. In these studies, different quantitative and qualitative criteria are taken into consideration and the best subcontractor is selected among the alternatives. That is, for a single company, a suitable subcontractor is selected. Our work is mainly different from the abovementioned studies in the sense that in the grand coalition (i.e., where all the companies cooperate), we can select the appropriate subcontractors for each company simultaneously. We do not need to determine the subcontractor of each company one by one. We also do not use multicriteria decision-making methods as in the abovementioned studies to choose the appropriate subcontractors, rather we assume that the companies cooperate to share their subcontractors and we assume that all other attributes of the subcontractors are similar except their distances and their capacities to satisfy demand requirements. We would like to see the result of horizontal cooperation in textile industry and propose a framework which requires the solution of heterogeneous fleet vehicle routing problem (SDHFVRP) when the companies cooperate. We also benefit from cooperative game theory to allocate the coalitional costs to each company.To the best of our knowledge, this study is the first collaboration work in the home textile sector. As opposed to previous studies from other sectors, the basis of cooperation is using the subcontractors jointly with keeping the same vehicle fleet. Therefore, each company collects its own fabrics by its own vehicles in order to protect customer information or special designs. When the table above is examined, establishing a coalition is quite sensible. This is because the total coalition cost (88.413) is less than the total individual costs (49.88 + 17.82 + 33.81 + 30.34 = 131.85). Mathematically,(18)vN≤∑j∈Nvj.Among the three methods, the most suitable cost allocation method is the weighted relative savings method because the cost reductions in this method are higher than the reductions found by the nucleolus method.


## 4.1. Case Data

There are four textile companies, namely,A, B, C, and D. Table 1 summarizes the descriptive statistics of these companies including the information about the number of subcontractors, the daily fabric requirements (demands of the companies), the numbers of vehicles, and the cost of each type of vehicle.Table 1
Descriptive statistics.
Company# of subcontractors# of vehiclesVehicleVehicle capacity (kg)Cost of each vehicle (t/km)A831, 2, 31500, 3000, 1500.48, 1.34, 0.24B1131, 2, 31250, 500, 5000.41, 0.30, 0.30C1231, 2, 32000, 1500, 10000.55, 0.48, 0.35D1031, 2, 31500, 1500, 12500.48, 0.48, 0.41The companies use their own vehicle fleets that are heterogeneous, i.e., the capacity of each vehicle differs from the other. The cost of each driver is fixed, i.e., the drivers are paid monthly and the salary is almost the same for all companies in the coalition. In the study, the transportation cost consists of fuel cost (t/km). The distances between companies are known. Each company records the information about their vehicles such as where and when they go and the distance the vehicle travels. The addresses of the companies are known, and “Google Maps” is used for obtaining the distance matrix. Each subcontractor’s daily production capacity is deterministic and is equal to the demand of the company. Semifinished goods can be transported at any time of the working hours. The objective is to minimize the transportation cost.


## 4.2. Individual Cost

The model is solved using LINGO version 9.0. The results before merging are shown in Table2.Table 2
Individual cost of each company.
Company# of subcontractors# of vehiclesVehicleTotal transportation cost (TL)A831, 249.88B1131, 2, 317.82C1231, 2, 333.81D1031, 2, 330.34Each company uses all its vehicles except companyA. Company A uses only two of its vehicles because the demand of company A is less than the total vehicle capacity of company A. Therefore, the use of two vehicles is sufficient to collect all of the fabrics of company A. In the next section, the coalitional costs will be calculated.


## 4.3. Coalitional Costs

In the coalition, we have more than one company (depot) and the companies carry their own fabrics back, so the problem type is multidepot heterogeneous fleet vehicle routing problem (MDHFVRP). Totally, there are 2N − 1 (24 − 1 = 15) vehicle routing problems in the coalition, including 4 SDHFVRPs and 11 MDHFVRPs. 4 SDHFVRPs were solved in the previous section. In this section, the solutions of the remaining 11 MDHFVRPs will be explained.The multidepot vehicle routing problem is NP-hard [26, 27]. Therefore, we solve it by a heuristic. The MDHFVRP can be solved as a clustering problem as the output becomes a set of vehicle routes clustered by depot.In our problem, two-player-coalition (e.g., companiesA and B use their subcontractors jointly) problems and three-player-coalition problems (e.g., companies A, B, and C use their subcontractors jointly) will be solved and finally the model of grand coalition (companies A, B, C, and D use their subcontractors jointly) will be solved. To solve two, three, and four-player-coalition models, we have to solve the MDHFVRP. The problem can be solved in two stages: firstly, subcontractors must be allocated to the companies (depots); then routes must be established that connect subcontractors assigned to the same company (depot). We solve the MDHFVRP by using the proposed “cluster first, route second” method.The proposed assignment algorithm allocates subcontractors to the companies taking into account that the subcontractor meets the demand of the company and the vehicle capacity is not exceeded. The proposed algorithm is explained with an example. Let us consider two companies (j = A, B): company A and company B. Assume that both companies have 3 subcontractors and the fabric production capacity of subcontractor i is denoted by pi. S1, S2, S3 belong to company A and S4, S5, S6 belong to company B. The fabric production capacities of company A’s subcontractors are 150 kg, 200 kg, and 150 kg, respectively. The fabric demand of company A is equal to sum of subcontractors’ capacities, i.e., 500 kg. The demand of company B is 1000 kg, and its subcontractors’ capacities are 450 kg, 300 kg, and 250 kg. To allocate the subcontractors to the companies in the coalition established by companies A and B, we follow the steps explained below:(1)
Firstly, the distances of all the subcontractors to each company in the coalition are found. The distances are given in Table3.For each subcontractor, the row minimum is found and each subcontractor (Si) is assigned to the closest company (see Table 4).(2)
After allocating all the subcontractors to the closest company, we check whether the assigned subcontractors meet the daily demand of each company. In other words, the sum of subcontractors’ capacity must be equal to or greater than the daily demand of company. If subcontractors meet the demand of the company (this condition is checked for each company in the coalition), clusters are obtained for the companies and vehicle routing problems can be solved. Otherwise, some of companies will have excess supply while others’ supply will be inadequate. In order to satisfy the requirements of all the partners in the coalition, some regulations are necessary about the allocation of the subcontractors. These details are explained in the next step. In Tables5 and 6, we have demand/capacity controls for company A and company B, respectively.(3)
The company that has insufficient supply chooses a new subcontractor from the company that has extra subcontractor. The subcontractors of the company which have extra supply become the candidates for the company which has inadequate supply. The problem is how to choose the subcontractors among the candidates. A candidate subcontractor may be close to the company, but its fabric production capacity may not be adequate. Meanwhile, there can be another candidate subcontractor that is farther, but its production capacity may be adequate. In this situation, a subcontractor selection criterion is needed. In this paper, a subcontractor selection criterion is defined by a ratio that is obtained by dividing the distance of the candidate subcontractor to the company, which has insufficient supply, to the fabric production capacity of the candidate subcontractor. This ratio corresponds to the distance to be visited for getting 1 kilogram from the subcontractor (kilometer per kilogram: km/kg). So, the less the rate is, the better it is. The subcontractor selection ratio (km/kg) is obtained for all the subcontractors of the company that needs extra subcontractors (see Table7).(4)
The subcontractor selection continues until the demands of all the companies in the coalition are satisfied. It is worth to note that as some subcontractors are used jointly by the companies, we assume that the capacities of some subcontractors can be split and they can serve more than one company.Table 3
Distances (km) to each company.
Distance toCADistance toCBS136S274S3210S459S583S672Table 4
Assignment of subcontractors.
Distance toCADistance toCBMinimum distanceAssigned companyS1363AS2744BS32102AS4595AS5833BS6722BTable 5
Demand/capacity control forCA.
Subcontractor ofCApi (kg)S1150S3150S4450Total capacity750The demand ofCA500Excess250Table 6
Demand/capacity control forCB.
Subcontractor ofCBpi (kg)S2200S5300S6250Total capacity750The demand ofCB1000Requirement250Table 7
Subcontractor selection amongCA’s subcontractors in order to meet the demand of CB.
Candidate subcontractor forCBDistance toCB for each subcontractor (X)Fabric production capacity (Y)Subcontractor selection criterion (Z = X/Y)S161500.040S3101500.066S494500.020Minimum rate0.020In the example, to meet the demand of companyB, S4 is selected among the candidate subcontractors. When subcontractor 4 produces for only company B, 250 kg of subcontractor 4’s capacity is consumed by company B and 200 kg becomes unused. On the other hand, company A needs 200 kg fabric if S4 works only for company B. Therefore, we propose to split the capacity of S4 among companies A and B in order to prevent the idle capacity and meet the demands of all companies. For this reason, S4 is used jointly.In the last situation, the subcontractors of companies are assigned as below.After obtaining the subcontractor clusters for each company (see Tables8 and 9) in the coalition, the vehicle routing problem is solved as SDHFVRP. The mathematical model given in the “Individual Cost Calculation” part is used in order to solve this problem. The following coalitional results are obtained (see Table 10).Table 8
Cluster of subcontractors forCA.
Subcontractor ofCApi (kg)S1150S3150(Jointly used)S4200Total capacity500The demand ofCA500Excess0Table 9
Cluster of subcontractors forCB.
Subcontractor ofCBpi (kg)S2200S5300S6250(Jointly used)S4250Total capacity1000The demand ofCB1000Requirement0Table 10
Coalitional costs.
Companies in the coalition# of subcontractorsTotal number of vehiclesCoalitional cost (TL)(A, B)19649.333(A, C)20659.753(A, D)18655.226(B, C)23657.161(B, D)21657.291(C, D)22653.483(A, B, C)31976.319(A, B, D)29982.055(A, C, D)30965.884(B, C, D)33963.124(A, B, C, D)411288.413Now, we have the individual and coalitional costs. In order to decide whether joining the coalition is beneficial or not, we need to know the shared cost of each company. So, the grand coalition cost must be allocated fairly among the companies. There are different methods to allocate coalitional costs. In the next section, the cost allocation methods used are explained and the numerical results are presented.


## 4.4. Cost Allocation Methods

In order to distribute the coalitional costs among the companies, three different methods are chosen: Shapley value, nucleolus, and weighted relative savings method. A coalition is denoted by “S” that is a subset of participants (companies) where “N” defines the grand coalition which includes all participants (companies). The cost of coalition S is shown by vS and the cost of grand coalition is vN. In cooperative game theory, vS is called the characteristic cost function and each participant (company) in the coalition is defined as a player.An allocation method that allocates the total cost,vN, among the companies (participants) j∈N is an efficient allocation method, if ∑j∈Nyj=vN, where yj is the allocated cost to company j. A cost allocation method should be individual rational if no player pays more than its individual cost. The individual rationality property is defined mathematically as yj ≤ vj.The core of the game is defined as those allocations,y, that satisfy the following conditions.(12)∑j∈Syj≤vS,S⊂N,∑j∈Nyj=vN,efficiency.A cost allocation in the core is said to be stable.The difference between the cost of coalitionS and the sum of the costs allocated to the companies in the coalition is defined as excess.Mathematically,eS,y=vS−∑j∈Syj. The meaning of at least one negative excess is that the cost allocation is not in the core.The cost allocation methods used in the paper are briefly explained below.


### 4.4.1. Shapley Value

This method allocates to each playerj an average of the marginal costs it implies when entering coalitions [21]. The cost allocated to the company j is equal to(13)yj=∅jN,v=∑S⊂N:j∈SS−1!N−S!N!vS−vS−j,where ∙ denotes the number of companies in the considered coalition. The summation is over all coalitions that contain participant j. Shapley value gives a unique solution. It is based on four axioms formulated by Shapley [28]. According to these axioms, a cost allocation computed by this solution concept satisfies the properties of efficiency, symmetry, dummy property, and additivity [6]. Symmetry means that if two arbitrary participants, i and j, have the same marginal cost with respect to all coalitions not containing i and j, the costs allocated to these two participants must be equal. The dummy property expresses that if there is a dummy player which means that he neither contributes to nor harms any coalition he may join, then the allocated cost to him should be zero. Finally, additivity states that given three different characteristic cost functions v1, v2, and v1+v2, for each participant, the allocated cost based on v1+v2 must be equal to the sum of the allocated costs based on v1 and v2, respectively. However, the property of “stability” is not guaranteed. For exact formulation of these axioms, refer to Shapley [28].


### 4.4.2. The Nucleolus

With this method, the worst inequity is tried to be minimized such that individual rationality is satisfied. This method was introduced by Schmeidler [29]. In this method, we ask each coalition S how dissatisfied it is with the proposed allocation y and we aim at minimizing the maximum dissatisfaction of any coalition. The dissatisfaction of a cost allocation y for a coalition S is expressed by the excess, which measures the amount by which coalition S falls short of its potential vS in the allocation y [6]. The nucleolus satisfies both the symmetry axiom and the dummy axiom. If the core is non-empty, the nucleolus is in the core, i.e., it represents a stable cost allocation. Mathematically, the nucleolus model can be expressed as(14)Minimizeεs.t.∑j∈Syj−vS≤ε,∀SS⊂N,S≠0,∑j∈Nyj=vN,yj≤vi.The solution of this method is also unique [30].


### 4.4.3. Weighted Relative Savings Method (WRSM)

It minimizes the largest difference between relative savings among the companies [30]. The relative savings of company i can be expressed as yi/vi.Therefore, the difference in relative savings between companiesi and j can be calculated as yi/vi−yj/vj. The contribution of each company to the grand coalition is different. So, it is necessary to find a contribution ratio weight wi. Each company’s contribution to the coalition cost is expressed as [vS−vS−i]. To find the weight of each company’s contribution, we use(15)wi=1−∑i∈SvS−vS−i∑i∈N∑i∈SvS−vS−i.The weighted relative savings of companyi are equal to wiyi/vi. The difference in relative savings between two companies i and j is written as(16)wiyivi−wjyjvj.The model is expressed as(17)Minimizefsubjecttowiyivi−wjyjvj≤f∀i,j∈N,∑j∈Nyj≥vS∀S⊂N∑j∈Nyj=vN,where f represents the objective function to minimize the maximum difference. The first constraint set calculates the difference between weighted relative savings. The rest of the constraints guarantee stable allocations [30].


## 4.4.1. Shapley Value

This method allocates to each playerj an average of the marginal costs it implies when entering coalitions [21]. The cost allocated to the company j is equal to(13)yj=∅jN,v=∑S⊂N:j∈SS−1!N−S!N!vS−vS−j,where ∙ denotes the number of companies in the considered coalition. The summation is over all coalitions that contain participant j. Shapley value gives a unique solution. It is based on four axioms formulated by Shapley [28]. According to these axioms, a cost allocation computed by this solution concept satisfies the properties of efficiency, symmetry, dummy property, and additivity [6]. Symmetry means that if two arbitrary participants, i and j, have the same marginal cost with respect to all coalitions not containing i and j, the costs allocated to these two participants must be equal. The dummy property expresses that if there is a dummy player which means that he neither contributes to nor harms any coalition he may join, then the allocated cost to him should be zero. Finally, additivity states that given three different characteristic cost functions v1, v2, and v1+v2, for each participant, the allocated cost based on v1+v2 must be equal to the sum of the allocated costs based on v1 and v2, respectively. However, the property of “stability” is not guaranteed. For exact formulation of these axioms, refer to Shapley [28].


## 4.4.2. The Nucleolus

With this method, the worst inequity is tried to be minimized such that individual rationality is satisfied. This method was introduced by Schmeidler [29]. In this method, we ask each coalition S how dissatisfied it is with the proposed allocation y and we aim at minimizing the maximum dissatisfaction of any coalition. The dissatisfaction of a cost allocation y for a coalition S is expressed by the excess, which measures the amount by which coalition S falls short of its potential vS in the allocation y [6]. The nucleolus satisfies both the symmetry axiom and the dummy axiom. If the core is non-empty, the nucleolus is in the core, i.e., it represents a stable cost allocation. Mathematically, the nucleolus model can be expressed as(14)Minimizeεs.t.∑j∈Syj−vS≤ε,∀SS⊂N,S≠0,∑j∈Nyj=vN,yj≤vi.The solution of this method is also unique [30].


## 4.4.3. Weighted Relative Savings Method (WRSM)

It minimizes the largest difference between relative savings among the companies [30]. The relative savings of company i can be expressed as yi/vi.Therefore, the difference in relative savings between companiesi and j can be calculated as yi/vi−yj/vj. The contribution of each company to the grand coalition is different. So, it is necessary to find a contribution ratio weight wi. Each company’s contribution to the coalition cost is expressed as [vS−vS−i]. To find the weight of each company’s contribution, we use(15)wi=1−∑i∈SvS−vS−i∑i∈N∑i∈SvS−vS−i.The weighted relative savings of companyi are equal to wiyi/vi. The difference in relative savings between two companies i and j is written as(16)wiyivi−wjyjvj.The model is expressed as(17)Minimizefsubjecttowiyivi−wjyjvj≤f∀i,j∈N,∑j∈Nyj≥vS∀S⊂N∑j∈Nyj=vN,where f represents the objective function to minimize the maximum difference. The first constraint set calculates the difference between weighted relative savings. The rest of the constraints guarantee stable allocations [30].


## 4.5. Allocated Costs

The allocated coalitional costs that are found by three different methods are given in Table11.Table 11
Allocated costs.
CompanyShapley valueThe nucleolusWRSMA30.34820.57936.128B18.65317.81911.250C19.69433.80921.627D19.71816.20619.408The nucleolus and the weighted relative savings methods are LP models and are solved by the LINGO software. In Table12, the allocated costs are compared with the individual costs of the companies.Table 12
Distribution of costs and savings.
CompanyIndividual costShapley valueCost reductionThe nucleolusCost reductionWRSMCost reductionA49.8830.34839%20.57959%36.12828%B17.8218.653−5%17.8190%11.25037%C33.8119.69442%33.8090%21.62736%D30.3419.71835%16.20647%19.40836%Sum131.8588.41388.41388.413StabilityNoYesYesTo our knowledge, no horizontal cooperation examples are available in the literature for the textile clusters, so no test instances are known for the collaboration problem studied in this paper. Therefore, the proposed framework is tested with real data obtained from the companies in textile clusters operating in Denizli.From Table12, we see that the total cost obtained after applying the proposed coalition framework decreases from 131.85 to 88.413. From Table 12, we also see that the cost value of all the companies improves when we use the WRSM method and decreases or stays the same in the grand coalition compared to individually determined costs when we use the nucleolus. The only exception is when the allocation is performed through the Shapley value. When the Shapley value is used, we see that the cost of city B increases by 5% which results in unstable solutions, and we can say that the Shapley values do not belong to the core. Guajardo and Ronnqvist [21] pointed out this and mentioned that Shapley values do not necessarily belong to the core. We see similar examples in [16, 18] where Shapley values do not generate stable solutions as in this paper.We see that the cost allocation computed according to Shapley satisfies efficiency property since the sum of the Shapley values of all companies is equal to value of the grand coalition.∑j∈Nyj = vN, i.e., (30.348 + 18.653 + 19.694 + 19.718) = v (A, B, C, D) = 88.413.According to symmetry property, if two players have equal contributions to every coalitionS, then their Shapley values must be the same, i.e., for all S which contains neither i nor j, vS∪i=vS∪j⟶∅c,Ni=∅c,Nj. In our example, we do not have interchangeable partners, so the Shapley values are different for each firm.According to dummy property, dummy firm entering or leaving the coalition brings neither loss nor profit. In this case, the Shapley value of that firm is 0. However, in our results, there is no company which does not contribute to the game, so we do not have a company with a Shapley value of 0. According to additivity, the sum of the Shapley values of the different games defined on a set of firms should be equal to the Shapley value of the sums of the games. But as we consider a single game, the proof of this axiom is out of the scope of this paper.The cost allocations computed according to Shapley value are not stable. Individual rationality is not satisfied. It is worth to note that according to Shapley values given in Table12, company B loses if it cooperates compared to the situation when it operates alone as 18.653 > 17.82. Therefore, we focus on the nucleolus and the WRSM as the allocations by these methods satisfy individual rationality and are stable. These allocations also satisfy the efficiency property; therefore, it can be said that the core is non-empty.The numerical results give rise to the following comments. First, horizontal collaboration through subcontracting sharing can hence produce large cost benefits to companies, and by the proposed collaborative approach, in the grand coalition where all the companies cooperate, we obtain significant cost reductions around 33% on average131.85−88.413/131.85×100. There are similar cooperative studies where significant cost reduction is possible through sharing of resources with collaborating partners, for example, Verdonck et al. [31] obtained an average of 17.9% decrease in transport costs by sharing distribution centers rather than vehicle capacity sharing mostly present in the literature. Second, we show that efficient cost allocation is possible by using the proposed framework. Although the individual savings by means of the nucleolus are 59% for company A and 47% for company D, there are no cost savings for company B and C. Besides its individual rationality property, some partners may receive a very small share of the gain which might give that partner a small incentive to collaborate. The individual savings by means of WRSM range from 28% to 37%, and as we see from Table 12, all other companies obtain cost savings. So, cost allocation through WRSM may be preferred by the cooperating companies.Existing studies on horizontal cooperation generally focus on collaboration opportunities within a transport context, especially when two or more partners collaborate through joint distribution [4, 14, 15, 32–35]. However, in this study, we present a new approach to textile firms operating in clusters: the sharing of subcontractors with collaborating companies. We propose that the companies cooperate with the other companies in the clustered region in order to share their subcontractors based on a distance and capacity related measure. In other words, there are predetermined subcontractors already that the companies work with, and by the proposed cooperation scheme, the companies select the appropriate subcontractors so that they gain cost advantages than their current non-cooperative scheme. In the literature, there are several subcontractor selection studies which consider various criteria in the decision-making process where multicriteria decision-making methods are employed. Many studies are carried out in different subjects [36–43], and only a few studies exist in textile industry although textile industry is one of the main industries in the world [44–47]. In these studies, different quantitative and qualitative criteria are taken into consideration and the best subcontractor is selected among the alternatives. That is, for a single company, a suitable subcontractor is selected. Our work is mainly different from the abovementioned studies in the sense that in the grand coalition (i.e., where all the companies cooperate), we can select the appropriate subcontractors for each company simultaneously. We do not need to determine the subcontractor of each company one by one. We also do not use multicriteria decision-making methods as in the abovementioned studies to choose the appropriate subcontractors, rather we assume that the companies cooperate to share their subcontractors and we assume that all other attributes of the subcontractors are similar except their distances and their capacities to satisfy demand requirements. We would like to see the result of horizontal cooperation in textile industry and propose a framework which requires the solution of heterogeneous fleet vehicle routing problem (SDHFVRP) when the companies cooperate. We also benefit from cooperative game theory to allocate the coalitional costs to each company.To the best of our knowledge, this study is the first collaboration work in the home textile sector. As opposed to previous studies from other sectors, the basis of cooperation is using the subcontractors jointly with keeping the same vehicle fleet. Therefore, each company collects its own fabrics by its own vehicles in order to protect customer information or special designs. When the table above is examined, establishing a coalition is quite sensible. This is because the total coalition cost (88.413) is less than the total individual costs (49.88 + 17.82 + 33.81 + 30.34 = 131.85). Mathematically,(18)vN≤∑j∈Nvj.Among the three methods, the most suitable cost allocation method is the weighted relative savings method because the cost reductions in this method are higher than the reductions found by the nucleolus method.


## 5. Conclusions

In the fierce competition conditions, supply chains compete against supply chains rather than companies due to globalization and scarce resources which require more control over the supply chain. The companies in the supply chain must provide cost efficiency to overcome high level of competition. To do this, cooperation among companies is necessary, especially for the companies in regional clusters.In this paper, we propose a game theoretic approach where textile companies cooperate to jointly use their subcontractors. The solution to the proposed coalition structure requires the solution of multidepot vehicle routing problem which is NP-hard. A clustering algorithm is proposed so that a subcontractor may serve more than one company in the coalition. In the proposed approach, the “cluster first, route second” method is used for the solution of the problem as long as the subcontractors have enough capacity to meet the demands of the companies. In case the customer demand is not met, some regulations are made about allocation of subcontractors and a subcontractor selection procedure is proposed to determine the cluster of the subcontractors for the companies in the cooperation.The methodology is validated through a case study in a textile cluster in Denizli. Four textile companies are considered to cooperate to work with each other’s subcontractors in order to obtain cost savings. They do not cooperate to share their vehicle fleet which is different from other horizontal research studies in the literature. The coalition cost arising from their horizontal cooperation is allocated among companies according to cooperative game theory. After cooperation, the companies obtain significant cost savings.This study also has some limitations. First, some assumptions are made in constructing the proposed framework. For example, we assume that the suppliers meet the demand of the companies unconditionally and we propose a supplier selection criterion based on the distance and the capacity. However, in real life, there may be other criteria to be considered such as the machinery of the subcontractors. In reality, the machinery of the suppliers may not be suitable for producing every fabric. In this case, there will be a capacity constraint according to the machine park. Therefore, the framework can be extended to include the machine park and capacity constraints in selecting the appropriate suppliers. Second, it is also assumed that the companies share their suppliers’ information with their partners. However, in organizations with horizontal cooperation, firms are not open and transparent in sharing information since they are independent entities and are also the competitors of each other. As a result, since they sell products to the same market, they do not want their products to be shared with other companies by suppliers or they do not want to share their know-how with other companies. This makes it difficult to collaborate in real life. In order to solve this situation, it is possible to work on how to construct structures and contracts that will establish a relationship of trust between suppliers and companies, which have sanctions. In addition, in order to support the implementation of horizontal cooperation, a decision support software program may be designed.In summary, there are some impediments to implement cooperation despite a significant cost reduction. For instance, partner selection is very important for the success of the cooperation. Collaborating with the right partners helps to increase speed, encourage innovation, and obtain market share. However, it requires knowledge about potential partner’s physical properties and intangible assets which makes finding right partners difficult [48]. Also, the establishment of a trustful environment is a precondition for clustering. Costs in terms of coordination, time, and resources can hold back cluster actors from collaborating. Future work studies might focus on partner selection and how to establish trustworthy relationships among the companies. It might also be interesting to study cooperation among companies which have different negotiation powers.



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*Source: 8256116-2023-02-07.xml*