Cooperative aggregate production planning: a game theory approach
Abstract
Production costs in general, and workforce and inventory costs in particular, constitute a large fraction of the operating costs of many manufacturing plants. We introduce cooperative aggregate production planning as a way to decrease these costs. That is, when production planning of two or more facilities (plants) is integrated, they can interchange workforce and products inventory; thus, their product demands can be satisfied at lower cost. This paper quantifies the cost saving and synergy of different coalitions of production plants by a new linear model for cooperative aggregate planning problem. The developed approach is explicated with a numerical example in which inventory and workforce levels of different coalitions of facilities are evaluated. Afterward, a key question would be how the cost saving of a coalition should be divided among members. We tackle the problem using different methods of cooperative game theory. These methods are implemented in the numerical example to gain an insight into properties of the corresponding game results.
Keywords
Aggregate production planning Cooperative mechanisms Game theory Reliable and stable production Costsaving opportunityIntroduction
In the real world, production plants (manufacturing facilities) often should take decisions regarding levels of inventory capacity, employment, and production levels, before demand is known. Aggregate production planning (APP) addresses the question, “How should a company best use the equipment and facilities that it currently has?” (Chopra and Meindl 2007). Accurate demand forecast and supply constraints are two key important inputs of APP. The goal of APP indeed is satisfying demands over a planning horizon with the minimum cost. APP, however, is an extrafirm operational problem rather than intrafirm one. Thus, modern production plans are viewed as a set of collaborative agreements between manufacturers’ network (Argoneto et al. 2008).
Aggregating production plan (APP) considers minimizing costs, levels of inventory, alteration of human resource level, wage of additional work for production, changes in production rate, number of machine startup, idle time of plant and work force and maximizing revenue and costumer services in high priority (Baykasoglu 2001).
Traditionally, much of APP is concentrated on a single company and may not always be focused on the cooperation among a set of production plants. Currently, many types of products are manufactured by cooperation of several producers. This means that most of plants use the various site production facilities in order to make economic competitive advantages. It is possible that particular segment of a family product is produced in different sites. The events like defection of device, the absence of operator, lack of expert work force can be of the reasons of allocation to different sites or plants. For example, agricultural production cooperatives, also called farmers coops, are activities in which a group of farmers that pool resources to improve their productivity and responsiveness to market demands (Cobia 1989). Several successful benchmarks of agricultural production cooperatives exist around the world such as Longo Mai cooperatives, Kibbutzim and Nicaraguan production cooperatives (Ruben and Lerman 2005).

How can cooperative aggregate production planning (CoAPP) be formulated?

How should benefits of cooperation of CoAPP be fairly distributed?
Cooperative game theory (CGT) primarily deals with coalition of players that coordinate their activities to enjoy the synergy of cooperation (Barron 2013; Branzei et al. 2008). CGT establishes a mathematical framework for fair and reasonable allocations of the cooperation benefits to each member of a coalition. By considering the production plants as a set of players, we first quantify the synergy of the CoAPP; then, we use CGT methods to assign the cooperation benefits to the companies. The difference of cooperation in this study with multisite studies is in terms of superadditivity feature (Asgharpour 2014) which is defined when two production plants make a cooperative coalition. In such relationship, for each members of coalition it is feasible to earn minimum effectiveness that each plant gained before arranging the coalition. However, in most of the time, the earned benefits after cooperation are more than the benefit of cooperation in multimodel. Indeed in multisite model, the features of each site optimize independently. However, cooperation of the sites causes the total revenue that is greater than or equal to independent revenue. In conventional approach of optimization, the problem will be optimized by different methods. In case of multiple problems, each of them has identical optimum solution. However in game theory’s problem, for each player there is a distinct optimization problem which is solved concurrently. The strategy of each player will affect both the other players’ problem and the final optimum solution simultaneously.

The inventory on each site in the multisite belongs to the same site, but in cooperation there is a possibility of replacement of inventory between them.

The workforces in each site in the multisite belong to the same site, but in cooperation there is the possibility of replacing the workers between them.
Therefore, these differences, while reducing costs, increase job security and workers’ skills and allow factories to use highertech equipment. The plants are moving with the maximum use of multiskill workforce and minimizing inventory for lean production.
The reminder of the paper is organized as follows. “Literature review” section reviews the related literature. “Prerequisites and assumptions” section describes the prerequisites and assumptions. “Mathematical programming approaches” section presents the formulations of basic APP and CoAPP. Costsaving opportunities of CoAPP are also examined using an illustrative example. “Collaborative frameworks for CoAPP” section discusses the methods of CGT for allocating the cost savings to production plants. “Conclusion and further research” section provides the conclusions and several directions for future research.
Literature review
Survey of aggregate production planning and its modeling approaches
Holt et al. (1955) introduced the concept of APP. From the mid1950s, a number of researchers have studied APP, its modeling techniques and the related applications. Nam and Logendran (1992) presented a detailed survey about APP models and methodologies.
Some researchers focused on application of APP in multisite environments. In the category of multisite APP models, various solution approaches exist to tackle the realworld industrial planning problems including robust optimization (Mirzapour AlEHashem et al. 2011), simple linear programming (Kanyalkar and Adil 2005; Proto and de Mesquita 2006; Wu and Williams 2003), stochastic programming (Leung et al. 2003b, 2006) and goal programming (Kanyalkar and Adil 2007; Leung et al. 2003a; Torabi and Hassini 2009). Kanyalkar and Adil (2005) generated a linear programming model for a multiplant aggregate production and dynamic distribution problem. Proto and de Mesquita (2006) developed a mixedinteger linear programming model to deal with an aggregate production and distribution planning problem considering multiple production and distribution sites for application in cement industry. Utilizing timestaged linear programming technique, Wu and Williams (2003) analyzed multisite APP problem. Leung et al. (2006) suggested a stochastic programming approach to consider multisite APP under an uncertain environment. Leung et al. (2003b) addressed a multisite APP problem employing a stochastic model under uncertain environment for application in a multinational lingerie company. Kanyalkar and Adil (2007) recommended a mixedinteger linear goal programming model to solve an aggregate multiitem, multiplant procurement, production and distribution problem. Leung et al. (2003a) developed a goal programming approach to handle the problem of APP for a multinational lingerie company with multiple manufacturing factories.
Several groups of researchers have considered supply chainoriented APP problems. Several solution techniques have been utilized by the researchers to manage practical APP problems in a supply chain network like robust optimization (Kanyalkar and Adil 2010; Mirzapour AlEHashem et al. 2011; Niknamfar et al. 2014), simple linear programming (Mirzapour Alehashem et al. 2012), stochastic programming (Mirzapour Alehashem et al. 2013), fuzzy programming (Aliev et al. 2007; Paksoy et al. 2010; Pathak and Sarkar 2012; Torabi and Hassini 2009; Yaghin et al. 2012), bilevel programming, system dynamics approach (Mendoza et al. 2014) and heuristic algorithm (Pal et al. 2011). Kanyalkar and Adil (2010) presented a robust optimization methodology for aggregate planning of a multisite procurement–production–distribution system. Mirzapour AlEHashem et al. (2011) considered a robust multiobjective mixedinteger nonlinear programming model to tackle a APP problem in a supply chain under uncertainty. Niknamfar et al. (2014) developed a robust optimization method for an aggregate production–distribution planning in a threelevel supply chain. Mirzapour Alehashem et al. (2012) suggested a mixedinteger linear programming model to solve an aggregate production–distribution planning problem in a green supply chain. Mirzapour Alehashem et al. (2013) proposed a stochastic programming approach to solve a multiperiod multiproduct multisite APP problem in a green supply chain for a mediumterm planning horizon under the assumption of demand uncertainty. Aliev et al. (2007) provided a genetic algorithm solution based on fuzzy programming for aggregate production–distribution planning in a supply chain. Paksoy et al. (2010) modeled a supply chain network design problem with fuzzy demand and capacities in production environment using the concept of aggregate production–distribution planning. Employing a fuzzy mixedinteger programming model, Pathak and Sarkar (2012) dealt with a supply chain network design problem in the field of aggregate production–distribution planning. Considering an interactive fuzzy goal programming approach, Torabi and Hassini (2009) tackled a multiobjective, multisite production planning problem by integrating procurement and distribution plans in a multiechelon supply chain network. Yaghin et al. (2012) utilized a hybrid fuzzy multiple objective technique to consolidate markdown pricing planning and APP in a two echelon supply chain. Mendoza et al. (2014) recommended a novel method to evaluate different APP strategies in a manpowerintensive supply chain considering system dynamics approach. Applying swarmbased heuristics, Pal et al. (2011) modeled a problem of aggregate purchasing, manufacturing and shipment planning for a supply chain extending over three echelons. Gholamian et al (2015) use the mixedinteger linear programming/fuzzy optimization for solution fuzzymathematical model to incorporate four objectives. Making tradeoff between production costs and green principles seen in research Entezaminia et al. (2016), Jing et al. (2016) use the GA for solution multisite problem.
Summary of conducted research on subject of multiple APP
Research  Space issue  Differentiation criteria  Problem attribute  Modeling approach  Solution technique 

Mirzapour AlEHashem et al. (2011)  Multisite  RO considering total losses (measured in cost) and customer satisfaction, consideration of productivity of different level workers and training and upgradation of employee  Multiperiod, multiproduct, multiobjective  Robust optimization  LPmetrics method 
Kanyalkar and Adil (2005)  Multisite  Different time grids and planning horizons (at tactical and operational level), substitutable capacities, forward coverage policy to avoid backorder  Multicustomer, multiproduct multiperiod  Simple linear programming  Deterministic 
Leung et al. (2006)  Multisite  Using stochastic programming to deal with uncertainty in demand, preference in selection of production plant  Multiproduct, multiperiod  Stochastic programming  Simplex method 
Leung et al. (2003b)  Multisite  More than one objective for the first time in MSAPP, utilization of import quota, nonhomogeneous value of workers (labor cost depends on location and experience)  Multiobjective, multiproduct, multiperiod  Preemptive GP  Simplex method (deterministic approach) 
Kanyalkar and Adil (2007)  Multiitem, multiplant procurement  Development of a single formulation for different levels of planning  Multiproduct, multiperiod, multiobjective  Mixedinteger linear goal programming (MILGP)  Weighing and preemptive GP 
Kanyalkar and Adil (2010)  Multisite  Robust optimization (RO) for integrated aggregate and detailed planning  Multiproduct, multiperiod, multiobjective  Mathematical  Robust optimization 
Mirzapour AlEHashem et al. (2011)  Multisite  RO considering total losses (measured in cost) and customer satisfaction, consideration of productivity of different level workers and training and upgradation of employee  Multiperiod, multiproduct, multiobjective  Robust optimization/mixedinteger nonlinear programming  LPmetrics method 
Niknamfar et al. (2014)  Multisite  Development of a RO model for a threelevel SC, weight of model robustness has been changed to perform sensitivity analysis  MultiDC, multiperiod multiproduct  Robust optimization  Scenariobased approach 
Mirzapour Alehashem et al. (2013)  Multisite  Consideration of green logistic in an uncertain APP model for the first time transportation mode choice with an objective to reduce green house gases (GHGs), multiple breakpoint discount and shortage penalty functions interrelationship between transportation cost and lead time  Multiperiod, multiproduct  MINLP converted to MILP  Twostage stochastic 
Aliev et al. (2007)  Multisite  Use of fuzzy model to deal with uncertainty in demand, production capacities and storage capacities tradeoff between profit maximization and fill rate  Single and multiperiod, multiproduct  Fuzzy mathematical  GA 
Pathak and Sarkar (2012)  Multisite  Development of a model to minimize the subcontracted units in each planning period and making sure of no inventory when planning horizon ends  Multiproduct, multiDC, multiperiod  Fuzzy programming  Possibilistic LP (fuzzy MILP) 
Gholamian et al. (2015)  Multisite  Development of fuzzymathematical model to incorporate four objectives  Multiobjective, multiproduct  Mixedinteger linear programming/fuzzy optimization  LZL approach, TH approach 
Gholamian et al. (2016)  Multisite  Developing a model that considers existing fuzziness (in data and inequalities) and insufficient knowledge of data  Multiobjective, multiproduct, multicustomer  Fuzzy mathematical  Deterministic 
Entezaminia et al. (2016)  Multisite  Making tradeoff between production costs and green principles  Multiobjective, multiperiod, multiproduct  Mathematical  LPmetrics method 
Jing et al. (2016)  Multisite  Consideration of remanufacturing and backordering  Multiproduct, multiperiod, multimarket  Mathematical  GA 
This research  Single, multiplants  Cooperation between manufacturers through labor and inventory to reduce costs and share profit of saving among colleagues  Multiperiod, multiplants  Game theory approach  Exact 
APP models classification based on the type of data and the number of objective functions
Deterministic/uncertainty  Objective function  Articles  

Deterministic  Single  Aghezzaf and Artiba (1998), Silva et al. (2000), Pradenas et al. (2004), Fahimnia et al. (2005), Piper and Vachon (2001), Singhvi and Shenoy (2002), Techawiboonwong and Yenradee (2003), Wang and Yeh (2014), Chaturvedi and Bandyopadhyay (2015), Erfanian and Pirayesh (2016), Chaturvedi (2017)  
Multiple  Leung and Chan (2009), Ismail and ElMaraghy (2009), Chakrabortty and Hasin (2013), Entezaminia et al. (2016), Abu Bakar et al. (2016), Mehdizadeh et al. (2018)  
Uncertain  Fuzzy  Single  Chen and Huang (2010), Liang et al. (2011), Chen and Huang (2014), Iris and Cevikcan (2014), Chakrabortty et al. (2015) 
Multiple  Madadi and Wong (2014), Gholamian et al. (2016), Fiasché et al. (2016), Chauhan et al. (2017), Zaidan et al. (2017), Mosadegh et al. (2017)  
Stochastic  Single  Mirzapour AleHashem et al. (2013), Jamalnia and Feili (2013), Ning et al. (2013), Entezaminia et al. (2016), Makui et al. (2016), Zhu et al. (2018)  
Multiple 
Survey of cooperative gametheoretic models
The methods of CGT can be used for assigning the cost saving to cooperating companies. Charles and Hansen (2008) proposed a theoretic costsaving framework based on CGT for global cost minimization and cost assignment in an enterprise network. They proved that under classical concave cost functions for all participants, the cost allocation computed by the activitybased costing method is rational and stable. Frisk et al. (2010) evaluated costsaving opportunity of cooperation among several forest companies in Sweden. They used the methods of CGT to fairly distribute total cost saving among participants. Lozano et al. (2013) adopted CGT to recognize the costsaving opportunities of different logistic companies that may be achieved when they merge their transportation requirements. They suggested a linear transportation problem to quantify the cost savings of the possible coalitions. Hennet and Mahjoub (2010) provided convincing interpretations of fair sharing of profit in a supply network formation. Hafezalkotob and Makui (2015) studied cooperation benefits of owners of logistic network under capacity uncertainty. They showed that the flow of logistic network becomes more reliable when the owners make coalitions. They also presented a number of CGT methods to allocate benefits of cooperation to the owners. Similarly, Naseri and Hafezalkotob (2016) evaluated cooperative network flow problem with pricing decisions. They investigated shapely value, τvalue, core center and minmax core methods to allocate the extra benefits of cooperation and compared the results. By considering total supply chain inventory costs, Mohammaditabar et al. (2016) used CGT to evaluate decentralized supplier selection problem between a buyer and a set of supplier. They found that a stable solution for the cooperative model exists that yields total supply chain cost as the centralized model. Zibaei et al. (2016) proposed a mathematical model for a vehicle routing problem that was managed by multiowners. The cost savings obtained from cooperation among owners were computed, and cooperative game theory methods were presented for allocating the cost savings to the cooperating owners. Fardi et al. (2019) developed a mixedinteger programming (MIP) formulation for a cooperative inventory routing problem (CoIRP) considering uncertainty, and the methods of CGT can be used for assigning the cost saving to cooperating companies. Baogui and Minghe (2017) developed a differential oligopoly game, and the impact of oligopoly which product prices are sticky and water right trading occurs is used to study. Cellini and Lambertini (2007) investigate a dynamic oligopoly game where goods are differentiated and prices are sticky, while profits are increasing in a larger level of production and the speed of price adjustment. Heidari Gharehbolagh et al. (2017) investigate the model of maximum flow problem in the presence of many unreliable sources with the objective of participating in the game.
The review table of the game theory in research dimensions
Articles  Allocation of profits  Space of the game  Research dimensions  

Transportation  Production planning  Inventory  
Fardi et al. (2019)  ✓  H  ✓  ✓  
Zibaei et al. (2016)  ✓  H  ✓  ✓  
Mohammaditabar et al. (2016)  H  ✓  
Naseri and Hafezalkotob (2016)  ✓  H  ✓  ✓  
Hafezalkotob and Makui (2015)  H  ✓  
Lozano et al. (2013)  ✓  H  ✓  
Frisk et al. (2010)  ✓  H  ✓  
Charles and Hansen (2008)  H  ✓  
Fei (2004)  ✓  H  
Baogui and Minghe (2017)  Oligopoly  ✓  
Cellini and Lambertini (2007)  ✓  Oligopoly  ✓  
René et al. (2018)  ✓  H  ✓  
FiestrasJaneiro et al. (2011)  H  ✓  
Razmi et al. (2018)  ✓  H  ✓  
Fathalikhani et al. (2018)  H  
Heidari Gharehbolagh et al. (2017)  ✓  H  ✓  
This research  ✓  H  ✓  ✓ 
Research gap
Reference to conducted literature review, there is lack of study based on aggregate product planning among manufacturing plants and the studies mostly focused on APP in a plant or multisite that aim to reduce the production cost in terms of labor cost, contractual cost, etc., satisfying the demands (public demands is dividable along sites) or can be executed by upgrading the logistic services such as transportation. Therefor this study tries to reduce the production cost using cooperation and coordination among multiplants regarding sharing of inventories and workforce to increase the satisfaction of demands (the demands of each plant are independent of the other plants) that eventually would be beneficial for whole plants and facilities, and various costs of production will contribute through them.
The continuous and unnecessary changes of workforce (hiring/firing) applied to previous models, cause some limitations and prohibitions by the governing rules of labor union and increase the cost at the national level (the rules for paying unemployment benefits; more info: Jimeno et al. 2018). In this model, CoAPP with possibility of workforce substitution among plants has advantages like preventing the multitude of recruitment and layoff, enhancement of the job security and reducing the hiring cost and firing’s penalties. On the other hand, training of expert workforce is a time and costconsuming task. The substitution of work force in addition to reducing the cost in terms of mentioned items will increase the production efficiency. The substitution of products among the plants in seasonal and emotional demands needs to be answered during the application period, or in cases where production facilities are not capable of producing high capacity due to lack of storage space, increased satisfaction of demand estimation and reduced accumulation capital and production costs and ultimately high profits (winwin games). Prior to commencement of the game, the plant by investigation about benefits of cooperation has the right to choose whether they want to cooperate or not. In multisite model regarding the cost of implementation such as the overhead cost of establishment, the declining of cooperation was impossible.
The cost saving of CoAPP is quantified by a new mathematical programming model for coalitions of production plants, which indicates the result of cooperation synergy. We propose several methods of CGT to calculate the positive results of saving.
Prerequisites and assumptions
The CoAPP is formulated based on the following assumptions:
Assumption 1
The horizon of production planning is finite and can be characterized by multiple periods of time. The time frameworks of production planning procedure are similar for the production plants.
Assumption 2
Parameters of production planning procedure of all plants are deterministic. Moreover, the forecasts of demands and parameters relating to employment, overtime and inventory levels about all companies are known in advance.
Assumption 3
Mathematical programming models for traditional and CoAPP are formulated based on the linear relationship in input data. We also assume that the plants desire very high level of customer service; thus, all demands of customers should be met. Consequently, revenues obtained over the planning horizon are fixed, and minimizing cost over the planning horizon in the objective function is equivalent to maximizing profit.
Assumption 4
The products of different firms (plants) are fully substitutable. That is, when several production plants form a coalition, the product demand from the coalition can be fulfilled by inventory of each member. Thus, the inventory of the members of a coalition can be managed in a centralized manner.
Assumption 5
We consider cooperation among homogeneous manufacturers. It means that the production process and technologies of plants are similar such that an employee of one plant can work in other plants. This assumption can be especially acceptable in lowtech industries such as brick production, canned fish production. When the production plants join the coalition, they may exchange their workforce to improve their production planning efficiency.
Assumption 6
According to the main assumption of CGT, the utility obtained from CoAPP model is considered as transferable. The simplifying assumption is frequently made in the CGT (Myerson 1991); thus, these games are also called transferable utility games (TUGs).
Mathematical programming approaches
Basic aggregate production planning problem
 t

the index of time period,
 \(C_{\text{H}}\)

the cost of hiring a worker,
 \(C_{\text{F}}\)

the cost of firing a worker,
 \(C_{\text{W}}\)

the wage cost of a worker in a period,
 \(C_{\text{R}}\)

the unit production cost per period on regular time,
 \(C_{\text{O}}\)

the unit production cost per period on overtime,
 \(C_{\text{I}}\)

the cost per period of carrying one unit of inventory,
 H _{ t}

the number of workers hired in period t,
 F _{ t}

the number of workers fired in period t,
 P _{ t}

the number of units produced on regular time in period t,
 O _{ t}

the number of units produced on overtime in period t,
 W _{ t}

the number of workers employed in period t,
 I _{ t}

the number of units stored in inventory (onhand inventory) at the end of period t,
 D _{ t}

the number of units of demand in period t,
 A _{1}

the number of units that one worker can produce in a period on regular time,
 A _{2}

the maximum number of units that one worker can produce in a period on overtime,
 A _{3}

the initial workforce level,
 A _{4}

the initial inventory level,
 A _{5}

the desired workforce level at the end of the planning horizon,
 A _{6}

the desired inventory level at the end of each period,
 T

the number of time periods over the planning horizon
Cooperative aggregate production planning problem
 t

the index of time period,
 k

the index of production plants (i.e., players),
 \(C_{{{\text{H}},k}}\)

the cost of hiring a worker by plant k,
 \(C_{{\text{F},k}}\)

the cost of firing a worker by plant k,
 \(C_{{{\text{W}},t}}\)

the wage cost of a worker in a period by plant k,
 \(C_{{\text{R},k}}\)

the production cost per period on regular time of plant k,
 \(C_{{\text{O},k}}\)

the production cost per period on overtime of plant k,
 \(C_{{{\text{I}},k}}\)

the cost per period of carrying one unit of inventory by plant k,
 H _{ k, t}

the number of workers hired in period t by plant k,
 F _{ k, t}

the number of workers fired in period t by plant k,
 P _{ k, t}

the number of units produced by plant k on regular time in period t,
 O _{ k, t}

the number of units produced by plant k on overtime in period t,
 W _{ k, t}

the number of workers employed by plant k in period t,
 I _{ k, t}

the number of units stored in inventory (onhand inventory) by plant k at the end of period t,
 D _{ k, t}

the number of units of demand for products of plant k in period t,
 A _{1}

the number of units that one worker can produce in a period on regular time,
 A _{2}

the maximum number of units that one worker can produce in a period on overtime,
 A _{ k,3}

the initial workforce level of plant k,
 A _{ k,4}

the initial inventory level of plant k,
 A _{ k,5}

the desired workforce level for plant k at the end of the planning horizon,
 A _{ k,6}

the desired inventory level for plant k at the end of each period,
 T

the number of time periods over the planning horizon
We note that APP (1)–(7) is a simple mathematical programming problem. Indeed, CoAPP (8)–(14) is a development of APP for multiple companies. Therefore, only one dimension (for companies as players) is added to the traditional APP model. Objective function (8) computes the sum of production costs for all cooperating companies, and constraints (9)–(14) are actually developed forms of (2)–(7) for the cooperating companies.
We note that traditional APP (1)–(7) is a linear programming (LP) problem with real decision variables. Thus, the problem can be effectively and quickly solved by solver package such as Lingo (or GAMS). CoAPP (8)–(14) also adds a dimension to the traditional APP; therefore, it is simple LP, as well. Because CoAPP is LP with real variables, it is not a complex problem (neither NPhard nor NPcomplete) and can be solved in a polynomial time with linear programming solver package. Therefore, CoAPP with high number of companies can be effectively solved.
The larger the cost saving CS(S_{m}), the higher the synergy of plants will be. Equation (17) can be adopted to evaluate and quantify the synergy of CoAPP of each coalition of plants. Thus, it can be utilized as an argument to motivate this type of cooperation.
We present a real numerical example to evaluate and quantify the synergy among production plants. We study a canned fishproducing company in the southern coast of Iran. This company has established three production plants that process the fresh tuna fishes. The company sells the products of the plants with three different brand names; however, these products are substitutable in the market. Because the production processes and products of the plants are almost similar, the plants can interchange inventories and workers to reduce the production costs.
The demand forecast (unit) of three plants in the numerical example
Production plant  January  February  March  April  May  June 

1  1000  3300  5800  3200  2200  1000 
2  1500  2400  3000  3500  4400  6000 
3  5500  3000  2500  2200  1700  1500 
The parameters of three plants in the numerical example
Production plant  C_{H,k} $/worker  C_{w,k} $/worker  C_{F,k} $/worker  C_{R,k} $/hour  C_{O,k} $/hour  C_{I,k} $/unit  A_{k,3} worker  A_{k,4} unit  A_{k,5} worker  A_{k,6} unit 

1  3500  1000  4000  30  40  15  100  1000  140  0 
2  3500  1000  4000  30  40  15  100  1200  190  0 
3  3500  1000  4000  30  40  15  100  1100  150  0 
Optimal results of CoAPP for each of the possible coalitions
Coalition  Months  

0  1  2  3  4  5  6  
S_{1} = {1}  D _{{1} ,t}  0  1000  3300  5800  3200  2200  1000 
I _{{1} ,t}  1000  3075  2850.00  125.00  0  0  0  
P _{{1} ,t}  0  1708.33  1708.33  1708.33  1708.33  1400.00  1000.00  
O _{{1} ,t}  0  1366.67  1366.67  1366.67  1366.67  800.00  0.00  
W _{{1} ,t}  100  170.83  170.83  170.83  170.83  140.00  140.00  
H _{{1} ,t}  0  70.83  0.00  0  0  0  0  
F _{{1} ,t}  0  0  0  0  0  30.83  0  
S_{2} = {2}  D _{{2} ,t}  0  1500  2400  3000  3500  4400  6000 
I _{{2} ,t}  1200  2200  3220.00  3640.00  3560  2580  0  
P _{{2} ,t}  0  1388.89  1900.00  1900.00  1900.00  1900.00  1900.00  
O _{{2} ,t}  0  1111.11  1520.00  1520.00  1520.00  1520.00  1520.00  
W _{{2} ,t}  100  138.89  190.00  190.00  190.00  190.00  190.00  
H _{{2} ,t}  0  38.89  51.11  0  0  0  0  
F _{{2} ,t}  0  0  0  0  0  0  0  
S_{3} = {3}  D _{{3} ,t}  0  5500  3000  2500  2200  1700  1500 
I _{{3} ,t}  1100  0  0.00  0.00  0  0  0  
P _{{3} ,t}  0  2444.44  1666.67  1500.00  1500.00  1500  1500.00  
O _{{3} ,t}  0  1955.56  1333.33  1000.00  700.00  200  0.00  
W _{{3} ,t}  100  244.44  166.67  150.00  150.00  150  150.00  
H _{{3} ,t}  0  144.44  0.00  0  0  0  0  
F _{{3} ,t}  0  0  77.78  16.67  0  0  0  
S_{4} = {1, 2}  D _{{1,2} ,t}  0  2500  5500  8800  6700  6600  7000 
I _{{1,2} ,t}  2200  4900  5340.00  2480.00  1720  1060  0  
P _{{1,2} ,t}  0  2888.889  3300.00  3300.00  3300.00  3300.00  3300.00  
O _{{1,2} ,t}  0  2311.111  2640.00  2640.00  2640.00  2640.00  2640.00  
W _{{1,2} ,t}  200  288.8889  330.00  330.00  330.00  330.00  330.00  
H _{{1,2} ,t}  0  88.88889  41.11  0  0  0  0  
F _{{1,2} ,t}  0  0  0  0  0  0  0  
S_{5} = {1, 3}  D _{{1,3} ,t}  0  6500  6300  8300  5400  3900  2500 
I _{{1,3} ,t}  2100  1933.33  1966.67  0.00  0  0  0  
P _{{1,3} ,t}  0  3518.52  3518.52  3518.52  3000.00  2900.00  2500.00  
O _{{1,3} ,t}  0  2814.81  2814.81  2814.81  2400.00  1000.00  0.00  
W _{{1,3} ,t}  200  351.85  351.85  351.85  300.00  290.00  290.00  
H _{{1,3} ,t}  0  151.85  0.00  0  0  0  0  
F _{{1,3} ,t}  0  0  0  0  51.85185  10.00  0  
S_{6} = {2, 3}  D _{{2,3} ,t}  0  7000  5400  5500  5700  6100  7500 
I _{{2,3} ,t}  2300  0  320.00  940.00  1360  1380  0  
P _{{2,3} ,t}  0  2611.11  3177.78  3400.00  3400.00  3400.00  3400  
O _{{2,3} ,t}  0  2088.89  2542.22  2720.00  2720.00  2720.00  2720  
W _{{2,3} ,t}  200  261.11  317.78  340.00  340.00  340.00  340  
H _{{2,3} ,t}  0  61.11  56.67  22.22222  0  0  0  
F _{{2,3} ,t}  0  0  0  0  0  0  0  
S_{7} = {1, 2, 3}  D _{{1,2,3} ,t}  0  8000  8700  11,300  8900  8300  8500 
I _{{1,2,3} ,t}  3300  2980  2920  260  0  0  0  
P _{{1,2,3} ,t}  0  4266.67  4800  4800  4800  4800  4800  
O _{{1,2,3} ,t}  0  3413.33  3840  3840  3840  3500  3700  
W _{{1,2,3} ,t}  300  426.67  480  480  480  480  480  
H _{{1,2,3} ,t}  0  126.67  53.33333  0  0  0  0  
F _{{1,2,3} ,t}  0  0  0  0  0  0  0 
Optimal cost of CoAPP and synergy for each of the possible coalitions
Coalition  TC(S_{m})  CS(S_{m})  Synergy (S_{m}) 

S_{1} = {1}  2,068,000.00  0  0 
S_{2} = {2}  2,425,000.00  0  0 
S_{3} = {3}  2,521,833.33  0  0 
S_{4} = {1, 2}  4,061,500.00  431,500.00  0.11 
S_{5} = {1, 3}  4,046,888.89  542,944.44  0.13 
S_{6} = {1, 3}  3,925,500.00  1,021,333.33  0.26 
S_{7} = {1, 2, 3}  5,631,900.00  1,382,933.33  0.25 
 (i)
In all periods of planning horizon, the plants’ workforce level of basic APP is higher than workforce level of CoAPP problem. Furthermore, the inventory of production plants reduces when the plants cooperate. Therefore, the exchanges of inventory and workforce decrease the total cost of CoAPP.
 (ii)
In the grand coalition, the fluctuation in workforce levels is lower than noncooperative situation. In particular, the number of dismissals reduces because of CoAPP (i.e., instead of dismissal, the workers can be exchanged among the cooperating plants). Therefore, the job security and satisfaction of workers can be dramatically increased because of plants’ cooperation.
 (iii)
Coalition among plants may result in a significant cost saving (i.e., 25%) which is a convincing argument to motivate cooperation.
 (iv)
Table 4 demonstrates that the collaborative effects of coalitions are not equal for their plants. For instance, considering the viewpoint of plant 1, joining to plant 2 generates lower synergy (11%) as compared to plant 3 (26%). These differences depend on the pattern of demands and cost parameters of the plants.
Collaborative frameworks for CoAPP
Once the total cost, cost saving and synergy are computed for all coalitions of the production plants, the problem is addressing this question “how to distribute the cost saving of the cooperation among different plants?” This is not a simple problem because it is not clear how much the contribution of each plant to the cost saving of a coalition is. Thus, we require a theoretically grounded approach and the one most appropriate and wellknown would be CGT (Reinhardt and Dada 2005; Bartholdi and KemahlıogluZiya 2005; Lozano et al. 2013; Asgari et al. 2013; Frisk et al. 2010; Hafezalkotob and Makui 2015; Mohammaditabar et al. 2016). For this purpose, some basic definitions and concepts related to CGT are briefly reviewed first; then, we will use these solution concepts in CoAPP. Even though several CGT solution concepts exist, we will focus here only on some of them including the Shapley value, the equal cost saving method (ECSM), the minmax core and the τvalue.
The main challenge of CGT is to assign payoff CS(K) among the players in a fair manner. Based on different interpretations of fairness, earlier researchers have proposed different solution concepts. We adopted some of them for the costsaving allocation problem of CoAPP, but the interested readers may refer to Barron (2013), Branzei et al. (2008), Gilles (2010) and Lozano et al. (2013) for more information.
The z variable measures the largest difference between costsaving assignments (see constraint (26)) that should be minimized in objective function (25). Constraints (27)–(28) ensure the stability of the assignment because it should belong to core space.
Assignment of the coalition cost saving according to methods of CGT
Plant  Shapley  M _{ k}  m _{ k}  τvalue  ECSM  The least core 

{1}  282,940.74  361,600  0  215,176.92  361,600  180,800 
{2}  522,135.19  839,988.89  6970  528,155.98  510,666.7  659,188.90 
{3}  577,857.41  951,433.33  181,344.44  639,600.42  510,666.7  542,944.40 
Stable  Yes  –  –  Yes  Yes  Yes 
Coalition satisfactions for different methods of CGT
Coalition  Shapley  τvalue  ECSM  The least core 

C_{1} = {1}  282,940.74  215,176.92  361,600.00  180,800.00 
(14%)  (10%)  (17%)  (9%)  
C_{2} = {2}  522,135.19  528,155.98  510,666.70  659,188.90 
(22%)  (22%)  (21%)  (27%)  
C_{3} = {3}  577,857.41  639,600.42  510,666.70  542,944.40 
(23%)  (25%)  (20%)  (22%)  
C_{4} = {1, 2}  373,575.93  311,832.90  440,766.70  408,488.90 
(9%)  (8%)  (11%)  (10%)  
C_{5} = {1, 3}  317,853.71  311,832.90  329,322.26  180,799.96 
(8%)  (8%)  (8%)  (4%)  
C_{6} = {2, 3}  78,659.27  146,423.07  0  180,799.97 
(2%)  (4%)  (0%)  (5%)  
Min \(F_{{s_{m} }} (CS,\overrightarrow {x} )\)  78,659.27  146,423.0  0  180,799.96 
(Min \(F_{{s_{m} }} (CS,\overrightarrow {x} )/TC(S_{m} )\))  (2%)  (4%)  (0%)  (4%) 
Max \(F_{{s_{m} }} (CS,\overrightarrow {x} )\)  577,857.41  639,600.42  510,666.70  659,188.90 
(Max \(F_{{s_{m} }} (CS,\overrightarrow {x} )/TC(S_{m} )\))  (23%)  (25%)  (21%)  (27%) 
Sum \(F_{{s_{m} }} (CS,\overrightarrow {x} )\)  2,153,022  2,153,022  2,153,022  2,153,022 
(Sum \(F_{{s_{m} }} (CS,\overrightarrow {x} )/TC(S_{m} )\))  (77%)  (77%)  (77%)  (77%) 
The least core method intends to maximize the minimum satisfaction of the coalitions. Table 9 also demonstrates the minimum satisfaction among the coalition for each method. Because of the definition of the least core, this method provides the largest minimum satisfaction (i.e., 180,799.96 (4%)). Therefore, the least core technique imposes fairness via the maximization of the minimum satisfaction of all coalitions of plants.
From the results of the numerical example, we know that CGT approach presents useful tools to assign extra benefits of cooperation of plants. The methods of CGT help to choose the best allocation system to maximize the plants’ satisfaction. The fair allocation of the extra benefits encourages the production plants to continue their participation.
Similarity between solutions of CGT methods, measured by MAD
Coalition  Shapley value  τvalue  ECSM  The least core 

Shapley value  –  0.29  0.34  0.59 
τvalue  –  –  0.64  0.57 
ESCM  –  –  –  0.78 
Least core  –  –  –  – 
Conclusion and further research
The traditional APP models have often been studied for analyzing the production planning of one production plant. This paper presented a new mathematical programming model for APP problems of multiple cooperating plants. We quantified the costsaving opportunity of the cooperation of plants caused by decreases in inventory and workforce levels. It was found that the job security and satisfaction of workers can be dramatically raised because of plants’ cooperation. Several methods of CGT including Shapley value, τvalue, the least core and equal cost saving methods were utilized for assignment of cost saving to cooperating plants. We found that fair allocation of cooperation cost saving can ensure the production plants satisfaction.
Various directions and suggestions exist for future research in the field. First of all, this study considers that inventories and workforce are fully interchangeable among cooperating production plants. However, in some real situations, products and workforce may be partially substitutable; thus, considering this assumption can be a fascinating extension of the study. Secondly, generalizing the proposed model to take account of uncertainty over cost and/or demand parameters is also an interesting extension. Finally, this study assumes that the cost parameters of the plants are common knowledge; however, it is unlikely that the plants would be privy to the real cost parameters. This situation would lead to a collaborative game model under asymmetric information that is interesting but challenging.
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