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A bi-objective model for collaborative planning in dyadic supply chain

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The collaborative planning and the management of production and storage processes are important components in supply chain management. The goal of this paper is to present the reliability of genetic algorithms on solving bi-objective models compared
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  DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633-638 A bi-objective model for collaborative planning in dyadic supply chain Hamza Ben Abdallah Université de Tunis El Manar, Faculté des Sciences de Tunis, LIP2-LR99ES18, 2092, Tunis, Tunisia Hamzabenabdallah88@yahoo.fr Zied Bahroun ESM Graduate Program, College of Engineering American University of Sharjah, P.O. Box 26666, Sharjah, United Aarab Emirates zbahroun@aus.edu  Naoufel Cheikhrouhou Laboratory for Production Management and Processes Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland naoufel.cheikhrouhou@epfl.ch Mansour Rached Université de Tunis El Manar, Faculté des Sciences de Tunis, LIP2-LR99ES18, 2092 Tunis, Tunisia mansour.rached@hotmail.com Abstract   —  The collaborative planning and the management of production and storage processes are important components in supply chain management. The goal of this paper is to   present the reliability of genetic algorithms on solving bi-objective models compared to mono-objective models. To do this we will be based initially on the mono- objective Dudek’s model and then we propose a division of the objective function in two objective functions. Finally we compare the results given by the genetic algorithms with the optimality result obtained using the LINGO solver on the mono- objective Dudek’s model.   . This model aims at simultaneously minimizing the total production cost and the total holding cost. To solve the proposed model, we use a genetic algorithm NSGA-II. The proposed several test provide results that demonstrate and validate the effectiveness of the multi-objective approach and elitists genetic algorithms in solving this type of problem, compared to the literature in the proposed test.   The validation of our approach will allow us later to use this algorithm in solving complex multi-objective models approaching the real context.   Keywords   —   Supply Chain Management, Collaborative Planning, Mathematical Programming, Multi-Objective Optimization Model, Elitist Genetic Algorithm NSGA II    I.   I  NTRODUCTION  Tactical planning is the determination of the quantities of  products to be manufactured per period in order to meet as well as possible the demand at lower costs. The issues differ mainly according to two criteria: mono-level planning (Master Production Scheduling of finished products) or multi-level  planning (finished products and components planning) and mono-site or multi-site planning. Commonly, the planning  problems are formalized by mathematical models known as "lot sizing" problems. Among them, the "Capacitated Lot Sizing Problem" (CLSP) is considered as a reference model to treat the problems of generation of master production scheduling in a mono-site context. For the multi-level  planning, "Multi Level Capacitated Lot Sizing Problem" (MLCLSP) is recognized as the reference model. If the mono-site problems were largely studied in the literature, the absence of a reference model for the multi-site issues is highlighted in [19]. This can be explained by the diversity of supply chains and treated problems. Nevertheless, from the multilevel nature of the multisite production, the models of the literature (see for instance [21] and [5]) are derived from the MLCLSP model. Integral part of the multisite planning  problems, collaboration of the production plans of the various  production sites has represented for several years a major stake of the supply chain optimization. The theoretical contribution of this work is to validate the elitist s’ genetic algorithms  to solving multi-objective problems in supply chain management, especially for the collaborative  planning problem . Given that the proposed model is that the Dudek’s model developed in 2007 with the duplication of the objective function so as to obtain a bi-objective model. So, our experimental study is based, firstly, on the comparison of results obtained from the resolution of the mono-objective Dudek’s model and the bi -objective Du dek’s model (which is the proposed model) by the NSGA-II algorithm to validate the  DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633-638  performance of the genetic approach in solving the multi-objective problems rather than mono-objective problems. Second, we tend to solve the mono- objective Dudek’s model  by LINGO solver to obtain an optimal solution and compare it with the solution obtained when solving the bi-objective model by the NSGA-II in order to judge the performance of adopting a multi-objective approach to this type of problem and assess the effectiveness of NSGA-II in solving this type of  problems. Thus, the purpose of this paper is to develop and to solve a multi-objective model for optimal collaborative multi- period planning in dyadic supply chain. The idea is to provide customer orders to the SC partners in order to coordinate and generate a global optimal production plan. In solving the multi-objective model, we use an elitist genetic algorithm  based on NSGA-II algorithm (Non-dominated Sorting Algorithm II) and compare it to a benchmark solution from the literature. The paper is further organized as follows. Related literature is reviewed in Section 2. Characteristics of the case  problem are presented in Section 3. The proposed model is detailed in Section 4. The resolution methodology is presented in section 4, followed by the computational results in section 5. A comparative study is done in section 6 to evaluate the  performance of the proposed model. Finally, summary and  potential research directions are drawn.   II.   M ULTI -O BJECTIVE P ROBLEMS IN S UPPLY C HAINS M ANAGEMENT :   A   L ITERATURE R  EVIEW  In the last decade, a large number of multi-objective optimization problems and resolution methods have been  presented in the literature on supply chain management. (Jayaraman, 1999) [29] developed a weighted multi-objective model for the equipment locating service problem to evaluate the compromise between the demand cover and the number of equipment. (Ertogral and Wu, 2000) [17] developed a coordination mechanism for the production planning in the various stages of the supply chain based on the Lagrangian relaxation of a global model of the chain. This global model is decomposed into specific sub-models with respect to the different actors. (Sabri and Beamon, 2000) [7] developed an integrated multi-objective model, for simultaneous, strategic and operational supply chain planning by taking account the supply chain cost, the customer service level and the flexibility of delivery as objectives. In [15], the authors developed a mathematical programming model in order to reduce the operating cost while maintaining the execution of customer orders at a high level for a supply chain network. In [4], the authors formulated production and distribution  planning model using multi-objective nonlinear mixed integer  programming for an equitable distribution of benefits through the supply chain. (Hugo et al., 2005) [1] proposed a based multi-objective optimization model for strategic investment  planning and supply chains design of hydrogen, considering  both investment and environmental criteria. In [18], the authors developed a multi-objective optimization method  based on simulation for the optimization problem of storage  policies in supply chains, taking into account the total holding cost and the service level. In [6], the authors solved a multi-objective linear programming problem at two levels for supply chains planning, using a technique adapted from fuzzy logic. (Chern and Hsieh, 2007) [2] proposed a heuristic algorithm for solving the central planning of supply chain problem, with three goals, including penalties, capacity utilization and the total cost. (Pokharel, 2008) [25] optimized the operation cost and reliability of delivery for supply chain design problem. (Raj and Lakshminarayanan, 2008) [28] examined three  performance criteria: customer satisfaction, breaking orders and excess inventories. [14] developed a multi-objective linear  programming model for production and distribution collaborative planning problem using the fuzzy goal  programming approach. (Liang, 2008) [26] developed a fuzzy multi-objective linear programming model to simultaneously minimize total costs and total delivery time in a supply chain  by adopting the fuzzy objective programming method. (You and Grossmann, 2008) [9] proposed a multi-period planning model for the design and the planning of supply chains under economic criteria with uncertainties in demand. (Torabi and Hassini, 2009) [23] considered the multi-objective optimization problem for the multi-level supply chain  planning. Franca et al. (2010) [22] proposed a multi-objective stochastic model considering both the benefits and defects of raw materials obtained from the supply chain. (Pinto-Varela et al. 2011) [27] used an optimization approach adapted from the symmetric fuzzy linear programming in order to solve a bi-objective model for the planning and design of supply chains  by considering the economic and environmental aspects. (Liu and Papageorgiou, 2012) [24] focused on the production  planning, the distribution, and global supply chains capacities, considering simultaneously the costs and the customer service level. They developed a multi-objective mixed linear  programming mixed approach with the total cost, the delivery time and the total lost sales as key objectives. (Ben Yahia et al., 2012) [30] studied the problem of a multi-objective optimization for multi-items cooperative supply chain multi- period planning. They have developed a bi-objective model for cooperative planning between different plants belonging to the same supply chain; this model aims at simultaneously minimizing the total production cost and the average inventory level considering multi-items production. We note that (Dudek and Stadtler, 2005) [10] proposed a coordination mechanism of a supply chain involving a supplier and a manufacturer, they have developed a mono-objective model for the Multi-Level Capacited Lot Sizing Problem. This model aims to minimize the total cost of supply chain which is equal to the sum of production cost and holding cost which are considered as objectives to our model that is described in the next sections. III.   P ROBLEM D ESCRIPTION  The most significant classical problem in relation to the type of problem under interest is the Multi-Level Capacited Lot-sizing Problem (MLCLSP).The objective is to determine, in a collaborative way, the quantities to be manufactured for  DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633-638 the various products over the planning horizon, in dyadic supply chain, in order to satisfy the demand (without breaking stock), with respecting the capacity constraints and to minimize the total cost by generating a global optimal  production plan. The supply chain considered is dyadic. Despite of the asymmetrical nature of information flows  between the actors of the supply chain, they share some, such as production capacity and production costs. Moreover, we assume for simplicity reasons, that the planning horizon of T time periods is the same for all supply chain partners and each  production site has eight work hours per day, but differs in the overtime available and the time of production of each product. IV.   P LANNING M ODEL  Based on the structure of the studied supply chain, assuming that each partner uses the model MLCLSP (multi-level capacitated lot-sizing problem) to generate its own local  production, neglecting the time setting as well as generate the global optimal production plan. The planning is based on reaching two main objectives; the first is the minimization of  production cost and the second is the minimization of holding cost.   Knowing that the objective function of the Dudek’s model is minimizing the total cost which is the sum of the  production cost and the holding cost and that both models have the same constraints, explains the equivalence between the bi-objective model detailed below and the mono-objective Dudek’s model . Our problem can be formulated as follows:    Set Indexes    T: Set of planning periods.    J: Set of operations.    M: Set of resources.    S  j : Set of direct successor operations of j.    Indexes    t: Planning period, t= l, ...,T.     j: O  peration, j = 1, …, J.      m: R  esource, m = 1, …, M.      Parameters    ch  j : Unit holding cost of operation j.    cv  j : Unit cost of operation j.    cf   j : Fixed setup cost of operation j.    co m : Unit cost of overtime (capacity expansion) at resource m.    D  j,t : (external) demand for operation j in period t.    B  j,t : Large constant.    C m,t : Capacity at resource m in period t.    I  j : Starting inventory of operation j.    a m,j : Unit requirement of resource m by operation j.    r   j,k  : Unit requirement of operation j by successor operation k.    Variables    C  p : Total production cost.    C s : Total holding cost.    x  j,t : Output of operation j in period t.    i  j,t : Inventory level of operation j at the end of  period t.    y  j,t : Setup variable of operation j in period t(y  j,t =1 if  product j is set up in period t ; y  j,t =0 otherwise).    o m,t : Overtime at resource m in period t.    The model Min (C  p , Cs) (1) S.t.   C  p = ∑∑[(cf  j ∗y j,t )+(cv j ∗x j,t )]+ ∈∈  ∑∑(co m ∗o m,t ) ∈∈   (1.1) C s  = ∑∑(ch j ∗i j,t ) ∈∈   (1.2) i j,t−1 +x j,t =D j,t +∑r j,k ∗x k,t +i j,t ∈    ∀  j ∈  J, t ∈  T   (2) ∑(a m,j ∗x j,t ) ∈ ≤C m,t ∗o m,t  ∀  m ∈  M, t ∈  T (3) x j,t ≤ , ∗y j,t  ∀  j ∈  J, t ∈  T (4) x j,t ≥0 ∀  j ∈  J, t ∈  T   (5) i j,t ≥0 ∀  j ∈  J, t ∈  T   (6) i j, =I j  ∀  j ∈  J   (7) o m,t ≥0 ∀  m ∈  M, t ∈  T   (8) y j,t ∈ {0, 1} ∀  j ∈  J, t ∈  T   (9) This is a program which aims to minimize the total  production cost (1.1), including manufacturing costs and execution costs, which results in a linear function of the changes of manufacturing (y  j,t ), the produced quantities (x  j,t ) and overtime on resources (o m,t ), and the total holding cost (1.2) which results in a linear function of inventory levels (i  j,t ). Constraints are used to describe:    evolution of inventory levels between two successive  periods (2): For each item, the sum of the stock at  period t-1 and the produced quantity at period t must satisfy the sum of internal (requirements of successor operations) and external demand (customer demand) of this article,    limit of production due to resources capacity (3),    coupling between decision variables of production and change series (assigning the value 1 to the variable execution when there is a production) (4),  DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633-638    that the variables representing the productions are  positive or null (5),    the fact that inventory levels are positive or null, it doesn’t allow as stock shortage (6),    The initialization of the opening inventories (7),    That overtime on resources are positive or null (8),    Whether there exists or not the production concerned over the period (9). V.   R  ESOLUTION OF T HE P ROPOSED B I -O BJECTIVE M ODEL  During the last decade, there has been a growing interest in using genetic algorithms (GA) to solve a variety of single as well as multi-objective problems in production and operations management that are considered as combinatorial and NP hard [3],[8], [13] and [20]. To solve our problem, we use a genetic algorithm based on NSGA-II (Non-dominated Sorting Algorithm II), (Deb 2002 [16]), which retains optimal Pareto solutions found over generations: Each population is distributed in several fronts by a fast sorting procedure. If the number of individuals in a new generation is less than the  population size, a crowding procedure is applied to the first next front F i  not included in this generation. Thus, a new  population is created by tournament selection, uniform crossover and mutation (binary and real).This GA has O (MN²) computational complexity (where M is the number of objectives and N is the population size) which is efficient from the computational point of view, compared to other Multi-objective Evaluation.  A.   Test Description The proposed test was developed using NSGA-II to compare the proposed model (which is the bi-objective Dudek’s model)  and the mono-objective Dudek’s model  which are equivalent. This test is designed where the demand for products is given and has to be fulfilled while facing finite capacities of staff and resources. Our model needs only 1000 iterations to converge to the optimal solution; however, Dudek’s model needs an important  number of generations to converge to the optimal solution. So, to work in the best condition for the comparison, the generation number used for the tests is equal to 2500 generations. Both of the resolution methods use the algorithm  NSGA-II. Set of tests: 2P.2P.3P In these tests 2P.2P.3P, the supply chain consists of two manufacturing Plants (2P) the planning is done over two  periods (2P) to provide three types of Products (3P) in order to satisfy the customers demand, respectively. The storage of finished products is done in the second  plant. The second product needs one item of the first operation to be produced. The third product needs one unit of operation 2 to be produced. The demand D  j,t , is represented in Table 1, where j is the product index and t is the period index. TABLE I. E XTERNAL DEMAND OF DIFFERENT ITEMS   D1,1 D2,1 D3,1 D1,2 D2,2 D3,2 90 40 50 80 60 70  B.    Experimental Results and Sensitivity Analysis   It should be noted that the proposed model means the model bi-objective Dudek  ’s model  detailed previously. To evaluate the proposed model and the solution quality provided  by the NSGA-II, a benchmark is done with respect to the mono-objective model developed by (Dudek 2007 [11]). So, in order to evaluate the performance of the adopted approach which is the use of genetic algorithms in solving multi-objective problems of supply chains management, we use the  NSGA-II algorithm to solve the models studied which are the mono- objective Dudek’s model  and the bi-objective proposed model equivalent to the Dudek’s one.  The results are interpreted in order to validate that the genetic algorithm  NSGA-II is more effective in solving the collaborative  planning problem by adopting a multi-objective approach than the resolution of the same problem by adopting a mono-objective approach. The generic solver LINGO 11.0 is used to provide the global optimal solution for the Dudek’s model in order to compare this solution to the obtained one for the proposed model using NSGA II: Since the optimal solution obtained by LINGO 11.0 provides the optimal total cost, which is the sum of the production cost and holding cost, then we can easily deduce the optimal values of two objective functions of the  proposed model so as to evaluate its performance and its optimality. We assume that all inventories at the starting  period are equal to zero. The constraints of the both models are the same. The objective function of Dudek’s model is to minimize the total cost (the sum total production cost C  p  and total holding cost C s  represented in equation (10). C=[cf  j ∗y j,t +cv j ∗x j,t +ch j ∗i j,t ] ∈∈ +co m ∗o m,t∈∈  (10) Thus, for this model, the total holding cost is optimized indirectly. When the total production cost is minimized and consequently the produced quantity level, the inventory level is indirectly minimized. With re spect to the Dudek’s model (2007), the model developed in our work gives an emphasis on minimizing the total holding cost by considering it as a second objective function. The results are visualized in Table 2 which contains the  produced quantities x  j,t , the used overtime o m,t , the inventory levels i  j,t , the objective functions (the total production cost C  p  and the total holding cost C s ) and the total cost C, both for Dudek’s model and the proposed model.    DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633-638 C.   Optimality of the Proposed Model It is clear that in the global optimal production plan  provided by the generic solver LINGO 11.0, the quantities  produced for the third item are equal to external demands. Moreover, the second resource does not need to use overtime in the two planning periods. It should be noted that in 75% of cases studied, the proposed model, solved with the NSGA-II algorithm gives an optimal total production cost equal to that  provided by the solver LINGO 11.0; this cost is equal to 13502.4 [UM]. One can deduce that starting from a population size greater than or equal to 150 and a number of generations equal to 1000, the proposed model provides an optimal  production cost. The best total holding cost provided by the  proposed model, in the studied cases, is equal to 12 [UM] with a population size equal to 300, while the optimal cost of holding is equal to 8 [UM]. So we can deduce that with  population sizes greater than 300, the proposed model will converge to the optimal solution with regard to the total holding cost Fig.1. TABLE II. T EST RESULTS FOR THE BI - OBJECTIVE MODEL ,  MONO - OBJECTIVE MODEL AND THE LINEAR PROGRAMMING LINGOPopulation size 100 150 200 300 Global optimal solution for mono-objective model provided by LINGO 11.0 solver Model Mono-objectve model with  NSGA-II Bi-objective model with NSGA-II Mono-objectve model with  NSGA-II Bi-objective model with NSGA-II Mono-objectve model with  NSGA-II Bi-objective model with NSGA-II Mono-objectve model with  NSGA-II Bi-objective model with NSGA-II  y  j,t    1 1 1 1 1 1 1 1 1  x 1,1   239 215 214 204 269 194 236 186 189  x 2,1   106 125 124 114 144 104 109 96 99  x 3,1   66 73 84 74 50 64 65 56 50  x 1,2   151 175 176 186 121 196 154 204 201  x 2,2   114 95 96 106 76 116 111 124 121  x 3,2   54 47 36 46 70 56 55 64 70 o 1,1   884 966 1080 780 1312 480 878 240 214 o 2,1   62 140 236 0 386 0 5 0 0 o 1,2 116 34 0 220 0 520 122 760 786 o 2,2   0 0 0 0 0 0 8 0 0 i 1,1   43 0 0 0 35 0 37 0 0 i 2,1   0 12 0 0 54 0 4 0 8 i 3,1   16 23 34 24 0 14 15 6 0 i 1,2   0 0 0 0 0 0 0 0 0 i 2,2   0 0 0 0 0 0 0 0 0 i 3,2   0 0 0 0 0 0 0 0 0 C  p 14432.4 15602.4 17842.4 13502.4 22412.4 13502.4 14372.4 13502.4 13502.4 C  s 96.5 58 68 48 106.5 28 89.5 12 8 C 14528.9 15660.4 17910.4 13550.4 22518.9 13530.4 14461.9 13514.4 13510.4 Figure 1. Convergence of the holding cost to the optimal solution  D.    Performance of the Adopted Approach One can note from these results that the total holding cost of proposed model is lower than the total holding cost of Dudek’s model by 100%. So, considering the total holding cost as an objective function and not as a constraint clearly improved the results. The per  formance of Dudek’s model is  better than the performance of our proposed model (that with regard to the total cost and in particular the total production cost). However, for all other tests and for all population sizes greater than or equal to 150, the performances reached by the  proposed model are bett er than those of Dudek’s model.  For example, for a population size equal to 200, the total holding cost (28) is lower than the Dudek’s model (106.5). Besides,
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