یک مدل برنامه ریزی آرمانی دوسویه و یک الگوریتم تجزیه بندرز اصلاح شده برای بازپیکربندی زنجیره تامین و انتخاب تامین کننده

The problem addressed in this paper is related to an aerospace company seeking to change its outsourcing strategies in order to meet the expected demand increase and customer satisfaction requirements regarding delivery dates and amounts. A bilinear goal programming model is developed to achieve the company's objectives. A modified Benders decomposition method is successfully applied to handle bilinear goal programming models in which the complicating binary variables affect the values of the deviational variables of goals attainment. This influence leads to formulate the master and the sub-problem as two goal programming models with different objective function decompositions as compared to the classical Benders one. Computational experiments show that the modified Benders algorithm outperforms a generic linearization method by reaching the optimal solution for larger problem with about 75% reduction in computation time.

Improving the efficiency of supply chain partners has become a major requirement of any supply chain due to highly competitive nature of the current marketplace. This requirement may lead the decision makers to reconfigure their supply chains towards achieving a better performance of their incorporated tiers. The decision of reconfiguring the supply chain includes which suppliers to select, how to distribute materials among them, and how to better allocate their capacities. Hence, the supply chain reconfiguration problem presents itself as a major challenge in this regard. There are many reasons to which the need for supply chain reconfiguration may be attributed. For example, the rapid advancement of technologies in the computer industry is the main driver behind reconfiguring the Digital Equipment Corporation supply chain, Arntzen et al. (1995). The new strategy reduced the cumulative cost by $1 billion, the assets by $400 million and increased the unit utilization by 500%. P&G's supply chain has been reconfigured to optimize the product sourcing problems, Camm et al. (1997). After two years of implementing modeling recommendations, 12 sites have been closed and annual savings have reached $250 million per year. The BASF North America's distribution system is also a good example for a company that realized great benefits from reconfiguring its network, Sery et al. (2001). The main objectives of reconfiguring their network are to reduce the distribution costs and provide a sufficient level of customer service. The proposed model outcomes resulted in cost savings of $10 millions and increasing the percentage of volume delivered within one day from 77% to 90%. Hewlett-Packard (HP) achieved cost savings of $10 million by reducing the number of contract manufactures, Laval et al. (2005). For a divergent supply chain reconfiguration, Vila et al. (2006) analyze the raw materials processing when there is a limited and regulated availability of raw materials. The study was applied in a partnership with three large Canadian lumber companies and a 15.4% increase of after tax profits was attained. In this paper, we are dealing with a supply chain that needs to be reconfigured as a result of increasing customer demand and poor on time delivery performance of the company. The company looks for the minimum cost policy that better utilizes suppliers’ capacities in order to face the demand increase, and improve the on time delivery performance of the chain. A bilinear goal programming reconfiguration model is developed to achieve these new objectives. The proposed model is solved through decomposition using a modified Benders decomposition technique that decomposes the model into a binary supplier selection model and a mixed-integer distribution planning model. The paper is organized as follows; the next section is a review of the related literature to the problem of interest and the used approach. The problem is defined in Section 3 while Section 4 describes the proposed goal programming model to handle this problem. Section 5 illustrates the generic linearization scheme proposed by Peterson (1971) and the proposed algorithm is explained in Section 6. Section 7 shows the computational efficiency of the developed method compared to Peterson's linearization scheme. Then the paper ends with the summary and conclusion.

In this paper, a supply chain reconfiguration and multi-criteria supplier selection problem is addressed. The problem is formulated as a bilinear goal programming model which aims at achieving three objectives. The model represents a real case of reconfiguring an existing supply chain of an aerospace company. The three objectives reflect the management vision for the new reconfigured supply chain that is required to meet the expected demand increase, overcome the customer dissatisfaction form late deliveries and minimize the distribution cost. The outcomes of the model suggest a new strategy that optimally utilizes suppliers’ capacities and assigns as much material as possible to reliable suppliers in order to improve the on-time delivery performance. These improvements may lead to an increase in the distribution cost as compared to the current strategy aiming at minimizing the cost as a single criterion. The distribution cost increase is expected as it conflicts with the other two goals that aim at improving the on-time delivery performance of the supply chain. Although the proposed model reconfigures an existing supply chain, it can be applied to configure new chains considering the suppliers as candidates among them the model selects the best. The proposed model is decomposed into a master and a sub-problem to resolve the bilinearity resulting from multiplying a binary variable by a continuous one. The master problem seeks the optimal values of the complicating binary variables while the sub-problem optimizes values of the non-complicating variables for those given values of the complicating variables. The objective functions of these master and sub-problem have different formulation from those objectives of the classical Benders approach. A modified Benders decomposition technique is developed to solve such goal programming models. The upper and lower bounds applied to the classical Benders approach do not apply in our case due to differences in objective functions. As such, an optimality condition is used to check for optimality at the end of each iteration. The modified technique efficiently reaches the optimal solution of larger size models with a reduction in solution time ranging from 56% to 89% as compared to a classical linearization approach. The algorithm can be generally applied to bilinear goal programming models in which the complicating variables directly affect the minimization of the deviational variable associated with each goal. This research overcomes the difficulties associated with direct implementation of classical Benders approach to solve such models by adapting the algorithm to cope with a master and a sub-problem formulated as two goal programming models having different objective function decomposition compared to the traditional Benders one.