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  1. 1. J Intell Manuf DOI 10.1007/s10845-015-1130-9 Bi-objective mixed-integer nonlinear programming for multi-commodity tri-echelon supply chain networks M. H. Alavidoost1 Mosahar Tarimoradi1 M. H. Fazel Zarandi1,2 Received: 6 April 2015 / Accepted: 17 July 2015 Springer Science+Business Media New York 2015 Abstract The competitive market and declined economy have increased the relevant importance of making supply chain network efcient. Up to now, this has resulted in great motivations to reduce the cost of services, and simul- taneously, to improve their quality. A mere network model, as a tri-echelon, consists of Suppliers, Warehouses or Dis- tribution Centers (DCs), and Retailer nodes. To bring it closer to reality, the majority of parameters in this network involve retailer demands, lead-time, warehouses holding and shipment costs, and also suppliers procuring and stocking costs which are all assumed to be stochastic. The aim is to determine the optimum service level so that total cost is minimized. Obtaining such conditions requires determin- ing which supplier nodes, and which DC nodes in network should be active to satisfy the retailers needs, an issue which is a network optimization problem per se. The pro- posed supply chain network for this paper is formulated as a mixed-integer nonlinear programming, and to solve this complicated problem, since the literature for the related benchmark is poor, three numbers of genetic algorithm called Non-dominatedSortingGeneticAlgorithm(NSGA-II),Non- dominated Ranking Genetic Algorithm (NRGA), and Pareto B Mosahar Tarimoradi M. H. Alavidoost M. H. Fazel Zarandi 1 Department of Industrial Engineering, Computational Intelligent Systems Laboratory, Amirkabir University of Technology, 424 Hafez Ave, P.O. Box 15875-4413, Tehran, Iran 2 Knowledge Intelligent Systems Laboratory, University of Toronto, Toronto, Canada Envelope-based Selection Algorithm (PESA-II) are applied and compared to validate the obtained results. The Taguchi method is also utilized for calibrating and controlling the parameters of the applied triple algorithms. Keywords Supply chain management Tri-echelon network Mixed-integer nonlinear programming NRGA NSGA-II PESA-II Taguchi method Introduction Supply Chain Management (SCM) is a comprehensive approach that contains the processes like retailer demand management, order fulllment, manufacturing management, procurement, product commercialization, returns manage- ment, etc. (Rogers & Leuschner 2004). From another point of view, it might also involve both internal and external func- tions of an company that enables its value chain to make products and provide services for the retailers (Handeld and Nichols 1999). SC usually consists of retailers, distribution centers (DCs), plants, and suppliers. In common SCs theses are supposed that raw material should be primed, products are manufactured at one or more plants, commodities are sent to warehouses and nally they might be shipped for the retailers. SCM is faced with handling a network of inter-connected businesses involved in the ultimate provision of commod- ity so that packaging services could be carried by the end retailers. Thus with such an aspect, SCM or in better terms Supply Chain Network (SCN), envelopes all the require- ments for synchronizing activities like material priming, work in processing to nal products, and distribution of the manufactured products to retailers. The goals of SCN are usually known such as minimizing the system costs and sat- 123
  2. 2. J Intell Manuf isfying the service level requirements. Such a comprehensive system is a draft that depicts the quantities of commodities, location of DCs, and even the time for the production process. There are numerous autonomous identities each of which tries to satisfy their own objective in an SCN. Thus, trying to solve a real SC problem might be very hard and requires more than one objective to be satised. Such a problem is called a multi-objective optimization problem that has numerous Pareto solutions. Attaining to matters like lower costs, shorter processing time and lead-time, lower stock, larger commod- ity diversities, better reliable delivery time, improved quality, and priming the coordination between demand, procurement and manufacturing that are all known as KPI1 for business owners, need a proper and well-devised SCN. SCM could be summarized into three main processes: SC structuring, SC programming, and SC control and mon- itoring. In SC structuring, we make strategic plan such as plant location, capacity of plants and the quantity of materi- als that are required in producing operations or distribution among facilities. The focus of structuring in traditional SCM is mainly devoted to a single objective, such as minimiz- ing the cost or maximizing the prot, whilst a real SC has often to be optimized considering more than one. In fact, real SC problems can usually be formulated as a case of a multi-objective problem requiring an algorithm capable for searching the space of objectives in a short run-time. In this paper, an attempt is made to optimize a bi-objective tri-echelon multi-commodity SC problem. The proposed net- work would of some suppliers, DCs, and retailers nodes. Putting the existing models into practice and bring them to reality is the contribution of this paper. This is attained using more realistic and applied assumptions in terms of uncertainties involved in all of the three strategic, tactical, and structuring the proposed SCN levels. Depicting it in a more specic manner, the xed and variable costs, retailers demand, total available production time for plants, setup and production time of producing products, are all assumed to be stochastic internal parameters following the uniform distrib- utions; a common probability model that is suitable for many natural stochastic processes based on the central limit theo- rem. Moreover, the goal is to determine the active suppliers and DCs assumed as Boolean variables so that the optimum paths for retailers demands satisfaction could be achieved. In other words, this paper aims to determine the optimum network for satisfying retailers demands subject to the two goals of minimum cost and maximum service levels. The problem is formulated to obtain the deterministic model of a bi-objective MINLP.2 The proposed mathemati- cal model of this work is hard to be resolved by the common analytical or exact approaches, thus three of MOGAs are 1 Key Performance Indicator. 2 Mixed-Integer Nonlinear Programming. utilized to nd Pareto Fronts; and since the literature for benchmarks to validate the obtained solutions is poor, these applied algorithms called NSGA-II,3 NRGA,4 and PESA-II5 are compared together via six numbers of the cited indexes. The Taguchi DOE6 method is used for controlling and cal- ibrating these applied algorithms parameters. Finally, the numerical example is presented and detailed comparison results are exposed and discussed. The rest of this paper aims to explain the problem back- ground in section Problem background. Afterward in section Formulating the proposed supply chain problem the proposed problem has been formulated. Then the solving procedure consisting of the current approaches for dealing with SCN problems, applied algorithms and their character- istics are considered in section Solving procedure. Indexes for multi-objective evolutionary algorithm provided in sec- tion Comparison measures. After that, the experimental results and a comparison between triplex calibrated algo- rithms based upon the dened indexes are considered in sectionTheexperiments.Finally,theconclusionsandsome guidelines for future studies are provided in section Con- cluding remarks and implications for future works. Problem background In the past few years, it has become obvious that many com- panies have reduced operational costs as much as possible. They are discovering that effective SCM is the next neces- sary step to take in order to increase prots and market share (Simchi-Levi et al. 2003). In many of the classical SCN structuring, the goal is sending/receiving merchandise from/to a layer to/from the other(s) so that procuring costs for both strategic and opera- tional functions are minimized. As an instance, Amiri (2006) structured SC model for making the best strategic decisions on locating the plants and DCs for dispatching commodi- ties from manifesting site to the retailers side, coincide with the goal of minimum total costs of the DCs in the net- work. In another work, Gebennini et al. (2009) offered a three-layered manufacturingdispatching system for mini- mum costs. Network designing faces with relations between various SC portions together, are mutually under the risks and uncertainties through the whole chain; an issue that created a controversial problem for the SC decision-making process, so that the recent goals are propounded. The uncertainties involved in SC networks could be depicted into three divi- sions based on the supplier layer, the receiver layer, and in the DC layer. Since the reversible logistic decisions and their 3 Non-Dominated Sorting Genetic Algorithm. 4 Non-Dominated Ranking Genetic Algorithm. 5 Pareto Envelope-Based Selection Algorithm. 6 Design of Experiments. 123
  3. 3. J Intell Manuf relation to the SCN scaffolding is very difcult and costly, the momentous of the interactions between these decisions is vastly enhanced under uncertainty. Mohammadi Bidhandi and Mohd Yusuff (2011) formulated a stochastic SCN model as a two-level program considering both strategic and tactical decisions. In their model, the retailer demands, the opera- tion cost, and the capacity of facilities can be uncertain as they can all have tardiness effects on the strategic decisions. For the strategic level, Snyder (2006) considered an RFLP7 for locating DCs level o