We propose an MILP model for designing a generic supply chain network. Our model is able to handle multi-commodity through supply chain from upstream to downstream. We also integrate transportation modes selection into designing supply chain network.
Number of constraints and decision variables highly impress by the number of transportation modes.
Let’s suppose that there is only one mode available between each pair of nodes through network. The model has (J+K) binary variables and ((IJ+J K+KL)P) continues variables. Besides, the number of constraints is ((J+K+L)P+(I+J+K)+2). First term refers to the flows conservation constraints. Second term refers to the capacity constraints. And last term mentions the two constraints for bounding the number of open plants and DCs. While by assuming 2 modes between each pair of nodes, total number of binary and continues variables would be double. Even the number of constraints would be increased more than double.
3.4. CONCLUSION 65 More precisely, It would be ((J+K+L)P + (I+J+K) + 2 + (IJ+J K+KL) + 4(IJ+J K+KL)).
Thus, transportation modes plays an important role in increasing the complexity of the model.
The model can apply either for designing supply chain from scratch or expanding an existing supply chain network. To expand the supply chain, If a given plant/DC was supposed to be opened, then the correspondingyj/yk variable can explicitly be set to 1 in the model. This is useful in the case of existing facilities which should remain active or former decisions that can not be changed. The same reasoning also applies to transportation mode variablestmij.
4
A Large Neighborhood Search for the SCND problem
The proposed SCND model reduces to a Capacitated Facility Location Problem (CFLP) by considering only flows conservations and capacity constraints between DCs and customers. Since the CFLP is NP- complete [Davis and Ray,1969], the proposed model is an NP-hard problem. Because of the computational complexity of the problem and its large number of variables and constraints, using commercial software and exact optimization methods may no longer be tractable. Thus the development of a meta-heuristic/heuristic method is suitable to find a near optimal solution for particularly large instances [Olivares-Benitez et al., 2013,Der-Horng and Meng,2008].
We propose an LNS heuristic able to deal with the three main types of decision variables: facility loca- tion, selection of transportation mode and calculation of optimal product flows. The location and transporta- tion modes are modeled by binary variables while the product flows are modeled by continuous variables.
A key issue is to determine the number and location of plants and DCs. We call thenetwork configura- tionthe number of plants and DCs open in the current solution, represented by the pair P
j∈J
yj, P
k∈K
yk
! . This network configuration has great influence on the whole solution. One of the main issues is that there are(Jmax−Jmin)×(Kmax−Kmin)possible network configurations. Good heuristic methods should explore all promising network configurations. The main challenges of the LNS algorithm are thus to handle both binary and continuous variables, to determine the strategy to visit and evaluate promising network configurations and lastly to select transportation modes.
The rest of this chapter is organized as follows. Section 4.1gives an overview of the general LNS ap- proach. Section4.2gives an overview of our LNS approach. Sections 4.2.1,4.2.2, and4.2.3describe the proposed removal and repair operators, and combinations of both. The management of network configura- tions is detailed in section4.3. The greedy heuristics to determine transportation modes and product flows are presented in section4.3.1.
4.1 A Large Neighborhood Search heuristic
Large Neighborhood Search (LNS) has been introduced by Shaw[1998] in a constraint programming framework. LNS is similar to theruin and recreatemethod introduced bySchrimpf et al.[2000]. Ropke and Pisinger[2010] present an extensive survey of the method and its application to combinatorial optimization
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problems, and more particularly vehicle routing problems. As depicted in Algorithm 1, the underlying principle of the LNS is to partially destroy and repair iteratively a solution in order to improve it.
Algorithm 1Main scheme of the Large neighborhood Search (LNS) Require: An initial solutionS0
1: BestSolution← S0
2: CurrentSolution← S0
3: whilethe termination criterion is not satisfieddo
4: Selection ofDestroy andRepairheuristics
5: S ←CurrentSolution
6: S ←Destroy(S)
7: S ←Repair(S)
8: ifS < BestSolutionthen
9: BestSolution ← S
10: CurrentSolution← S
11: else
12: ifN ewSolutionAccepted(S)then
13: CurrentSolution← S
14: end if
15: end if
16: end while
17: return BestSolution
LNS relies on repetitive use of problem dependent heuristics for destroying and repairing the current solution (lines 4–7). The resulting solution is saved when it dominates the preceding ones (lines 8–10) and it may be accepted even if it deteriorates the objective function during the search (line 11–13). In this case, the most common acceptance criteria come either from Simulated Annealing [Kirkpatrick et al., 1983] or from the Record-to-Record Travel algorithm [Dueck,1993].
Recent heuristic in the literature based on Large neighborhood Search achieves a remarkable success in the fields of vehicle routing problems and scheduling. Generally, the key idea of the LNS algorithm in those problems is to remove a number of customers or tasks with a destroy operator and then re-insert them with a repair operator.
To our knowledge, the use of LNS for solving SCND problems is still very scarce. Copado-Méndez et al.[2013] model two cases studies in chemical engineering and solve them with an hybrid LNS algorithm.
They combine LNS with standard branch-and-cut techniques. They randomly choose a set of variables to remove and invoke a commercial MIP solver to improve the solution changing the value of the variables removed. They consider several stopping criteria such as maximum number of iterations and maximum execution time of the algorithm.