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...In particular, high-tech industries, such as the electronics, semiconductor and telecommunications industries, which have capital intensive equipment and short technology life cycle, struggle with the production and transportation planning problem in their complex and rapid changing environment (Wu and Golbasi, 2004). The difficulty of the combined problem is one of the reasons why for many years practitioners and academics have looked at these problems separately.
The supply chain we analyze is a two-stage, dynamic supply chain that consists of a number of facilities, which can be viewed as a plant with an associated warehouse, and a set of retailers. The goal is to find a minimum-cost production and transportation schedule. We propose a network flow formulation and use a Lagrangean decomposition procedure to provide solutions for this integrated production and transportation planning problem.
A special case of this problem is the multi-commodity lot sizing problem. A great deal of work has been devoted to the multi-commodity lot sizing problem since it is the core of "aggregated production planning" models. Solutions to lot sizing problems are often inputs to "master production schedules" and subsequently, to "materials requirements planning" systems in a "push" type manufacturing environment (Nahmias, 1997). Chen and Thizy (1990) show that the multi-commodity lot sizing problem is strongly NP-hard. Since the multi-commodity lot sizing problem is a special case of our problem, this means that our problem is also NP-hard.
Eppen and Martin (1987) provide a shortest-path formulation for the multi-commodity lot sizing problem. They show that the Linear Programming (LP) relaxation of the shortest-path formulation is very effective in generating lower bounds and that the bounds are equal to those that could be generated using Lagrangean relaxation or column generation methods. Most of the promising heuristic approaches to solve the multi-commodity lot sizing problem seem to be based on Lagrangean relaxation. Thizy and van Wassenhove (1985) and Trigeiro et al. (1989) propose a Lagrangean relaxation of the capacity constraints. They update the Lagrangean multipliers using a subgradient approach and propose a heuristic to obtain feasible solutions. Chen and Thizy (1990) analyze and compare the quality of different lower bounds using relaxation methods such as Lagrangean relaxations with respect to different sets of constraints and the LP relaxation. Stadtler (1996) introduces novel formulations of the multi-commodity lot sizing problem and compares their suitability when used to solve problems with different characteristics.
The problem we study is related to the ones studied by Chandra and Fisher (1994), Wu and Golbasi (2004) and Eksioglu et al. (2005). Chandra and Fisher (1994) were among the first to study integrated production-distribution planning problems. They investigate the effect of coordinating production and distribution on a single-plant, multi-commodity, multi-period scenario. Their computational study shows that the coordinated approach can yield up to 20% in cost savings. Eksioglu et al. (2005) analyze the single-commodity case of the problem studied in this paper. They propose a primal-dual algorithm that gives tight lower and upper bounds. Wu and Golbasi (2004) study a similar multi-facility, single-retailer, multi-commodity, multi-period production and transportation planning problem. Their production model, unlike ours, considers a bill of materials for each end-item. They use a Lagrangean decomposition procedure to solve the problem.
2. Problem formulation
The problem studied here involves the coordination of production, inventory and transportation decisions of K commodities over a planning horizon of T periods. There are F facilities that produce and deliver the final product to R retailers. The facilities have identical production capabilities in the sense that each product can be produced in each facility. However, the unit production, transportation and inventory holding costs differ between facilities. The proposed models can be used for strategic and tactical decisions. Therefore, the time period we refer to in our model could be as long as a month or more. For this reason, it is assumed that the cost parameters vary between periods. Commodities share a common production resource with item-specific setup costs. The goal is to decide on a production schedule for each commodity, so that the sum of the production, transportation and inventory costs in all of the facilities is minimized, demand is satisfied and bundle capacity constraints are not violated. It is assumed that the initial and ending inventories are zero for all products, all costs are non-negative, and backlogging is not allowed. If the initial or ending inventories are not equal to zero then they can be set to zero by adjusting the demands for either the first or last period respectively. Also, the backlogging assumption can be easily relaxed by adding new arcs to the network.
[FIGURE 1 OMITTED]
The resulting problem is formulated as a network flow problem. The structure of this network is illustrated in Fig. 1 for a single-commodity problem. The network consists of a source node, T layers, and F facilities and R retailers in each layer. The arcs connecting each facility to each retailer represent the transportation network. The source node s supplies the total demand, and the arcs connecting facilities in one time period to the same facilities in the next time period are the inventory arcs.
The following notation is used for problem formulation:
Problem data
[p.sub.itk] = unit production cost for commodity k at facility i in period t;
[s.sub.itk] = production set-up cost for commodity k at facility i in period t;
[h.sub.itk] = inventory holding cost for commodity k at facility i in period t;
[c.sub.ijtk] = unit transportation cost for commodity k from facility i to retailer j in period t;
[b.sub.jtk] = demand of retailer j for commodity k in period t;
[v.sub.it] = production (bundle) capacity at facility i in period t.
Decision variables
[q.sub.itk] = production quantity of commodity k at facility i in period t;
[x.sub.ijtk] = amount of commodity k transported from facility i to retailer j in period t;
[I.sub.itk] = amount of commodity k in inventory at facility i at the end of period t;
[y.sub.itk] = a binary variable that is equal to one if production of commodity k occurs at facility i in period t, and is equal to zero otherwise.
The following is a Mixed-Integer Linear Programming (MILP) formulation of the problem:
(P):
min [F.summation over (i=1)][T.summation over (t=1)][K.summation over (k=1)] {[s.sub.itk][y.sub.itk] + [p.sub.itk][q.sub.itk] + [h.sub.itk][I.sub.itk] + [R.summation over (j=1)] [c.sub.ijtk][x.sub.ijtk]},
subject to
[I.sub.i,t-1,k] + [q.sub.itk] = [R.summation over (j=1)] [x.sub.ijtk] + [I.sub.itk], i = 1,..., F, t = 1,..., T, k = 1,..., K, (1)
[F.summation over (i=1)] [x.sub.ijtk] = [b.sub.jtk], j = 1,..., R, t = 1,..., T, k = 1,..., K, (2)
[K.summation over (k=1)] [q.sub.itk] [less than or equal to] [v.sub.it], i = 1,..., F, t = 1,..., T, (3)
[q.sub.itk] [less than or equal to] [R.summation over (j=1)] [b.sub.jtTk][y.sub.itk], i = 1,..., F,...
NOTE: All illustrations and photos
have been removed from this article.
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