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This paper addresses a multi-period fixed charge transportation problem (MPFCTP), which includes transportation, inventories and backlog. The transportation costs are represented in the form of a nonlinear function. The objective is to determine the size of the shipments in each period, inventory at the end of each period and backlog at the end of each period, so that, the total cost of transportation, inventory and backlog are minimized. A constructive heuristic and an equivalent variable cost heuristic are proposed here as a solution methodology to find the lower bound and approximate solutions.

Keywords: Multi-period, Fixed charge, Equivalent variable cost, Approximate solution

- Abstract
- Introduction
- Problem description
- Solution methodologies
- Constructive heuristics by relaxing fixed cost (HU1)
- Equivalent variable cost heuristics (HU2)
- Numerical illustration

- Computational results and performance analysis
- Conclusion
- References

[...] The problem includes both fixed and variable transportation costs, inventory holding costs (during excess supply), and backlog penalty cost (during excess demand). Products are produced and distributed from suppliers to customers, simultaneously held in inventory at supplier side or customer side or both, if excess supply happens. The cost of holding inventory is different form supplier to supplier as well as customer to customer because of the nature of the inventory location. Similarly, when excess demand or scarce supply occurs, the backlog and corresponding penalty cost will also play a role in the total distribution cost. [...]

[...] Poh.K.L, Choo.K.W, Wong.C.G, (2005), A heuristic approach to the multi-period multicommodity transportation problem, Journal of the Operational Research Society, 56: 708–718. Qiu-Hong Zhao, Shou-Yang Wang , K-K Lai and Guo-Ping Xia, (2001), Dynamic Multi-period Transportation Model for Vehicle Composition with Transshipment Points , Advanced Modeling and Optimization : 17-28. Ragsdale, C.T., McKeown, P.G., (1991), An Algorithm for solving fixed charge problems using surrogate constraints. Computers & Operations Research 87–96. Raj.K.A.A.D and Rajendran.C (2008), Fast heuristic algorithms to solve a single stage fixed charge transportation problem. [...]

[...] (2007), Nonlinear fixed charge transportation problem by spanning tree-based genetic algorithm', Computers & Industrial Engineering 290–298. Kennington, J.L., and Unger, V.E., (1973), The group-theoretic structure in the fixed-charge transportation problem, Operation Research, 1142–1153. Kennington, J.L., and Unger, V.E., (1976), A new branch-and-bound algorithm for the fixedcharge transportation problem, Management Science (10):1116–1126. Kim.J.U, and Kim.Y.D, (2000), A Lagrangian relaxation approach to multi-period inventory/distribution planning, Journal of the Operational Research Society, 51: 364-370 King, J.R., (1975), Production Planning and Control. Pergamon Press, New York. [...]

[...] Substituting the results obtained in equations 3 and the lower bound and approximate solution are found as follows: ZL = (Lower bound by eqn.3) Z2 = (Approximate solution by HU2) Table 2a: Transportation cost data (Cij & FCij) j i Cij 20 FCij Table 2b: Suppliers' data (Pit, SHi & SIi0) i - Pit SHi SIi0 t Table 2c: Customers' data (Djt, CHj, BCj, BLj0 & CIj0) j - Djt t CHj BCj BLj0 CIj0 Table Optimal distribution schedule for the relaxed problem 2 j i BLjt CIit SIi1 t SIi2 SIi3 Table Equivalent variable cost (EVCijt) matrix j i t Table Optimal distribution schedule for the relaxed problem 3 j i BLjt CIit SIi1 t SIi2 SIi3 Computational results and performance analysis The capability of the proposed heuristics in handling the MPFCTP problem is analysed by comparing the approximate solutions (Z1 and Z2) with lower bound value (ZL). [...]

[...] Figure 1 shows the operational elements of the multi-period distribution problem under consideration n n i Pi t + SIi t-1 Cumulative supply 1 j 2 j Xijt Cij /FCij j Dj t + BLj CIj t-1 Cumulative Demand 1 m n Suppliers j n-1 n Customers Figure Operational elements of multi-period fixed charge transportation problem during the period t This model attempts to integrate transportation, inventories and backlog decisions monolithically from a centralized planning point of view with objective equation as given below. [...]

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- Number of pages7 pages
- LanguageEnglish
- Formatpdf
- Publication date24/05/2010
- Updated on24/05/2010

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