http://hdl.handle.net/1765/1353
series: EI 9526-/A

# An Algorithm for Single-item Capacitated Economic Lot Sizing with Piecewise Linear Production Costs and General Holding Costs

### Shaw, D.X. Wagelmans, A.P.M.

Research Paper
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We consider the Capacitated Economic Lot Size problem with piecewise linear production costs and general holding costs, which is an NP-hard problem but solvable in pseudo-polynomial time. A straightforward dynamic programming approach to this problem results in an [TeX: $O(n^2 \\bar{c} \\bar{d} )$] algorithm, where [TeX: $n$] is the number of periods, and [TeX: $\\bar d$ and $\\bar c$] are the average demand and the average production capacity over the $n$ periods, respectively. However, we present a dynamic programming procedure with complexity [TeX: $O(n^2 \\bar{q} \\bar{d} )$], where [TeX: $\\bar q$] is the average number of pieces of the production cost functions. In particular, this means that problems in which the production functions consist of a fixed set-up cost plus a linear variable cost are solved in [TeX: $O(n^2 \\bar{d})$] time. Hence, the running time of our algorithm is only <it>linearly</it> dependent on the magnitude of the data. This result also holds if extensions such as backlogging and start-up costs are considered. Moreover, computational experiments indicate that the algorithm is capable of solving quite large problem instances within a reasonable amount of time. For example, the average time needed to solve test instances with 96 periods, 8 pieces in every production function and average demand of 100 units, is approximately 40 seconds on a SUN SPARC 5 workstation.

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