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On the optimum capacity of capacity expansion problems

Author

Listed:
  • Chao Yang
  • Chunyan Hao
  • Jianzhong Zhang

Abstract

In this paper we consider problems of the following type: Let E = { e 1 , e 2 ,..., e n } be a finite set and $${\mathcal {F}}$$ be a family of subsets of E. For each element e i in E, c i is a given capacity and $${\mathcal {w}}$$ i is the cost of increasing capacity c i by one unit. It is assumed that we can expand the capacity of each element in E so that the capacity of family $${\mathcal {F}}$$ can be expanded to a level r. For each r, let f (r) be the efficient function with respect to the capacity r of family $${\mathcal {F}}$$ , and $${\phi(r)}$$ be the cost function for expanding the capacity of family $${\mathcal {F}}$$ to r. The goal is to find the optimum capacity value r * and the corresponding expansion strategy so that the pure efficency function $${f(r^*)-\phi(r^*)}$$ is the largest. Firstly, we show that this problem can be solved efficiently by figuring out a series of bottleneck capacity expansion problem defined by paper (Yang and Chen, Acta Math Sci 22:207–212, 2002) if f (r) is a piecewise linear function. Then we consider two variations and prove that these problems can be solved in polynomial time under some conditions. Finally the optimum capacity for maximum flow expansion problem is discussed. We tackle it by constructing an auxiliary network and transforming the problem into a maximum cost circulation problem on the auxiliary network. Copyright Springer-Verlag 2007

Suggested Citation

  • Chao Yang & Chunyan Hao & Jianzhong Zhang, 2007. "On the optimum capacity of capacity expansion problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(2), pages 225-233, October.
  • Handle: RePEc:spr:mathme:v:66:y:2007:i:2:p:225-233
    DOI: 10.1007/s00186-007-0147-z
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    References listed on IDEAS

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    1. Zhang, Jianzhong & Liu, Zhenhong & Ma, Zhongfan, 2000. "Some reverse location problems," European Journal of Operational Research, Elsevier, vol. 124(1), pages 77-88, July.
    2. James B. Orlin, 1993. "A Faster Strongly Polynomial Minimum Cost Flow Algorithm," Operations Research, INFORMS, vol. 41(2), pages 338-350, April.
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