Selecting a Portfolio of Suppliers Under Demand and Supply Risks

We analyze a planning model for a firm or public organization that needs to cover uncertain demand for a given item by procuring supplies from multiple sources. Each source faces a random yield factor with a general probability distribution. The model considers a single demand season. All supplies need to be ordered before the start of the season. The planning problem amounts to selecting which of the given set of suppliers to retain, and how much to order from each, so as to minimize total procurement costs while ensuring that the uncertain demand is met with a given probability. The total procurement costs consist of variable costs that are proportional to the total quantity delivered by the suppliers, and a fixed cost for each participating supplier, incurred irrespective of his supply level. Each potential supplier is characterized by a given fixed cost and a given distribution of his random yield factor. The yield factors at different suppliers are assumed to be independent of the season's demand, which is described by a general probability distribution. Determining the optimal set of suppliers, the aggregate order and its allocation among the suppliers, on the basis of the exact shortfall probability, is prohibitively difficult. We have therefore developed two approximations for the shortfall probability. Although both approximations are shown to be highly accurate, the first, based on a large-deviations technique (LDT), has the advantage of resulting in a rigorous upper bound for the required total order and associated costs. The second approximation is based on a central limit theorem (CLT) and is shown to be asymptotically accurate, whereas the order quantities determined by this method are asymptotically optimal as the number of suppliers grows. Most importantly, this CLT-based approximation permits many important qualitative insights.

[1]  George L. Nemhauser,et al.  Note--On "Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms" , 1979 .

[2]  Awi Federgruen,et al.  The Greedy Procedure for Resource Allocation Problems: Necessary and Sufficient Conditions for Optimality , 1986, Oper. Res..

[3]  David D. Yao O𝕡timal Run Quantities for an Assembly System with Random Yields , 1988 .

[4]  Laurence A. Wolsey,et al.  Best Algorithms for Approximating the Maximum of a Submodular Set Function , 1978, Math. Oper. Res..

[5]  R. Akella,et al.  Diversification under supply uncertainty , 1993 .

[6]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[7]  A. Mokkadem Mixing properties of ARMA processes , 1988 .

[8]  Nils Rudi,et al.  Newsvendor Networks: Inventory Management and Capacity Investment with Discretionary Activities , 2002, Manuf. Serv. Oper. Manag..

[9]  G. Nemhauser,et al.  Exceptional Paper—Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms , 1977 .

[10]  Yigal Gerchak,et al.  Multiple Lotsizing in Production to Order with Random Yields: Review of Recent Advances , 2004, Ann. Oper. Res..

[11]  J. V. Mieghem Investment Strategies for Flexible Resources , 1998 .

[12]  Ram Akella,et al.  A wafer design problem in semiconductor manufacturing for reliable customer service , 1990 .

[13]  Hau L. Lee,et al.  Lot Sizing with Random Yields: A Review , 1995, Oper. Res..

[14]  Paul H. Zipkin,et al.  Simple Ranking Methods for Allocation of One Resource , 1980 .

[15]  Brian Tomlin,et al.  On the Value of Mitigation and Contingency Strategies for Managing Supply Chain Disruption Risks , 2006, Manag. Sci..

[16]  J. Lehoczky,et al.  Supply management in assembly systems with random yield and random demand , 2000 .

[17]  H. Thorisson Coupling, stationarity, and regeneration , 2000 .

[18]  H. Teicher,et al.  Probability theory: Independence, interchangeability, martingales , 1978 .

[19]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[20]  M. Parlar,et al.  Diversification under yield randomness in inventory models , 1993 .

[21]  J. George Shanthikumar,et al.  Supplier diversification: effect of discrete demand , 1999, Oper. Res. Lett..

[22]  Yigal Gerchak,et al.  Yield randomness, cost tradeoffs, and diversification in the EOQ model , 1990 .

[23]  Candace Arai Yano,et al.  Lot sizing in assembly systems with random component yields Candace Arai Yano, Yigal Gerchak. , 1989 .

[24]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[25]  H. White Asymptotic theory for econometricians , 1985 .

[26]  J. Michael Harrison,et al.  Multi-Resource Investment Strategies: Operational Hedging Under Demand Uncertainty , 1997, Eur. J. Oper. Res..

[27]  Yigal Gerchak,et al.  The Structure of Periodic Review Policies in the Presence of Random Yield , 1990, Oper. Res..

[28]  Hau L. Lee,et al.  Input Control for Serial Production Lines Consisting of Processing and Assembly Operations with Random Yields , 1996, Oper. Res..

[29]  Paul H. Zipkin,et al.  Foundations of Inventory Management , 2000 .

[30]  Chelsea C. White,et al.  An Inventory Control Model with Possible Border Disruptions , 2005 .

[31]  Mark A. McComb Comparison Methods for Stochastic Models and Risks , 2003, Technometrics.

[32]  J. Lehoczky,et al.  Optimal order policies in assembly systems with random demand and random supplier delivery , 1996 .