Abstract
This paper relies on a decision-tree approach to aid a buying firm in determining the optimal size of its supply base in the presence of risks. The risk under consideration refers to any unpredictable operations interruptions caused by all suppliers being unavailable to satisfy the buying firm's demand. The relationship between the levels of risk and associated trade-offs is captured by a decision-tree model, from which the expected cost function is formulated. The exact or approximate optimal solutions for various scenarios, as well as their sensitivity, are obtained and examined.
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Appendices
Appendix A
This appendix shows how the optimal solution with a normally distributed financial loss is obtained. The logarithmic expected utility function with normally distributed financial loss is given by (refer to (19) and (4))
The first derivative of (A.1) with respect to the decision variable is
Then, the second derivative of (A.1) can be found as
Therefore, any extreme point is a min point. The minimum is found by setting the first derivative in (A.2) to zero; that is, letting S n=x,
The function in (A.3) has two roots, but only one root is feasible, which is shown below after all the simplifications,
Then after substituting n=ln x/ln S, the optimal solution is obtained.
Appendix B
This appendix shows how the optimal solution with a gamma distributed financial loss is obtained. The logarithm of the expected utility function with gamma distributed financial loss is given by (refer to (24))
Assuming n to be continuous, we can obtain the first derivative of (B1) as
and the second derivative is
Since λ−θp>0, (B.3) is also positive. Therefore, the function in (B.1) (or (21)) is convex, and, again, any extreme point is a min point. The minimum is found by setting the first derivative in (B.2) to zero, that is,
or
which is equivalent to
Solving (B.4) for n yields
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Berger, P., Zeng, A. Single versus multiple sourcing in the presence of risks. J Oper Res Soc 57, 250–261 (2006). https://doi.org/10.1057/palgrave.jors.2601982
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DOI: https://doi.org/10.1057/palgrave.jors.2601982