# Case Study: Optimal Allocation of Stock Levels and Stochastic Customer Demands to a Capacitated Resource

** Case study background and problem formulations**

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

**PROBLEM: problem_m_n**

Minimize Avg_pm_pen_ni + Linear (minimizing expected cost)

subject to

Linear ≤ Const (vehicle capacity)

Box constraints (bounds on decision variables)

Variables: real, boolean

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Avg_pm_pen_ni = Average Partial Moment Penalty Normal Independent for Loss

Box constraints = constraints on individual decision variables

Variables = define type of variables

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# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset1 | 8 | N/A | -882.130402177 | 0.02 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

Instructions for importing problems from Run-File to PSG MATLAB.

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset2 | Problem Statement | Data | Solution | 8 | N/A | -351.266372731684 | 0.02 |

Dataset3 | Problem Statement | Data | Solution | 13 | N/A | -1373.06576243 | 0.09 |

Dataset4 | Problem Statement | Data | Solution | 13 | N/A | -973.823585968 | 0.06 |

Dataset5 | Problem Statement | Data | Solution | 20 | N/A | -3583.1141819 | 0.25 |

Dataset6 | Problem Statement | Data | Solution | 20 | N/A | -4119.2165147 | 0.15 |

Dataset7 | Problem Statement | Data | Solution | 40 | N/A | -11888.6588017 | 0.70 |

Dataset8 | Problem Statement | Data | Solution | 40 | N/A | -11601.4992279 | 0.45 |

Dataset9 | Problem Statement | Data | Solution | 50 | N/A | -21216.3525975 | 0.71 |

Dataset10 | Problem Statement | Data | Solution | 50 | N/A | -15597.7658522 | 0.73 |

Dataset11 | Problem Statement | Data | Solution | 100 | N/A | -91627.93190268 | 4.09 |

Dataset12 | Problem Statement | Data | Solution | 100 | N/A | -64760.2978986 | 4.34 |

**CASE STUDY SUMMARY**

This case study optimizes an allocation of stochastic demands to a capacitated resource.

This case study is based on the paper by Chen, S. and J. Geunes (2010). A single-period model is defined to solve a joint customer demand allocation and multiple-item stock level problem for a resource that must respond to uncertain customer demands. It is assumed that customer demands are statistically independent and that the demands for items are revealed upon being visited by a repair person (after problem diagnosis). There are m potential customers; the supplier must choose a subset of customers to serve using a capacity constrained resource (e.g., a vehicle) and the optimal resource stock level for each item. It is assumed that during a given customer-service visit, any item required for the customer’s service that is not available results in an item-specific penalty cost for an inability to complete service. For each item carried on the vehicle, a variable cost is incurred (e.g., for loading/unloading and/or transporting the part). Because the vehicle stocking decisions must be determined prior to actual customer demand realizations, it may be practical in some contexts to consider a salvage value for each unused item carried on the vehicle (this might correspond to a reduction in future loading/unloading costs; alternatively, a negative value would correspond to an opportunity cost of the vehicle capacity usage). A customer-specific revenue is gained for each customer visit. The objective is to maximize the expected profit, or equivalently, to minimize the expected cost (equal to the negative of expected profit). We state our objective in minimization form and refer to this objective as the expected cost.