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

Back to main page

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
——————————————————————–———————–——————–
Avg_pm_pen_ni = Average Partial Moment Penalty Normal Independent for Loss
Box constraints = constraints on individual decision variables
Variables = define type of variables
——————————————————————–———————–——————–

Dataset1 8 N/A -882.130402177 0.02 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments 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.