Background

Problem. Supply Chain Planning

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

References

 

 

Background

 

This case study models the supply chain design problem as a sequence of splitting and combining processes. The problem is formulated as a two-stage stochastic program. The first-stage decisions are strategic location decisions, whereas the second stage consists of operational decisions. The objective is to maximize the expected profits over the planning horizon. Another variant of objective is to minimize the sum of investment costs and expected costs of operating the supply chain.

 

The solved problem instance is a real-life problem from the Norwegian meat industry. The task is to balance supply and demand on a weekly basis, ensuring that the right raw materials are available at the right production facilities in order to satisfy demand. The planning horizon is four weeks. Demand for the first week is known with certainty, whereas planning of production and material flow for the remaining three weeks is based on predicted demand. For more details see Schütz (2011).

 

Below we give a general formulation of the problem. It is reformulated as minimization one. Solutions of the second stage problem for different scenarios give the values of the stochastic Recourse Function.

 

The solved stochastic linear programming problem contains 22,676 columns and 11,978 rows for the first stage and 74,398 columns and 37,893 rows for every subproblem of the second stage. The number of scenarios is 75. So in the equivalent LP formulation the problem contains 5,602,526 columns and 2,853,953 rows.

 

Problem

 

Simplified Problem Statement

 

Minimize Avg (Recourse) (minimizing average of recourse function)

subject to

ConstVector1 ≤ Linearmulti ≤ ConstVector2 (linear constraints on the first stage variables)

Box constraints (bounds on the first stage variables)

 

where

 

Avg = Average for Recourse

Linearmulti = Linear Multiple

Box constraints = constraints on individual decision variables

Recourse = Minimal value of the following second stage subproblem depending on scenarios for the given first stage variables

Minimize Linear (minimizing linear objective of the second stage subproblem)

subject to

ConstVector3 ≤ Linearmulti ≤ ConstVector4 (linear constraints on the second stage variables depending on scenarios)

Box constraints (bounds on the second stage variables)

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB function tbpsg_run (General (Text) Format of PSG in MATLAB):

 

 

Input Files to run CS:

 

 

References

 

[1] Rockafellar, R.T., and S. Uryasev (2000): Optimization of conditional value-at-risk. Journal

of Risk 2, 21–41.