Case Study: The Brewery (Logistics) Problem

Submitted: 2 Apr 2008

Operations Research Topics: LinearProgramming, IntegerProgramming, NetworkOptimisation, TransportationProblem

Application Areas: Logistics

Contents

• Problem Description
• Problem Formulation
• Computational Model
• Results
• Conclusions
• StudentTasks

Problem Description

A boutique brewery has two warehouses from which it distributes beer to five carefully chosen bars. At the start of every week, each bar sends an order to the brewery's head office for so many crates of beer, which is then dispatched from the appropriate warehouse to the bar. The brewery would like to have an interactive computer program which they can run week by week to tell them which warehouse should supply which bar so as to minimize the costs of the whole operation.

For example, suppose that at the start of a given week the brewery has 1000 cases at warehouse A, and 4000 cases at warehouse B, and that the bars require 500, 900, 1800, 200, and 700 cases respectively.

The transportation costs are given in Table 1.

Table 1 Transportation costs from warehouses to bars

Given this data, the brewery management want to know which warehouse should supply which bar.

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Problem Formulation

Since crates of beer are being shipped from the warehouses to the bars, this problem is a bipartite network and can be modelled as a transportation problem.

The supply nodes are the warehouses, the demand nodes are the bars. The total supply is 1,000 + 4,000 = 5,000 and the total demand is 500 + 900 + 1,800 + 200 + 700 = 4,100, so the problem is unbalanced. We can either alter the balanced formulation to incorporate this or add a dummy demand node to receive the extra supply.

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Computational Model

Since this is a transportation problem we can use the AMPL model file for transportation problems.

The data file will have the supply nodes as the warehouses:

set SUPPLY_NODES := A B ;

set DEMAND_NODES := 1 2 3 4 5 ;

The supply and demand is given and can be specified:

param Supply :=
A          1000
B          4000
 ;

param Demand :=
1            500
2            900
3          1800
4            200
5            700
 ;

Finally, the transportation costs are easier to specify using a transposed table:

param Cost (tr) :
     A    B :=
1    2    3
2    4    1
3    5    3
4    2    2
5    1    3
 ;

Note that since the problem is not balanced, the AMPL script file transportation.run (see Transportation Problems in AMPL for details) automatically adds a dummy demand node to the problem.

The AMPL model file transportation.mod also ensures the supply, demand and bound variables are integer so that only linear programming will be necessary (as we will get naturally integer solutions, see The Transportation Problem for details).

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Results

By inserting

data brewery.dat;

Figure 1 AMPL output for the Brewery Problem

This shows that the optimal solution is to make the shipments shown in Table 1.

Table 1 Optimal shipments for the Brewery Problem

There are 900 crates left at Warehouse B.

Notice that CPLEX did not need to use any branch-and-bound nodes, i.e., no integer programming was required. This confirms that a transportation problem has naturally integer solutions. By specifying the supply, demand and bound parameters be integer we guarantee that only linear programming needs to be used.

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Conclusions

There are many ways to present the solution to the Brewery Problem: as a list of shipments, in a table, etc. You can use AMPL's display, print and printf commands to create formatted output that can be easily used in your management summary.

For example, brewery.out (shown below) was produced by an AMPL script file:

TRANSPORTATION SOLUTION -- Non-zero shipments
TotalCost = $8600

Ship 300 crates of beer from warehouse A to pub 1
Ship 700 crates of beer from warehouse A to pub 5
Ship 200 crates of beer from warehouse B to pub 1
Ship 900 crates of beer from warehouse B to pub 2
Ship 1800 crates of beer from warehouse B to pub 3
Ship 200 crates of beer from warehouse B to pub 4

Also, showing the solution as a graphical output can be informative.

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Student Tasks

1. Solve the Brewery Problem. Write a management summary of your solution.

   What to hand in Hand in your management summary.

2. The next week the brewery has 3000 and 1500 crates of beer at Warehouses A and B respectively. The demand from Bars 1, 2, 3, 4 and 5 is 700, 1200, 1000, 800, 700. The transportation costs have not changed.

   Solve the Brewery Problem using this new data. Write a management summary of your solution.

   What to hand in Hand in your management summary.

3. Write a script file to generate the output shown in brewery.out. Use this script file to create a new output file with the solution to Task 1 displayed.

   What to hand in Hand in your script file and you new output file.

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