# 3.6.1 Scenario: Location Configuration with Linear Programming

### Intended learning outcomes: Disclose and solve the planning problem at the Ironer Company, which produces two products for two markets, given a maximum total capacity in a year.

The Ironer Company, a manufacturer of ironing machines, has its facilities at one single location. The Ironer Company markets two different products in two regions. Once a year, the company performs rough-cut capacity planning based on sales forecasts. In addition, it must answer the following important question for marketing: With the given capacity situation, what quantity of what product should be offered in which market in order to maximize the contribution margin? While demand for New Product P1 is increasing sharply in Market M2, sales of Predecessor Product P2 are declining as the market becomes saturated (decline stage). Here the assumed market demand reflects the maximum saleable number of pieces. The contribution margins of the two markets differ, in part considerably, due to the differing cost and price structures. Figure 3.6.1.1 shows the details:

Fig. 3.6.1.1       Input data for the planning problem at the Ironer Company.

Ironer Company requires 4 hours to manufacture Product P1 and 2.4 hours to manufacture Product P2. The total capacity in a year is 15,000 hours. Please answer the following questions:

1. What quantities of P1 and P2 should be put on the two markets in order to maximize the contribution margin?
2. A consulting firm is proposing, by introducing lean-/just-in-time concepts (lean/ JIT), to increase the contribution margin by 5% and lower the capacity required for P1 by 60 minutes and the capacity required for P2 by 24 minutes. What should the maximum cost of introducing JIT concepts be? And how will this improve the company situation?
3. In addition, the marketing department decides to increase market penetration of P1 and, to maximize profits, to intensify the decline of P2. To do this, sales of P2 in Market M1 must rise to 4,000, while for Market M2 complete product withdrawal is planned. What are the advantages and disadvantages of this strategy?

Proceed as follows:

1. A) Define the decision variables. Possible solution:

X_Pi_Mj , 1<= j <= 2, 1 <= i <= 2 stands for the number of products P1 that will be delivered to Market Mj

1. B) Formulate the target function. Possibility: contribution margin = max!

= (DB_P1_M1 · X_P1_M1) + (DB_P1_M2 · X_P1_M2) + (DB_P2_M1 · X_P2_M1) + (DB_P2_M2 · X_P2_M2)

Figure 3.6.1.2 shows how you can perform these first steps utilizing MS Excel Solver, Microsoft Excel’s tool for solving linear optimization.

Fig. 3.6.1.2        Solver tool in MS Excel, part 1.

Formulate all side conditions:

##### (a.) Demand: Maximum coverage of market demands
• X_P1_M1 <= maximum demand for Product P1 in Market M1=1000
• X_P1_M2 <= maximum demand for Product P1 in Market M2=5000
• X_P2_M1 <= maximum demand for Product P2 in Market M1=3000
• X_P2_M2 <= maximum demand for Product P2 in Market M2=2000
##### (b.) Capacity: Restricted total capacity
• X_P1_M1 · capacity requiredP1 + X_P1_M2 · capacity requiredP1
+ X_P2_M1 · capacity requiredP2 + X_P2_M2 · capacity requiredP2
<= total capacity
##### (c.) Variable non-negativity
• X_P1_M1 >= 0;   X_P1_M2  >= 0;   X_P2_M1  >= 0;   X_P2_M2  >= 0

Figure 3.6.1.3 shows how MS Excel Solver handles the formulation of side conditions. The constraint operators (e.g. , <=, >=) were entered only as text, for purposes of clarity to the reader.

Fig. 3.6.1.3        Solver tool in MS Excel, part 2.

Figure 3.6.1.4 shows how you must actually enter the decision variables, target function, and side conditions using the Solver tool in MS Excel.

Fig. 3.6.1.4        Solver tool in MS Excel, part 3.

• Click Solve (see Figure 3.6.1.4) to display the results. If you have entered everything correctly, the maximum contribution margin achievable should be 356.500€, as it is in Figure 3.6.1.2.
##### Task 2), the introduction of JIT:
• Start out from the basic case in Task 1, copying and pasting it into a new Excel spreadsheet. Now change the values accordingly.
• Through the higher contribution margins and lower capacity required, the total contribution margin and service level for P1 can be increased.
• If everything is set correctly, you can determine the cost ceiling for introducing the JIT concept as the contribution margin difference: 451.500€ - 356.500€ = 94.900€.
• Start out from the basic case in Task 2) (JIT), copying and pasting it into a new Excel spreadsheet. Now change the values accordingly.
• By intensifying the decline of Product P2, the total contribution margin can be incre­ased even more, to 476.000€. The service level for P1, however, drops in compa­rison, since the available capacities produce Product P2 for Market M1. For this reason, it is necessary to consider to what extent the increase in the number of pieces of P1 will take place and the extent to which P1 will at best replace product P2.

## Course section 3.6: Subsections and their intended learning outcomes

• ##### 3.6 Scenarios and Exercises

Intended learning outcomes: Elaborate an example of a location configuration with linear programming.

• ##### 3.6.1 Scenario: Location Configuration with Linear Programming

Intended learning outcomes: Disclose and solve the planning problem at the Ironer Company, which produces two products for two markets, given a maximum total capacity in a year.