Minimize the cost of operating 3 different types of machines while meeting product demand. | ||||||||

Each machine has a different cost and capacity. There are a certain number of machines | ||||||||

available for each type. | ||||||||

Information on machines | ||||||||

Initial cost per day | Additional cost per product | Products per day (Max) | Number of machines | |||||

Alpha-1000 | $200 | $1.50 | 40 | 8 | ||||

Alpha-2000 | $275 | $1.80 | 60 | 5 | ||||

Alpha-3000 | $325 | $1.90 | 85 | 3 | ||||

Number of machines to use | ||||||||

Alpha-1000 | 1 | |||||||

Alpha-2000 | 1 | |||||||

Alpha-3000 | 1 | |||||||

Number of products to make per day | ||||||||

Alpha-1000 | 300 | |||||||

Alpha-2000 | 300 | |||||||

Alpha-3000 | 300 | |||||||

Total | 900 | |||||||

Demand | 750 | |||||||

Maximum number of products that can be made per day | ||||||||

Alpha-1000 | 40 | |||||||

Alpha-2000 | 60 | |||||||

Alpha-3000 | 85 | |||||||

Cost | $2,360.00 | |||||||

Problem | ||||||||

A company has three different types of machines that all make the same product. Each | ||||||||

machine has a different capacity, start-up cost and cost per product. How should the company | ||||||||

produce its product with the available machines to meet the daily demand? | ||||||||

Solution | ||||||||

1) The variables are the number of machines to use and the number of products to make on | ||||||||

each machine. In worksheet Alloc1, these are given the names Products_made and | ||||||||

Machines_used. | ||||||||

2) First, there are the logical constraints. These are | ||||||||

Products_made >= 0 via the Assume Non-Negative option | ||||||||

Machines_used >= 0 via the Assume Non-Negative option | ||||||||

Machines_used = integer. | ||||||||

Second, there are the demand and capacity constraints. These are: | ||||||||

Machines_used <= Machines_available | ||||||||

Products_made <= Maximum_products | ||||||||

Total_made >= Demand | ||||||||

3) The objective is to minimize cost. This is defined on the worksheet as Total_cost. | ||||||||

Remarks | ||||||||

Notice that we used an integer constraint to make sure no fractions of machines were used. | ||||||||

This has the usual drawback; the problem is much more difficult to solve than the 'relaxed' | ||||||||

version without the integer constraint. When large numbers of machines are involved, the | ||||||||

integer constraint can often be dropped. It is most often not critical whether 1586 or 1587 | ||||||||

machines are used, for example. This means that a number of 1586.4 would be acceptable. | ||||||||

However, in this case there are only a few machines and it does make a big difference whether | ||||||||

2 or 3 machines are used. An answer of 2.5 would not be satisfactory. | ||||||||