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A MINIMIZATION PROBLEM 57

2.5 A Minimization Problem

So far, we have looked at problems where we want to maximize the objective
function. LP problems where we seek to minimize the objective function are just
as common. M&D Chemicals is based in Germany and manufactures a variety of
chemical products that are sold to other companies who produce bath soaps and
shower gels. Based on an analysis of current inventory levels and potential demand
for the coming week, M&D’s management specified that the combined production
for products A and B must total at least 350 litres. Separately, a major customer’s
order for 125 litres of product A must also be satisfied. Product A requires two
hours of processing time per litre and product B requires one hour of processing
time per litre. For the coming week, 600 hours of processing time are available.
M&D’s objective is to satisfy these requirements at a minimum total production
cost. Production costs are E2 per litre for product A and E3 per litre for product B.
To find the minimum-cost production schedule, we will formulate the M&D
Chemicals problem as a linear programme. Following a procedure similar to the
one used for the GulfGolf problem, we first define the decision variables and the
objective function for the problem. Let:

A ¼ number of litres of product A
B ¼ number of litres of product B
With production costs at E2 per litre for product A and E3 per litre for product B,
the objective function that corresponds to the minimization of the total production
cost can be written as:

Min 2A þ 3B
Next consider the constraints placed on the M&D Chemicals problem. To satisfy the
major customer’s demand for 125 litres of product A, we know A must be at least
125. Thus, we write the constraint:
1A 125
For the combined production for both products, which must total at least 350 litres,
we can write the constraint:
1A þ 1B 350
Finally, for the limitation of 600 hours on available processing time, we add the
constraint:
2A þ 1B 600
After adding the nonnegativity constraints (A, B 0), we arrive at the following
linear programme for the M&D Chemicals problem:
Min 2A þ 3B
s:t:
1A 125 Demand for product A
1A þ 1B 350 Total production
2A þ 1B 600 Processing time
A; B 0
Because the linear programming model has only two decision variables, the graph-
ical solution procedure can be used to find the optimal production quantities. The

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