Question

A company manufactures four products (1,2,3,4) on two machines (X and Y). The time (in minutes) to process one unit of each product on each machine is shown below:,

Product 1 in X = 10 min , in Y= 27 min

Product 2 in X = 12 min , in Y =19 min

Product 3 in X =13 min,in Y =33 min

Product 4 in X = 8 min,in Y =23 min

The profit per unit for each product (1,2,3,4) is £10, £12, £17 and £8 respectively.

• Product 1 must be produced on both machines X and Y but products 2, 3 and 4 can be produced on either machine.

• The factory is very small and this means that floor space is very limited.

• Only one week's production is stored in 50 square meters of floor space where the floor space taken up by each product is 0.1, 0.15, 0.5 and 0.05 (square meters) for products 1, 2, 3 and 4 respectively

Customer requirements mean that, the amount of product 3 produced should be related to the amount of product 2 produced. Over a week approximately twice as many units of product 2 should be produced as product 3.

• Machine X is out of action (for maintenance /because of breakdown) 5% of the time and machine Y 7% of the time.

• Assuming a working week 35 hours long formulate the problem of how to manufacture these products to maximize total profit?

Can you explain why the objective function is • Objective Function: max Z= 10x1 + 12(x2 + y2 ) + 17(x3 + y3 ) + 8(x4 + y4 ) , I couldn't understand why we used x2+y2 instead of using only the one that produces in shortest time? Thank you.

Answer 1

In the problem above, the following is the description of the decision variables:

x2 indicates the number of products of type B produced in the machine x

y2 indicates the number of products of type B produced in the machine y.

As mentioned in the question except for Product A all the other products can be made in either of the machines.

"(x2 + y2)" This will ensure that we count all those products of type B produced in machine x and y.

If we judge the number of products made based on only the shortest time, we might not utilize the other resource in case the current shortest time taking resource is used up. The above will ensure we use both the machines are used at optimal.

Imagine all the processing time in Machine X is used up and there is some processing time left in Machine Y, if we used the decision variable based on only the shortest time we might not use up the resource of leftover processing time of Machine Y

Also as it can be observed for the case of product A, only "x" is used because. both machine X and machine y is required.