Problem on Decision variables

Question

A factory has three distinct systems for making similar product:

System 1: Worker runs 3 machines of type-A, each of which costs $20 per day to run, each generates 100 units per day and the worker is paid $40 per day.
System 2: Worker runs 5 machines of type-A, each of which costs $20 per day to run, generates 70 units per day and the worker is paid $50 per day.
System 3: Worker runs 2 machines of type-B, each of which costs $35 per day to run, generates 160 units per day and the worker is paid $60 per day.
There are 45 machines of type-A, 10 of type-B and 16 workers. Each and every unit can be sold for $50.

Give answers to the questions below:

1. Give a model which explains how production must be organized so as to maximize gain. Clearly explain all computations, formulas and model.

2. Give an optimal solution to your model supposing continuous decision variables (explain the solution).

3. Determine the maximum amount you would pay for an extra machine type A, B? How did you come up to this conclusion? Would you hire additional workers? Determine the maximum you would pay per day?

Answer 1

(1) Decision variables:

X1 - Number of setups as system 1
X2 – Number of setups as system 2
X3 – Number of setups as system 3

Objective function
Max Z = $15000X1 + $17500X2 + $16000X3 - $40X1 - $60X1 - $50X2 - $100X2 - $60X3 - $70X3
Max Z = $14900X1 + $17350X2 + $15870X3
For X1, the revenue = 3machines*100units*selling price$50 = $15,000 and similarly for others.

Constraints
3X1 + 5X2 ≤ 45 (constraint for type A machines)
2X3 ≤ 10 (constraint for type B machines)
X1 + X2 + X3 ≤ 16 (constraint for workers)
X1, X2, X3 ≥ 0 (continuous decision variable and so integer is not assumed)

(2) The optimal solution was found using excel solver and it was found to be 5 setups of system 1, 6 setups of system 2 and 5 setups of system 3 to achieve a maximum profit of $257,950.

(3) The maximum amount that could be paid for an extra machine for type A is $1225 and that for type B is $2322.5, since increase in these availability values by 1 unit will increase the total profit by $1225 and $2322.5 respectively (meaning they are Lagrange multipliers for type A and B machines). Yes, it is profitable to hire extra workers. An increase in the number of workers by 1 per day can increase the profit by $11225 (Lagrange multiplier for worker usage). Hence the maximum that could be paid per day is $11225.