A chemical manufacturer produces two products, which we shall call chemical X and chemical Y. Each product is manufactured by a two step process that involves blending and mixing in machine A and packaging on machine B. The two products complement each other because the same production facilities can be used for both products, thus achieving better utilization of these facilities.
Definition of Decision Variables:
Since the facilities are shared and the costs and profits from each product are different, there is a question about how to utilize the available machine time in the most profitable way. Chemical X is seemingly more profitable, and the manager once tried producing the maximum amount of chemical X within market limitations and using the balance of his capacity to produce chemical Y. The result, however, was that this allocation of machine time resulted in poor profit performance. The manager now feels that some appropriate balance between the two products is best, and he wishes to determine the production races for each product per two week period.
Thus, the decision variables are,
x, the number of units of chemical X to be produced.
y, the number of units of chemical Y to be produced.
Definition of Objective Function:
The physical plant and basic organization represent the fixed costs of the organization. These costs are irrelevant to the production scheduling decision, so they are ignored. The manager has, however, obtained price and variable cost information and has computed the contribution to profit and overhead per unit for each product sold as shown Table below. The objective is to maximize profit, and the contribution rates have a linear relationship to this objective. Therefore, the objectives functions is to maximum the sum of the total contribution for chemical X (60x) plus the total contribution from chemical Y (50y), or
Maximize Z = 60x + 50y
Definition of Constraints:
The processing times for each unit of the two products on the mixing machine A and the packaging machine, B, are as follows:
Machine A Machine B
Product (hours/unit) (hours/unit)
Chemical X 2 3
Chemical Y 4 2
For the upcoming two week period, machine A has 80 hours of processing time available B has 60 hours available.
Sales Prices, variable costs, and contributions per unit for chemical X and Y
Sales Price Variable Costs Contribution to
Profit and Overhead
(p) ( c) (r = p – r)
Chemical X $ 350 $290 $60
Chemical Y 450 400 50
Machine A Constraints:
Since we are limited by the 80 hours available on the mixing machine A, the total time spent in the manufacturing of chemical X and chemical Y cannot exceed the total time available. For machine A, because chemical X requires two hours per unit and chemical Y requires four hours per unit, the total time spent on the two products must be less than or equal to 80 hours; that is.
2x + 4y ≤80
Machine B Constraints:
Similarly, the available time on the packaging machine B is limited to 60 hours. Because chemical X requires three hours per unit and chemical Y requires two hours per unit, the total hours two products must be less than or equal to 60 hours; that is,
3x + 2y ≤60
Market forecasts indicate that we can expect to sell a maximum of 16 units of chemical X and 18 units of chemical Y. Therefore,
x ≤ 16
y ≤ 18
Minimum Production Constraints
The minimum production for each product is zero. Therefore,
x ≥ 0
y ≥ 0
The Linear Optimization Model:
We can now summarize a statement of the linear optimization model for the two-product chemical company in the standard linear programming format as follows:
Maximize Z = 60x + 50 y
2x + 4y ≤ 80 (machine A)
3x + 2y ≤ 60 (machine B)
x ≤ 16 ( demand for chemical X)
y ≤ 18 ( demand for chemical Y)
x ≥ 0 ( minimum production for chemical X)
y ≥ 0 ( minimum production for chemical Y)