Maximize 120P1 + 150P2 + 180P3 + 105P4 Total profit Subject to 4P1 + 12P2 + 10P3
ID: 3356487 • Letter: M
Question
Maximize 120P1 + 150P2 + 180P3 + 105P4 Total profit
Subject to
4P1 + 12P2 + 10P3 + 8P4 12000 Production budget constraint
4P1 + 3P2 + 2P3 + 3P4 4000 Labor hours constraint
P3 > 200 Minimum quantity needed for Product 1 constraint
P4 > 100 Minimum quantity needed for Product 2 constraint
And P1, P2, P3, P4 0 Non-negativity constraints
The QM for Windows output for this problem is given below.
Linear Programming Results:
Variable Status Value
P1 Basic 456.25
P2 NONBasic 0
P3 Basic 937.5
P4 Basic 100
slack 1 NONBasic 0
slack 2 NONBasic 0
surplus 3 Basic 737.5
surplus 4 NONBasic 0
Optimal Value (Z) 234000
Original problem w/answers:
P1 P2 P3 P4 RHS Dual
Maximize 120 150 180 105
Constraint 1 4 12 10 8 <= 12000 15
Constraint 2 4 3 2 3 <= 4000 15
Constraint 3 0 0 1 0 >= 200 0
Constraint 4 0 0 0 1 >= 100 -60
Solution-> 456.25 0 937.5 100 Optimal Z-> 234000
Ranging Results:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
P1 456.25 0 120 72 360
P2 0 75 150 -Infinity 225
P3 937.5 0 180 113.3333 300
P4 100 0 105 -Infinity 165
Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Constraint 1 15 0 12000 6100 19300
Constraint 2 15 0 4000 2540 9900
Constraint 3 0 737.5 200 -Infinity 937.5
Constraint 4 -60 0 100 0 1142.857
3. (a) What are the ranges of optimality for the profit of Product 1, Product 2, Product 3, and Product 4?
(b) Find the dual prices of the four constraints and interpret their meanings. Determine the ranges in which each of these dual prices is valid.
(c) If the profit contribution of Product 2 changes from $150 per unit to $200 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the ranging results given above).
(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given above.).
Explanation / Answer
3 a) Ranges of optimality for P1, P2, P3 & P4 are:
Range of optimality for P1: 72 - 360 in the sense that the objective function coefficient which is 120 now can lower till 72 or can be increased till 360. It is as good as saying that the profit/unit for P1 which is 120 can be changed to 72 - 360 without changing the current optimal values.
Range of optimality for P2: -ve infinity to 225. Since the profit/unit cannot go beyond 0, here -ve infinity can be considered as 0 profit/unit. It can be decreased to zero profits or can be increased till 225 without affecting the current optimal values.
Range of optimality for P3: 113.33 - 300, the current objective function coefficient of Product P3 which is 180 can be lowered till 113.33 or can be increased till 300 without affecting the current optimal results.
Range of optimality for P4: -Infinity to 165, current coeff value which is 108 can be lowered till 0 or can be increased till 165 for product P4 without affecting the current optimal values.
3 b) Dual prices of the four constraints and the ranges within which these dual prices hold good and beyond these limits, dual prices will also change.
Shadow price of 15 for constraint 1 means that for every one monetary unit increase in production budget, profit will increase by 15 monetary units. It is as good as saying that if the production unit somehow arranges $1 for production purpose, profits would increase by $15. And this would go on till $19300. And for every $1 decrease, the profits would shrink by $15 and this would go on till $6100. Beyond $6100 and $19300, the shadow price of $15 will also change.
Shadow price of 15 for constraint 2 means that for every one labor hour extra gained, the profits would increase by 15 monetary units and the limits which this holds good is 2540 - 9900, beyond these limits, the shadow price of 15 will change.
Shadow price of 0 for constraint 3 is because there are already 737.5 units as slack (excess). So, if somehow the manufacturing unit makes Product P3, profit would not increase but would increase the slack by so much. This is till 737.5 units, beyond this the dual price of 0 will not hold good.
Shadow price of -60 for constraint 4 means that if manufacturing unit makes even 1 extra unit of P4, it would lose 60 monetary units. And this holds good till the limits 0 - 1142.857, beyond these, the shadow price of -60 will change.
3c) If the profit contribution of product P2 changes from 150 to 200, current optimal values would not change as this increase is well within the limits (upper bound of 225), but the new total profit would be again same as we are currently not manufacturing anything of P2 which is 0.
120 * 456.25 + 200 * 0 + 180 * 937.5 + 108 * 100 = 234000
3d) As dual price of constraint 1 & 2 are same, resource 1 should be obtained in large qty as the upper bound of the shadow price being 15 is till 19300, this means that the profit can be increased by almost 7300 *15 = 109500 monetary units, which is really very huge.
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Constraints Dual Prices Lower Limit Upper limit 1. Production Budget 15 6100 19300 2. Labor Hrs 15 2540 9900 3. Min Qty for P3 0 -Infinity 937.5 4. Min Qty for P4 -60 0 1142.857