Based on the information below, please answer the following: (a) Determine the o
ID: 446150 • Letter: B
Question
Based on the information below, please answer the following:
(a) Determine the optimal solution and optimal value and interpret their meanings.
(b) Determine the slack (or surplus) value for each constraint and interpret its meaning.
(c) What are the ranges of optimality for the profit of Product 1, Product 2, Product 3, and Product 4?
(d) Find the dual prices of the four constraints and interpret their meanings. What are the ranges in which each of these dual prices is valid?
(e) If the profit contribution of Product 2 changes from $100 per unit to $130 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 below).
(f) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given below.).
Let P1 = number of Product 1 to be produced
P2 = number of Product 2 to be produced
P3 = number of Product 3 to be produced
P4 = number of Product 4 to be produced
Maximize 80P1 + 100P2 + 120P3 + 70P4 Total profit
Subject to
10P1 + 12P2 + 10P3 + 8P4 3200 Production budget constraint
4P1 + 3P2 + 2P3 + 3P4 1000 Labor hours constraint
5P1 + 4P2 + 3P3 + 3P4 1200 Material constraint
P1 > 100 Minimum quantity needed for Product 1 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 100
P2 NONBasic 0
P3 Basic 220
P4 NONBasic 0
slack 1 NONBasic 0
slack 2 Basic 160
slack 3 Basic 40
surplus 4 NONBasic 0
Optimal Value (Z) 34400
Original problem w/answers:
P1 P2 P3 P4 RHS Dual
Maximize 80 100 120 70
Constraint 1 10 12 10 8 <= 3200 12
Constraint 2 4 3 2 3 <= 1000 0
Constraint 3 5 4 3 3 <= 1200 0
Constraint 4 1 0 0 0 >= 100 -40
Solution-> 100 0 220 0 Optimal Z-> 34400
Ranging Results:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
P1 100 0 80 -Infinity 120
P2 0 44 100 -Infinity 144
P3 220 0 120 87.5 Infinity
P4 0 26 70 -Infinity 96
Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Constraint 1 12 0 3200 1000 3333.333
Constraint 2 0 160 1000 840 Infinity
Constraint 3 0 40 1200 1160 Infinity
Constraint 4 -40 0 100 0 120
Explanation / Answer
(a) The optimal solution is X1 = 0, X2 = 60, X3 = 160, and the optimal value is 4100.
Interpretation: The Company should manufacture 60 units of product 2, 160 units of product 3, and 0 units of product 1. If they do that, then they will make a total profit of $4100.
(b) The surplus variable is S1 = 20. The slack variables are S2 = 0, S3 = 20, and S4 = 0.
Interpretation: There are 20 units produced in excess of the required 200 units. There are 20 units of unused resource 2. All available units of resource 1 and resource 3 are being used.
(c) The ranges of optimality for the objective function coefficients are given below:
Range of optimality for the objective function coefficient of X1: From no lower limit to 13.
Range of optimality for the objective function coefficient of X2: From 13.3333 to 30.
Range of optimality for the objective function coefficient of X3: From 16.25 to 22.5.
(d) The dual prices for the four constraints are 0, 1, 0, and 6 respectively.
Interpretation: The dual price for an additional unit to total quantity produced is worth $0 to the company. This value is valid only in the range of no lower limit to 220 total units. The dual price for the total quantity produced is not worth any profit to the company because they are already producing more than the specified total number of units.
The dual price for an additional unit of resource 1 is worth $1 to the company. This additional profit is valid only in the range of 400 to 900 units of resource 1.
The dual price for an additional unit of resource 2 is worth $0 to the company. This value is valid only in the range of 380 to no upper limit units of resource 3. The dual price for resource 2 is not worth any profit to the company because there is already unused amount of resource 2 available.
The dual price for an additional unit of resource 3 is worth $6 to the company. This additional profit is valid only in the range of 500 to 625 units of resource 3.
(e) If the profit contribution of a unit of Product 2 changed from $15 to $25, it is still within the range of optimality (from 13.3333 to 30) for the profit contribution of Product 2. Therefore, the optimal solution remains the same. The optimal total profit will change.
New optimal total profit = 10X1 + 25X2 + 20X3 = 10(0) + 25(60) + 20(160) = $4700.
The total profit has increased by
4700 – 4100 = $600.
(f) If the availability of Resource 3 changed from 600 units to 620 units, it is still within the range of optimality (from 500 to 625) for the range of feasibility for Resource 2. Therefore, the company will still manufacture Product 2 and Product 3. It will still not manufacture Product 1. The optimal solution will change because the quantities of Product 2 and Product 3 manufactured will change. The optimal total profit will also change.
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