1) Determine the optimal solution and optimal value and interpret their meanings
ID: 3122429 • Letter: 1
Question
1) Determine the optimal solution and optimal value and interpret their meanings.
2) Determine the slack (or surplus) value for each constraint and interpret its meaning.
PLEASE TYPE ALL ANSWERS - NO WRITTEN ANSWERS
Maximize 40X + 50Y + 20Z Total Revenue
Subject to X + Y + Z 400 Total quantity produced
X + 3Y + 2Z 500 Resource 1
2X + Y + 2Z 400 Resource 2
3X + Y + 2Z 600 Resource 3
And X, Y, Z 0
Where X, Y, and Z represent the number of units of Product 1, Product 2, and Product 3 to be manufactured.
The QM for Windows output for this problem is given below.
Linear Programming Results:
Variable Status Value
X Basic 140
Y Basic 120
Z NONBasic 0
slack 1 Basic 140
slack 2 NONBasic 0
slack 3 NONBasic 0
slack 4 Basic 60
Optimal Value 11600
Ranging Results:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
X 140 0 40 16.67 100
Y 120 0 50 20 120
Z 0 32 20 -Infinity 52
Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Constraint 1 0 140 400 260 Infinity
Constraint 2 12 0 500 200.00 1200
Constraint 3 14 0 400 166.67 437.5
Constraint 4 0 60 600 540 Infinity
Explanation / Answer
Solution
Linear Programming Results as given part of the question are reproduced below:
Variable Status Value
X Basic 140
Y Basic 120
Z NONBasic 0
slack 1 Basic 140
slack 2 NONBasic 0
slack 3 NONBasic 0
slack 4 Basic 60
Optimal Value 11600
Now, to work out solution,
Q 1
From the final results reproduced above,
optimal solution is X = 140, Y = 120, Slack 1 = 140, Slack 4 = 60 and optimal value is 11600.
Interpretation of their meanings:
Since X and Y represent quantity of Product 1 and Product 2 respectively, the final solution stipulates that 140 units of Product 1 and 120 units of Product 2 must be produced. Also, the other variable in the optimal solution being Slack 1 and Slack 4, which respectively refer to the Constraint 1 pertaining to production capacity,and Constraint 4 pertaining to Resource 3, production capacity is under-utilised to the extent of 140 units (Slack 1) and Resource 3 is under-utilised to the extent of 60 units (Slack 4). Further, Z is non-basic with value 0 => Product 3 will not be produced.
Optimal value is 11600 => by producing 140 units of Product 1 and 120 units of Product 2 (and no production of Product 3), would yield a total revenue of 11600.
Q2
Slack (or surplus) value for each constraint:
Since all constraints are of less-than-or-equal-to type, there are no surplus variables. All constraints have only slack variables. Of the four slack variables, only Slack 1 and Slack 4, which respectively refer to the Constraint 1 pertaining to production capacity and Constraint 4 pertaining to Resource 3, appear in the basis and their values are 140 and 60. Other two slack variables are zero.
Interpretation of their meanings:
Slack 1 with a value of 140 => production capacity is under-utilised to the extent of 140 units.
Slack 4 with a value of 60 => Resource 3 is under-utilised to the extent of 60 units.
Other slacks are zero => the other two constraints pertaining to Resources 2 and 3 are fully utilized.
DONE
[Additional inputs:
1. Substituting x = 140 and y = 120 in the objective function (Total Revenue) would give 11600 which is the optimal value.
2. Substituting x = 140 and y = 120 in Constraint 1 (Total quantity produced), LHS = 260, while RHS = 400 giving a difference of 140 which represents the value of Slack 1.
3. Substituting x = 140 and y = 120 in Constraint 2 and 3, LHS = RHS implying Resources 2 and 3 are fully utilized.]
4. Substituting x = 140 and y = 120 in Constraint 4. LHS = 540, while RHS = 600 giving a difference of 60 which represents the value of Slack 4.
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