Question 1

The improvement in the value of the optimal solution per-unit increase in a constraint's right-hand side is

Accepted Answer

A)  the slack value. 
B)  the dual price. 
C)  never negative. 
D)  the 100% rule. 
A)  the slack value. 
B)  the dual price. 
C)  never negative. 
D)  the 100% rule. B

Question 2

Creative Kitchen Tools manufactures a wide line of gourmet cooking tools from stainless steel. For the coming production period, there is demand of 1200 for 8 quart stock pots, and unlimited demand for 3 quart mixing bowls and large slotted spoons. In the following model, the three variables measure the number of pots, bowls, and spoons to make. The objective function measures profit. Constraint 1 measures steel, constraint 2 measures manufacturing time, constraint 3 measures finishing time, and constraint 4 measures the stock pot demand.
Max
5x1 + 3x2 + 6x3
s.t.
3x1 + 1x2 + 2x3 ≤ 15000
4x1 + 4x2 + 5x3 ≤ 18000
2x1 + 1x2 + 2x3 ≤ 10000
x1 ≤ 1200
x1, x2, x3 ≥ 0
​
The final tableau is:   ​ 
a.
Calculate the range of optimality for c1, c2, and c3.
b.
Calculate the range of feasibility for b1, b2, b3, and b4.
c.
​
Suppose that the inventory records were incorrect and the company really has only 14000 units of steel. What effect will this have on your solution?
d.
​
Suppose that a cost increase will change the profit on the pots to $4.62. What effect will this have on your solution?
e.
​
​
Assume that the cost of time in production and finishing is relevant. Would you be willing to pay a $1.00 premium over the normal cost for 1000 more hours in the production department? What would this do to your solution?

Accepted Answer

a.4.80 ≤ c₁ < ∞
−∞ < c₂ ≤ 5
−∞ < c₃ ≤ 6.25
B.13500 ≤ b₁ < ∞
480 ≤ b₂ ≤ 20000
9000 ≤ b₃ < ∞
−∞ < b₄ ≤ 4500
C.This would affect only the amount of slack, decreasing it from 1500 to 500.
D.This change is out of the range of optimality so the basis would changE.E.
An increase of 1000 hours is within the range of feasibility and the cost is less than the dual price, so it makes sense to do this. The new solution would be
s₁ = 1500 − .75(1000) = 750
s₄ = 3300 + .25(1000) = 3550
s₃ = 1000 − .5(1000) = 500
x₁ = 4500 + .25(1000) = 4750
Z = 22500 + (1.25 − 1)(1000) = 22750

Question 3

Given the simplex tableau for the optimal primal solution

Accepted Answer

A)  the values of the dual variables can be found from the cj − zj values of the slack/surplus variable columns. 
B)  the values of the dual surplus variables can be found from the cj − zj values of the primal decision variable columns. 
C)  the value of the dual objective function will be the same as the objective function value for the primal problem. 
D)  each of the above is true. 
A)  the values of the dual variables can be found from the cj − zj values of the slack/surplus variable columns. 
B)  the values of the dual surplus variables can be found from the cj − zj values of the primal decision variable columns. 
C)  the value of the dual objective function will be the same as the objective function value for the primal problem. 
D)  each of the above is true. D

Question 4

A linear programming problem with the objective function 3x1 + 8x2 has the optimal solution x1 = 5, x2 = 6. If c2 decreases by 2 and the range of optimality shows 5 ≤ c2 ≤ 12, the value of Z

Accepted Answer

A)  will decrease by 12. 
B)  will decrease by 2. 
C)  will not change. 
D)  cannot be determined from this information. 
A)  will decrease by 12. 
B)  will decrease by 2. 
C)  will not change. 
D)  cannot be determined from this information.

Question 5

Write the dual of the following problem
Min Z
= 2x1 − 3x2 + 5x3
s.t.
−3x1 + 2x2 + 5x3 ≥ 7
2x1 − x3 ≥ 5
4x 2 + 3x3 ≥ 8.

Accepted Answer

Max Z₂ = 7y₁ + 5y₂ +

Question 6

The primal problem is
Min
2x1 + 5x2 + 4x3
s.t.
x1 + 3x2 + 3x3 ≥ 30
3x1 + 7x2 + 5x3 ≥ 70
x1, x2, x3 ≥ 0
​
The final tableau for its dual problem is   ​
Give the complete solution to the primal problem.

Accepted Answer

The complete primal

Question 7

The range of optimality for a basic variable defines the objective function coefficient values for which the variable will remain part of the current optimal basic feasible solution.

Accepted Answer

A) True 
 B)False

Question 8

A one-sided range of optimality

Accepted Answer

A)  always occurs for non-basic variables. 
B)  always occurs for basic variables. 
C)  indicates changes in more than one coefficient. 
D)  indicates changes in a slack variable's coefficient. 
A)  always occurs for non-basic variables. 
B)  always occurs for basic variables. 
C)  indicates changes in more than one coefficient. 
D)  indicates changes in a slack variable's coefficient.

Question 9

For the basic feasible solution to remain optimal

Accepted Answer

A)  all cj − zj values must remain ≤ 0. 
B)  no objective function coefficients are allowed to change. 
C)  the value of the objective function must not change. 
D)  each of the above is true. 
A)  all cj − zj values must remain ≤ 0. 
B)  no objective function coefficients are allowed to change. 
C)  the value of the objective function must not change. 
D)  each of the above is true.

Question 10

Write the dual to the following problem.
Min
12x1 + 15x2 + 20x3 + 18x4
s.t.
x1 + x2 + x3 + x4 ≥ 50
3x1 + 4x3 ≥ 60
2x2 + x3 − 2x4 ≤ 10
x1, x2, x3, x4 ≥ 0

Accepted Answer

Max 50u₁ + 60u₂ − 10

Question 11

Dual prices and ranges for objective function coefficients and right-hand side values are found by considering

Accepted Answer

A)  dual analysis. 
B)  optimality analysis. 
C)  ranging analysis. 
D)  sensitivity analysis. 
A)  dual analysis. 
B)  optimality analysis. 
C)  ranging analysis. 
D)  sensitivity analysis.

Question 12

​Explain how to put an equality constraint into canonical form and how to calculate its dual variable value.

Accepted Answer

To put an equality constraint into canon

Question 13

If the dual price for b1 is 2.7, the range of feasibility is 20 ≤ b1 ≤ 50, and the original value of b1 was 30, which of the following is true?

Accepted Answer

A)  There currently is no slack in the first constraint. 
B)  We would be willing to pay up to $2.70 per unit for up to 20 more units of resource 1. 
C)  If only 25 units of resource 1 were available, profit would drop by $13.50. 
D)  Each of the above is true. 
A)  There currently is no slack in the first constraint. 
B)  We would be willing to pay up to $2.70 per unit for up to 20 more units of resource 1. 
C)  If only 25 units of resource 1 were available, profit would drop by $13.50. 
D)  Each of the above is true.

Question 14

The number of constraints to the dual of the following problem is: Max Z
= 3x1 + 2x2 + 6x3
S.t.
4x1 + 2x2 + 3x3 ≥ 100
2x1 + x2 − 2x3 ≤ 200
4x2 + x3 ≥ 200

Accepted Answer

A)  1. 
B)  2. 
C)  3. 
D)  4. 
A)  1. 
B)  2. 
C)  3. 
D)  4.

Question 15

The range of optimality is useful only for basic variables.

Accepted Answer

A) True 
 B)False

Question 16

The range of optimality is calculated by considering changes in the cj − zj value of the variable in question.

Accepted Answer

A) True 
 B)False

Question 17

Given the following linear programming problem
Max
10x1 + 12x2
s.t.
1x1 + 2x2 ≥ 40
5x1 + 8x2 ≤ 160
1x1 + 1x2 ≤ 40
x1, x2 ≥ 0
​
the final tableau is   ​ 
a.
Find the range of optimality for c1 and c2.
b.
Find the range of feasibility for b1, b2, and b3.
c.
Find the dual prices.

Accepted Answer

a.7.5 ≤ c₁ < ∞ −∞ <

Question 18

The linear programming problem:
Max
6x1 + 2x2 + 3x3 + 4x4
s.t.
x1 + x2 + x3 + x4 ≤ 100
4x1 + x2 + x3 + x4 ≤ 160
3x1 + x2 + 2x3 + 3x4 ≤ 240
x1, x2, x 3, x4 ≥ 0
​
has the final tableau:   ​
Fill in the table below to show what you would have found if you had used The Management Scientist to solve this problem.
LINEAR PROGRAMMING PROBLEM
MAX
6X1+2X2+3X3+4X4
S.T.
1) 1X1 + 1X2 + 1X3 + 1X4 < 100
2) 4X1 + 1X2 + 1X3 + 1X4 < 160
3) 3X1 + 1X2 + 2X3 + 3X4 < 240
​
OPTIMAL SOLUTION
Objective Function Value =   ​   ​
OBJECTIVE COEFFICIENT RANGES   ​
RIGHT HAND SIDE RANGES

Accepted Answer

LINEAR PROGRAMMING PROBLEM
MAX
6X1 + 2X2

Question 19

For this optimal simplex tableau the original right-hand sides were 100 and 90. The problem was a maximization.   ​ 
a.What would the new solution be if there had been 150 units available in the first constraint?
b.What would the new solution be if there had been 70 units available in the second constraint?

Accepted Answer

a.x₃ = 14.4, x₁ = 6, Z

Question 20

There is a dual price associated with each decision variable.

Accepted Answer

A) True 
 B)False

The improvement in the value of the optimal solution per-unit increase in a constraint's right-hand side is

Given the simplex tableau for the optimal primal solution

A linear programming problem with the objective function 3x₁ + 8x₂ has the optimal solution x₁ = 5, x₂ = 6. If c₂ decreases by 2 and the range of optimality shows 5 ≤ c₂ ≤ 12, the value of Z

Write the dual of the following problem Min Z = 2x₁ − 3x₂ + 5x₃ s.t. −3x₁ + 2x₂ + 5x₃ ≥ 7 2x₁ − x₃ ≥ 5 4x ₂ + 3x₃ ≥ 8.

The primal problem is Min 2x₁ + 5x₂ + 4x₃ s.t. x₁ + 3x₂ + 3x₃ ≥ 30 3x₁ + 7x₂ + 5x₃ ≥ 70 x₁, x₂, x₃ ≥ 0 The final tableau for its dual problem is Give the complete solution to the primal problem.

The range of optimality for a basic variable defines the objective function coefficient values for which the variable will remain part of the current optimal basic feasible solution.

A one-sided range of optimality

For the basic feasible solution to remain optimal

Write the dual to the following problem. Min 12x₁ + 15x₂ + 20x₃ + 18x₄ s.t. x₁ + x₂ + x₃ + x₄ ≥ 50 3x₁ + 4x₃ ≥ 60 2x₂ + x₃ − 2x₄ ≤ 10 x₁, x₂, x₃, x₄ ≥ 0

Dual prices and ranges for objective function coefficients and right-hand side values are found by considering

Explain how to put an equality constraint into canonical form and how to calculate its dual variable value.

If the dual price for b₁ is 2.7, the range of feasibility is 20 ≤ b₁ ≤ 50, and the original value of b₁ was 30, which of the following is true?

The number of constraints to the dual of the following problem is: Max Z = 3x₁ + 2x₂ + 6x₃ S.t. 4x₁ + 2x₂ + 3x₃ ≥ 100 2x₁ + x₂ − 2x₃ ≤ 200 4x₂ + x₃ ≥ 200

The range of optimality is useful only for basic variables.

The range of optimality is calculated by considering changes in the c_j − z_j value of the variable in question.

There is a dual price associated with each decision variable.

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance, and Operations Management

Advanced Linear Programming Applications

Distribution and Network Models

Integer Linear Programming

Nonlinear Optimization Models

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Decision Analysis

Multicriteria Decisions

Time Series Analysis and Forecasting

Markov Processes

Linear Programming: Simplex Method

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters

Exam 18: Simplex-Based Sensitivity Analysis and Duality

The improvement in the value of the optimal solution per-unit increase in a constraint's right-hand side is

Given the simplex tableau for the optimal primal solution

A linear programming problem with the objective function 3x1 + 8x2 has the optimal solution x1 = 5, x2 = 6. If c2 decreases by 2 and the range of optimality shows 5 ≤ c2 ≤ 12, the value of Z

Write the dual of the following problem Min Z = 2x1 − 3x2 + 5x3 s.t. −3x1 + 2x2 + 5x3 ≥ 7 2x1 − x3 ≥ 5 4x 2 + 3x3 ≥ 8.

The primal problem is Min 2x1 + 5x2 + 4x3 s.t. x1 + 3x2 + 3x3 ≥ 30 3x1 + 7x2 + 5x3 ≥ 70 x1, x2, x3 ≥ 0 ​ The final tableau for its dual problem is ​ Give the complete solution to the primal problem.

The range of optimality for a basic variable defines the objective function coefficient values for which the variable will remain part of the current optimal basic feasible solution.

A one-sided range of optimality

For the basic feasible solution to remain optimal

Write the dual to the following problem. Min 12x1 + 15x2 + 20x3 + 18x4 s.t. x1 + x2 + x3 + x4 ≥ 50 3x1 + 4x3 ≥ 60 2x2 + x3 − 2x4 ≤ 10 x1, x2, x3, x4 ≥ 0

Dual prices and ranges for objective function coefficients and right-hand side values are found by considering

​Explain how to put an equality constraint into canonical form and how to calculate its dual variable value.

If the dual price for b1 is 2.7, the range of feasibility is 20 ≤ b1 ≤ 50, and the original value of b1 was 30, which of the following is true?

The number of constraints to the dual of the following problem is: Max Z = 3x1 + 2x2 + 6x3 S.t. 4x1 + 2x2 + 3x3 ≥ 100 2x1 + x2 − 2x3 ≤ 200 4x2 + x3 ≥ 200

The range of optimality is useful only for basic variables.

The range of optimality is calculated by considering changes in the cj − zj value of the variable in question.

Given the following linear programming problem Max 10x1 + 12x2 s.t. 1x1 + 2x2 ≥ 40 5x1 + 8x2 ≤ 160 1x1 + 1x2 ≤ 40 x1, x2 ≥ 0 ​ the final tableau is ​ a. Find the range of optimality for c1 and c2. b. Find the range of feasibility for b1, b2, and b3. c. Find the dual prices.

There is a dual price associated with each decision variable.

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance, and Operations Management

Advanced Linear Programming Applications

Distribution and Network Models

Integer Linear Programming

Nonlinear Optimization Models

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Decision Analysis

Multicriteria Decisions

Time Series Analysis and Forecasting

Markov Processes

Linear Programming: Simplex Method

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters

A linear programming problem with the objective function 3x₁ + 8x₂ has the optimal solution x₁ = 5, x₂ = 6. If c₂ decreases by 2 and the range of optimality shows 5 ≤ c₂ ≤ 12, the value of Z

Write the dual of the following problem Min Z = 2x₁ − 3x₂ + 5x₃ s.t. −3x₁ + 2x₂ + 5x₃ ≥ 7 2x₁ − x₃ ≥ 5 4x ₂ + 3x₃ ≥ 8.

The primal problem is Min 2x₁ + 5x₂ + 4x₃ s.t. x₁ + 3x₂ + 3x₃ ≥ 30 3x₁ + 7x₂ + 5x₃ ≥ 70 x₁, x₂, x₃ ≥ 0 The final tableau for its dual problem is Give the complete solution to the primal problem.

Write the dual to the following problem. Min 12x₁ + 15x₂ + 20x₃ + 18x₄ s.t. x₁ + x₂ + x₃ + x₄ ≥ 50 3x₁ + 4x₃ ≥ 60 2x₂ + x₃ − 2x₄ ≤ 10 x₁, x₂, x₃, x₄ ≥ 0

Explain how to put an equality constraint into canonical form and how to calculate its dual variable value.

If the dual price for b₁ is 2.7, the range of feasibility is 20 ≤ b₁ ≤ 50, and the original value of b₁ was 30, which of the following is true?

The number of constraints to the dual of the following problem is: Max Z = 3x₁ + 2x₂ + 6x₃ S.t. 4x₁ + 2x₂ + 3x₃ ≥ 100 2x₁ + x₂ − 2x₃ ≤ 200 4x₂ + x₃ ≥ 200

The range of optimality is calculated by considering changes in the c_j − z_j value of the variable in question.