Deck 3: Introduction to Optimization Modeling

Question 1

Shadow prices are associated with nonbinding constraints,and show the change in the optimal objective function value when the right side of the constraint equation changes by one unit.

Answer 1

False

Question 2

When the profit increases with a unit increase in a resource,this change in profit will be shown in Solver's sensitivity report as the:

A) add-in price 
B) sensitivity price 
C) shadow price 
D) additional profit

Answer 2

C

Question 3

Suppose the allowable increase and decrease for an objective coefficient of a decision variable that has a current value of $50 are $25 (increase)and $10 (decrease).If the coefficient were to change from $50 to $65,the optimal value of the objective function would not change.

Answer 3

False

Question 4

If the objective function has the equation {4x1+ 2x2},then the slope of the objective function line is −2.

Question 5

All linear programming problems should have a unique solution,if they can be solved.

Question 6

If a constraint has the equation 5x1+ 2x2 ≤ 60,then the constraint line passes through the points (0,12)and (30,0).

Question 7

When formulating a linear programming spreadsheet model,there is a set of designated cells that play the role of the decision variables.These are called the changing variable cells.

Question 8

Common errors in LP models that exhibit unboundedness are a constraint that has been omitted or an input which is incorrect.

Question 9

The feasible region in a graphical solution of a linear programming problem will appear as some type of polygon,with lines forming all sides.

Question 10

If a manufacturing process takes 4 hours per unit of x1and 2 hours per unit of x2 and a maximum of 100 hours of manufacturing process time are available,then an algebraic formulation of this constraint is:

A) 4x1 + 2x2 ≥ 100 
B) 4x1 − 2x2 ≤ 100 
C) 4x1 + 2x2 ≤ 100 
D) 4x1 − 2x2 ≥ 100

Question 11

Suppose a company sells two different products,x1and x2,for net profits of $6 per unit and $3 per unit,respectively.The slope of the line representing the objective function is:

A) 0.5 
B) −0.5 
C) 2 
D) −2

Question 12

Consider the following linear programming problem:The above linear programming problem:Maximize4x1 + 2x2Subject to:4x1 + 2x2 ≤ 402x1 + x2≥ 20x1,x2 ≥ 0

A) has only one feasible solution 
B) has more than one optimal solution 
C) exhibits infeasibility 
D) exhibits unboundedness

Question 13

The proportionality property of LP models implies that the sum of the contributions from the various activities to a particular constraint equals the total contribution to that constraint.

Question 14

In using Excel to solve linear programming problems,the changing variable cells represent the:

A) value of the objective function 
B) constraints 
C) decision variables 
D) total cost of the model

Question 15

The feasible region in all linear programming problems is bounded by:

A) corner points 
B) hyperplanes 
C) an objective line 
D) all of these options

Question 16

Linear programming models have three important properties.They are:

A) optimality, additivity and sensitivity 
B) optimality, linearity and divisibility 
C) divisibility, linearity and nonnegativity 
D) proportionality, additivity and divisibility

Question 17

The condition of nonnegativity requires that:

A) the objective function cannot be less that zero 
B) the decision variables cannot be less than zero 
C) the right hand side of the constraints cannot be greater then zero 
D) the reduced cost cannot be less than zero

Question 18

In an optimization model,there can only be one:

A) decision variable 
B) constraint 
C) objective function 
D) shadow price

Question 19

The equation of the line representing the constraint 4x1+ 2x2 ≤ 100 passes through the points:

A) (25,0) and (0,50) 
B) (0,25) and (50,0) 
C) (−25,0) and (0,−50) 
D) (0,−25) and (−50,0)

Question 20

Suppose the shadow price for a constraint is $12 and the allowable increase and decrease for the right hand side of the constraint are 25 (increase)and 10 (decrease).If the right hand side of that constraint were to increase by 10 the shadow price changes.

Question 21

Exhibit 3-2
Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin (contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold.
Refer to Exhibit 3-2.What is the incremental contribution associated with adding an hour of assembly time
Over what range of increase is the marginal value valid

Question 22

Exhibit 3-2
Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin (contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold.
Refer to Exhibit 3-2.What is the value of additional capacity on the polisher
How much increase and decrease in this capacity is possible before a change occurs in the optimal production schedule

Question 23

Exhibit 3-2
Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin (contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold.
[Part 2] Refer to Exhibit 3-2.Obtain a sensitivity report for the solution reported in Part 1.Which constraints are binding

Question 24

Exhibit 3-2
Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin (contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold.
[Part 1] Refer to Exhibit 3-2.Find an optimal solution to Western's problem.What is the production plan,and what is the total revenue

Question 25

Exhibit 3-1
A winemaker in California's Napa Valley must decide how much of two types of wine she will produce from a particular variety of grapes. Each liter of table wine yields $8 profit, while each liter of dessert wine produces $3 profit. The labor hours and bottling process time used for type of wine are given in the table below. Resources available include 200 labor hours and 80 hours of bottling process time. Assume the winemaker has more than enough grapes available to supply any feasible production plan.
​
[Part 2] Refer to Exhibit 3-1.Using the graphical solution method,find an optimal solution to the model in Part 1 and determine the maximum profit.

Question 26

Exhibit 3-1
A winemaker in California's Napa Valley must decide how much of two types of wine she will produce from a particular variety of grapes. Each liter of table wine yields $8 profit, while each liter of dessert wine produces $3 profit. The labor hours and bottling process time used for type of wine are given in the table below. Resources available include 200 labor hours and 80 hours of bottling process time. Assume the winemaker has more than enough grapes available to supply any feasible production plan.
​
[Part 3] Refer to Exhibit 3-1.Implement the model in Part 1 in Excel Solver and obtain an answer report.Which constraint(s)are binding on the optimal solution

Question 27

Exhibit 3-1
A winemaker in California's Napa Valley must decide how much of two types of wine she will produce from a particular variety of grapes. Each liter of table wine yields $8 profit, while each liter of dessert wine produces $3 profit. The labor hours and bottling process time used for type of wine are given in the table below. Resources available include 200 labor hours and 80 hours of bottling process time. Assume the winemaker has more than enough grapes available to supply any feasible production plan.
​
[Part 1] Refer to Exhibit 3-1.Formulate a linear programming model that will enable the winemaker to determine the number of liters of each type of wine to produce in order to maximize her profit.

Question 28

Exhibit 3-1
A winemaker in California's Napa Valley must decide how much of two types of wine she will produce from a particular variety of grapes. Each liter of table wine yields $8 profit, while each liter of dessert wine produces $3 profit. The labor hours and bottling process time used for type of wine are given in the table below. Resources available include 200 labor hours and 80 hours of bottling process time. Assume the winemaker has more than enough grapes available to supply any feasible production plan.
​
[Part 4] Refer to Exhibit 3-1.Obtain a sensitivity report for the model in Part 1.How much should the winemaker be willing to pay for an additional labor hour

Question 29

Exhibit 3-1
A winemaker in California's Napa Valley must decide how much of two types of wine she will produce from a particular variety of grapes. Each liter of table wine yields $8 profit, while each liter of dessert wine produces $3 profit. The labor hours and bottling process time used for type of wine are given in the table below. Resources available include 200 labor hours and 80 hours of bottling process time. Assume the winemaker has more than enough grapes available to supply any feasible production plan.
​
[Part 5] Refer to Exhibit 3-1.Suppose the winemaker can obtain 100 addition labor hours.Can you use the sensitivity analysis obtained for Part 4 to determine her expected profit
Would her bottling plan change
Explain your answer.

Question 30

Exhibit 3-2
Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin (contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold.
Refer to Exhibit 3-2.An advertising agency has devised a marketing plan for the Western Chassis Company that will increase the market for Deluxe chassis.The plan will increase demand by 75 Deluxe chassis per month at a cost of $100 per month.Should Western adopt the plan
Briefly explain why.