Quiz
Question 1 (1 point)
The specification or description of the relationship between the dependent and independent variables is generally called
Question 1 options:
a)
a constraint.
b)
a declaration.
c)
a function.
d)
a mathematical model.
Question 2 (1 point)
The ultimate goal of the problem identification step of the problem-solving process is
Question 2 options:
a)
collecting lots of information.
b)
helping the decision maker realize there is a problem.
c)
identifying the root problem or problems causing the mess.
d)
convincing the decision maker the mess is really a problem that can be solved.
Question 3 (1 point)
To be effective, a modeler must
Question 3 options:
a)
be an effective presenter of results.
b)
collect the proper input data for the model.
c)
understand how modeling fits into the problem-solving process.
d)
apply the correct modeling technique.
Question 4 (1 point)
The goal of the modeling approach to problem solving is to
Question 4 options:
a)
help individuals make good decisions.
b)
ensure optimality of decisions.
c)
determine a set of optimal decisions.
d)
determine feasibility of decisions.
Question 5 (1 point)
A mathematical model is considered to be "valid" when
Question 5 options:
a)
it accurately represents the relevant characteristics of the object or decision.
b)
it has passed a validation test.
c)
it replicates all aspects of the object or decision.
d)
the left-hand and right-hand sides of expressions are equal.
Question 6 (1 point)
Variables are termed independent when they satisfy which of the following?
Question 6 options:
a)
The function value depends upon their values.
b)
The decision maker has no control over them.
c)
The variables have no relationship to one another.
d)
The variable is described as an output of the spreadsheet model.
Question 7 (1 point)
The best models
Question 7 options:
a)
accurately reflect relevant characteristics of the real-world object or decision.
b)
are mathematical models.
c)
replicate all aspects of the real-world object or decision.
d)
replicate the characteristics of a component in isolation from the rest of the system.
Question 8 (1 point)
In which of the following categories of modeling techniques do the independent variables have unknown or uncertain values or coefficients?
Question 8 options:
a)
Descriptive models
b)
Predictive models
c)
Prescriptive models
d)
Probabilistic models
Question 9 (1 point)
Which of the following fields of study is defined in Chapter One as the one that "uses computers, statistics, and mathematics to solve business problems"?
Question 9 options:
a)
Accounting
b)
Information systems
c)
Business analytics
d)
Scientific management
Question 10 (1 point)
Solutions to which of the following categories of modeling techniques indicate a course of action to the decision maker?
Question 10 options:
a)
Descriptive models
b)
Predictive models
c)
Prescriptive models
d)
Preventive models
Question 11 (1 point)
Which of the following statements is true of using models in problem solving and decision analysis?
Question 11 options:
a)
It is a fairly new idea.
b)
It is required in order to find good solutions.
c)
It is something everyone has done before.
d)
It is tied to the use of computers.
Question 12 (1 point)
Why would someone wish to use a spreadsheet model?
Question 12 options:
a)
To implement a computer model.
b)
Because spreadsheets are convenient.
c)
To analyze decision alternatives.
d)
All of these.
Question 13 (1 point)
A company makes two products, X1 and X2. They require at least 20 of each be produced. Which set of lower bound constraints reflect this requirement?
Question 13 options:
a)
X1 ≥ 20, X2 ≥ 20
b)
X1 + X2 ≥ 20
c)
X1 + X2 ≥ 40
d)
X1 ≥ 20, X2 ≥ 20, X1 + X2 ≤ 40
Question 14 (1 point)
A diet is being developed which must contain at least 100 mg of vitamin C. Two fruits are used in this diet. Bananas contain 30 mg of vitamin C and Apples contain 20 mg of vitamin C. The diet must contain at least 100 mg of vitamin C. Which of the following constraints reflects the relationship between Bananas, Apples and vitamin C?
Question 14 options:
a)
20 A + 30 B ≥ 100
b)
20 A + 30 B ≤ 100
c)
20 A + 30 B = 100
d)
20 A = 100
Question 15 (1 point)
A company uses 4 pounds of resource 1 to make each unit of X1 and 3 pounds of resource 1 to make each unit of X2. There are only 150 pounds of resource 1 available. Which of the following constraints reflects the relationship between X1, X2 and resource 1?
Question 15 options:
a)
4 X1 + 3 X2 ≥ 150
b)
4 X1 + 3 X2 ≤ 150
c)
4 X1 + 3 X2 = 150
d)
4 X1 ≤ 150
Question 16 (1 point)
The desire to maximize profits is an example of a(n)
Question 16 options:
a)
decision.
b)
constraint.
c)
objective.
d)
parameter.
Question 17 (1 point)
The first step in formulating a linear programming problem is
Question 17 options:
a)
Identify any upper or lower bounds on the decision variables.
b)
State the constraints as linear combinations of the decision variables.
c)
Understand the problem.
d)
Identify the decision variables.
e)
State the objective function as a linear combination of the decision variables.
Question 18 (1 point)
The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits.
X1 = number of product 1 produced in each batch
X2 = number of product 2 produced in each batch
MAX: 150 X1 + 250 X2
Subject to: 2 X1 + 5 X2 ≤ 200
3 X1 + 7 X2 ≤ 175
X1, X2 ≥ 0
How much profit is earned if the company produces 10 units of product 1 and 5 units of product 2?
Question 18 options:
a)
750
b)
2500
c)
2750
d)
3250
Question 19 (1 point)
The constraint for resource 1 is 5 X1 + 4 X2 ≥ 200. If X1 = 40 and X2 = 20, how many additional units, if any, of resource 1 are employed above the minimum of 200?
Question 19 options:
a)
0
b)
20
c)
40
d)
80
Question 20 (1 point)
Which of the following actions would expand the feasible region of an LP model?
Question 20 options:
a)
Loosening the constraints.
b)
Tightening the constraints.
c)
Multiplying each constraint by 2.
d)
Adding an additional constraint.
Question 21 (1 point)
A common objective in the product mix problem is
Question 21 options:
a)
maximizing cost.
b)
maximizing profit.
c)
minimizing production time.
d)
maximizing production volume.
Question 22 (1 point)
The objective function for a LP model is 3 X1 + 2 X2. If X1 = 20 and X2 = 30, what is the value of the objective function?
Question 22 options:
a)
0
b)
50
c)
60
d)
120
Question 23 (1 point)
What is the goal in optimization?
Question 23 options:
a)
Find the decision variable values that result in the best objective function and satisfy all constraints.
b)
Find the values of the decision variables that use all available resources.
c)
Find the values of the decision variables that satisfy all constraints.
d)
None of these.
Question 24 (1 point)
Linear programming problems have
Question 24 options:
a)
linear objective functions, non-linear constraints.
b)
non-linear objective functions, non-linear constraints.
c)
non-linear objective functions, linear constraints.
d)
linear objective functions, linear constraints.
Question 25 (1 point)
Which type of spreadsheet cell represents the left hand sides (LHS) formulas in an LP model?
Question 25 options:
a)
Target or set cell
b)
Changing variable cell
c)
Constraint cell
d)
Constant cell
Question 26 (1 point)
Which type of spreadsheet cell represents the objective function in an LP model?
Question 26 options:
a)
Objective cell
b)
Changing variable cell
c)
Constraint cell
d)
Constant cell
Question 27 (1 point)
Which type of spreadsheet cell represents the decision variables in an LP model?
Question 27 options:
a)
Target or set cell
b)
Variable cell
c)
Constraint cell
d)
Constant cell
Question 28 (1 point)
The constraints X1 ≥ 0 and X2 ≥ 0 are referred to as
Question 28 options:
a)
positivity constraints.
b)
optimality conditions.
c)
left hand sides.
d)
nonnegativity conditions.
Question 29 (2 points)
Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited, so at most 8 will be produced.
Let X1 = Number of Beds to produce
X2 = Number of Desks to produce
The LP model for the problem is
MAX: 30 X1 + 40 X2
Subject to: 6 X1 + 4 X2 ≤ 36 (carpentry)
4 X1 + 8 X2 ≤ 40 (varnishing)
X2 ≤ 8 (demand for desks)
X1, X2 ≥ 0
A B C D E
1 Jones Furniture
2
3 Beds Desks
4 Number to make: Total Profit:
5 Unit profit: 30 40
6
7 Constraints: Used Available
8 Carpentry 6 4 36
9 Varnishing 4 8 40
10 Desk demand 1 8
Refer to Exhibit 3.1. What formula should be entered in cell E5 in the accompanying Excel spreadsheet to compute total profit?
Question 29 options:
a)
=B4*B5+C4*C5
b)
=SUMPRODUCT(B8:C8,$B$4:$C$4)
c)
=SUM(B5:C5)
d)
=SUM(E8:E10)
Question 30 (2 points)
Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited, so at most 8 will be produced.
Let X1 = Number of Beds to produce
X2 = Number of Desks to produce
The LP model for the problem is
MAX: 30 X1 + 40 X2
Subject to: 6 X1 + 4 X2 ≤ 36 (carpentry)
4 X1 + 8 X2 ≤ 40 (varnishing)
X2 ≤ 8 (demand for desks)
X1, X2 ≥ 0
A B C D E
1 Jones Furniture
2
3 Beds Desks
4 Number to make: Total Profit:
5 Unit profit: 30 40
6
7 Constraints: Used Available
8 Carpentry 6 4 36
9 Varnishing 4 8 40
10 Desk demand 1 8
Refer to Exhibit 3.1. What formula should be entered in cell D8 in the accompanying Excel spreadsheet to compute the amount of carpentry used?
Question 30 options:
a)
=B4*B5+C4*C5
b)
=SUMPRODUCT(B8:C8,$B$4:$C$4)
c)
=SUM(B5:C5)
d)
=SUM(E8:E10)
Question 31 (2 points)
Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited, so at most 8 will be produced.
Let X1 = Number of Beds to produce
X2 = Number of Desks to produce
The LP model for the problem is
MAX: 30 X1 + 40 X2
Subject to: 6 X1 + 4 X2 ≤ 36 (carpentry)
4 X1 + 8 X2 ≤ 40 (varnishing)
X2 ≤ 8 (demand for desks)
X1, X2 ≥ 0
A B C D E
1 Jones Furniture
2
3 Beds Desks
4 Number to make: Total Profit:
5 Unit profit: 30 40
6
7 Constraints: Used Available
8 Carpentry 6 4 36
9 Varnishing 4 8 40
10 Desk demand 1 8
Refer to Exhibit 3.1. Which cells should be changing cells in this problem?
Question 31 options:
a)
B4:C4
b)
E5
c)
D8:D10
d)
E8:E10
Question 32 (2 points)
Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited, so at most 8 will be produced.
Let X1 = Number of Beds to produce
X2 = Number of Desks to produce
The LP model for the problem is
MAX: 30 X1 + 40 X2
Subject to: 6 X1 + 4 X2 ≤ 36 (carpentry)
4 X1 + 8 X2 ≤ 40 (varnishing)
X2 ≤ 8 (demand for desks)
X1, X2 ≥ 0
A B C D E
1 Jones Furniture
2
3 Beds Desks
4 Number to make: Total Profit:
5 Unit profit: 30 40
6
7 Constraints: Used Available
8 Carpentry 6 4 36
9 Varnishing 4 8 40
10 Desk demand 1 8
Refer to Exhibit 3.1. Which of the following statements represent the carpentry, varnishing and limited demand for desks constraints?
Question 32 options:
a)
B4:C4 ≤ B5:C5
b)
E5 ≤ 0
c)
D8:D10 ≤ E8:E10
d)
E8:E10 ≤ D8:D10
Question 33 (1 point)
Given an objective function value of 150 and a shadow price for resource 1 of 5, if 10 more units of resource 1 are added (assuming the allowable increase is greater than 10), what is the impact on the objective function value?
Question 33 options:
a)
increase of 50
b)
increase of unknown amount
c)
decrease of 50
d)
increase of 10
Question 34 (1 point)
Which of the following statements is false concerning either of the Allowable Increase and Allowable Decrease columns in the Sensitivity Report?
Question 34 options:
a)
The values equate the decision variable profit to the cost of resources expended.
b)
The values give the range over which a shadow price is accurate.
c)
The values give the range over which an objective function coefficient can change without changing the optimal solution.
d)
The values provide a means to recognize when alternate optimal solution exist.
Question 35 (1 point)
A binding less than or equal to (≤) constraint in a maximization problem means
Question 35 options:
a)
that all of the resource represented by the constraint is consumed in the solution.
b)
it is not a constraint that the level curve contacts.
c)
another constraint is limiting the solution.
d)
the requirement for the constraint has been exceeded.
Question 36 (1 point)
The allowable decrease for a changing cell (decision variable) is
Question 36 options:
a)
the amount by which the constraint coefficient can decrease without changing final optimal solution.
b)
an indication of how many more units to produce to maximize profits.
c)
the amount by which objective function coefficient can decrease without changing the final optimal solution.
d)
an indication of how much to charge to get the optimal solution.
Question 37 (1 point)
The allowable increase for a constraint is
Question 37 options:
a)
how many more units of resource to purchase to maximize profits.
b)
the amount by which the resource can increase given shadow price.
c)
how much resource to use to get the optimal solution.
d)
the amount by which the constraint coefficient can increase without changing the final optimal value.
Question 38 (1 point)
A binding greater than or equal to (≥) constraint in a minimization problem means that
Question 38 options:
a)
the variable is up against an upper limit.
b)
the minimum requirement for the constraint has just been met.
c)
another constraint is limiting the solution.
d)
the shadow price for the constraint will be positive.
Question 39 (1 point)
The allowable increase for a changing cell (decision variable) is
Question 39 options:
a)
how many more units to produce to maximize profits.
b)
the amount by which the objective function coefficient can increase without changing the optimal solution.
c)
how much to charge to get the optimal solution.
d)
the amount by which constraint coefficient can increase without changing the optimal solution.
Question 40 (1 point)
If the allowable increase for a constraint is 100 and we add 110 units of the resource what happens to the objective function value?
Question 40 options:
a)
increase of 100
b)
increase of 110
c)
decrease of 100
d)
increases but by unknown amount
Question 41 (2 points)
Exhibit 4.1
The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report.
Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week
Trucking Limits
Railway Limits
Air Cargo Limits
1
45
60
15
2
50
55
10
3
55
45
5
Costs ($ per 1000 tons)
$200
$140
$400
The following is the LP model for this logistics problem.
Let Xij = amount shipped by mode i in week j
where i = 1(Truck), 2(Rail), 3(Air)
and j = 1, 2, 3
Let WLij = weekly limit of mode i in week j (as provided in above table)
MIN: 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
Subject to:
Xij ≤ WL ij for all i and j Weekly limits by mode
X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 Total at end of three weeks
X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of two weeks
X11 + X21 + X31 ≥ 120 Total at end of first week
X11 + X12 + X13 ≥ 0.45*250 Truck mix requirement
X21 + X22 + X23 ≥ 0.40*250 Rail mix requirement
X31 + X32 + X33 ≤ 0.15*250 Air mix limit
Xij ≥ 0 for all i and j
Final
Reduced
Objective
Allowable
Allowable
Cell Name
Value
Cost
Coefficient
Increase
Decrease
$D$6 Week 1 by Truck
45
0
200
360
1E+30
$E$6 Week 1 by Rail
60
0
140
360
1E+30
$F$6 Week 1 by Air
15
0
500
1E+30
360
$D$7 Week 2 by Truck
50
0
200
0
1E+30
$E$7 Week 2 by Rail
55
0
140
0
1E+30
$F$7 Week 2 by Air
0
360
500
1E+30
360
$D$8 Week 3 by Truck
13
0
200
1E+30
0
$E$8 Week 3 by Rail
12
0
140
60
0
$F$8 Week 3 by Air
0
360
500
1E+30
360
Constraints
Final
Shadow
Constraint
Allowable
Allowable
Cell Name
Value
Price
R.H. Side
Increase
Decrease
$D$18 Week 1 by Truck
45
−360
45
13
0
$E$18 Week 1 by Rail
60
−360
60
15
0
$F$18 Week 1 by Air
15
0
15
1E+30
0
$D$19 Week 2 by Truck
50
0
50
13
25
$E$19 Week 2 by Rail
55
0
55
12
25
$F$19 Week 2 by Air
0
0
10
1E+30
10
$D$20 Week 3 by Truck
13
0
55
1E+30
42
$E$20 Week 3 by Rail
12
0
45
1E+30
33
$F$20 Week 3 by Air
0
0
5
1E+30
5
$D$9 Shipped by Truck
108
60
108
12
13
$E$9 Shipped by Rail
127
0
100
27
1E+30
$F$13 Total Shipped Tons
250
140
250
33
0
$F$9 Shipped by Air
15
0
37.5
1E+30
22.5
$G$6 Week 1 Totals
120
360
120
0
15
$G$7 Week 2 Totals
225
0
200
25
1E+30
$G$8 Week 3 Totals
250
0
250
0
1E+30
Refer to Exhibit 4.1. The Week 1 by Truck and Week 1 by Rail constraints each have a shadow price of −360. What do these values imply?
These values imply that increasing the weekly limits on these two modes will reduce total cost by $360 per unit increase in limit.
Question 41 options:
Question 42 (2 points)
Exhibit 4.1
The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report.
Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week
Trucking Limits
Railway Limits
Air Cargo Limits
1
45
60
15
2
50
55
10
3
55
45
5
Costs ($ per 1000 tons)
$200
$140
$400
The following is the LP model for this logistics problem.
Let Xij = amount shipped by mode i in week j
where i = 1(Truck), 2(Rail), 3(Air)
and j = 1, 2, 3
Let WLij = weekly limit of mode i in week j (as provided in above table)
MIN: 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
Subject to:
Xij ≤ WL ij for all i and j Weekly limits by mode
X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 Total at end of three weeks
X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of two weeks
X11 + X21 + X31 ≥ 120 Total at end of first week
X11 + X12 + X13 ≥ 0.45*250 Truck mix requirement
X21 + X22 + X23 ≥ 0.40*250 Rail mix requirement
X31 + X32 + X33 ≤ 0.15*250 Air mix limit
Xij ≥ 0 for all i and j
Final
Reduced
Objective
Allowable
Allowable
Cell Name
Value
Cost
Coefficient
Increase
Decrease
$D$6 Week 1 by Truck
45
0
200
360
1E+30
$E$6 Week 1 by Rail
60
0
140
360
1E+30
$F$6 Week 1 by Air
15
0
500
1E+30
360
$D$7 Week 2 by Truck
50
0
200
0
1E+30
$E$7 Week 2 by Rail
55
0
140
0
1E+30
$F$7 Week 2 by Air
0
360
500
1E+30
360
$D$8 Week 3 by Truck
13
0
200
1E+30
0
$E$8 Week 3 by Rail
12
0
140
60
0
$F$8 Week 3 by Air
0
360
500
1E+30
360
Constraints
Final
Shadow
Constraint
Allowable
Allowable
Cell Name
Value
Price
R.H. Side
Increase
Decrease
$D$18 Week 1 by Truck
45
−360
45
13
0
$E$18 Week 1 by Rail
60
−360
60
15
0
$F$18 Week 1 by Air
15
0
15
1E+30
0
$D$19 Week 2 by Truck
50
0
50
13
25
$E$19 Week 2 by Rail
55
0
55
12
25
$F$19 Week 2 by Air
0
0
10
1E+30
10
$D$20 Week 3 by Truck
13
0
55
1E+30
42
$E$20 Week 3 by Rail
12
0
45
1E+30
33
$F$20 Week 3 by Air
0
0
5
1E+30
5
$D$9 Shipped by Truck
108
60
108
12
13
$E$9 Shipped by Rail
127
0
100
27
1E+30
$F$13 Total Shipped Tons
250
140
250
33
0
$F$9 Shipped by Air
15
0
37.5
1E+30
22.5
$G$6 Week 1 Totals
120
360
120
0
15
$G$7 Week 2 Totals
225
0
200
25
1E+30
$G$8 Week 3 Totals
250
0
250
0
1E+30
Refer to Exhibit 4.1. Of the three percentage of effort constraints, Shipped by Truck, Shipped by Rail, and Shipped by Air, which should be examined for potential cost reduction?
In this case, the percentage by Truck, Shipped by Truck, should be examined. Decreasing the percentage by truck will decrease cost as the shadow price is 60.
Question 43 (2 points)
Exhibit 4.1
The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report.
Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week
Trucking Limits
Railway Limits
Air Cargo Limits
1
45
60
15
2
50
55
10
3
55
45
5
Costs ($ per 1000 tons)
$200
$140
$400
The following is the LP model for this logistics problem.
Let Xij = amount shipped by mode i in week j
where i = 1(Truck), 2(Rail), 3(Air)
and j = 1, 2, 3
Let WLij = weekly limit of mode i in week j (as provided in above table)
MIN: 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
Subject to:
Xij ≤ WL ij for all i and j Weekly limits by mode
X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 Total at end of three weeks
X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of two weeks
X11 + X21 + X31 ≥ 120 Total at end of first week
X11 + X12 + X13 ≥ 0.45*250 Truck mix requirement
X21 + X22 + X23 ≥ 0.40*250 Rail mix requirement
X31 + X32 + X33 ≤ 0.15*250 Air mix limit
Xij ≥ 0 for all i and j
Final
Reduced
Objective
Allowable
Allowable
Cell Name
Value
Cost
Coefficient
Increase
Decrease
$D$6 Week 1 by Truck
45
0
200
360
1E+30
$E$6 Week 1 by Rail
60
0
140
360
1E+30
$F$6 Week 1 by Air
15
0
500
1E+30
360
$D$7 Week 2 by Truck
50
0
200
0
1E+30
$E$7 Week 2 by Rail
55
0
140
0
1E+30
$F$7 Week 2 by Air
0
360
500
1E+30
360
$D$8 Week 3 by Truck
13
0
200
1E+30
0
$E$8 Week 3 by Rail
12
0
140
60
0
$F$8 Week 3 by Air
0
360
500
1E+30
360
Constraints
Final
Shadow
Constraint
Allowable
Allowable
Cell Name
Value
Price
R.H. Side
Increase
Decrease
$D$18 Week 1 by Truck
45
−360
45
13
0
$E$18 Week 1 by Rail
60
−360
60
15
0
$F$18 Week 1 by Air
15
0
15
1E+30
0
$D$19 Week 2 by Truck
50
0
50
13
25
$E$19 Week 2 by Rail
55
0
55
12
25
$F$19 Week 2 by Air
0
0
10
1E+30
10
$D$20 Week 3 by Truck
13
0
55
1E+30
42
$E$20 Week 3 by Rail
12
0
45
1E+30
33
$F$20 Week 3 by Air
0
0
5
1E+30
5
$D$9 Shipped by Truck
108
60
108
12
13
$E$9 Shipped by Rail
127
0
100
27
1E+30
$F$13 Total Shipped Tons
250
140
250
33
0
$F$9 Shipped by Air
15
0
37.5
1E+30
22.5
$G$6 Week 1 Totals
120
360
120
0
15
$G$7 Week 2 Totals
225
0
200
25
1E+30
$G$8 Week 3 Totals
250
0
250
0
1E+30
Refer to Exhibit 4.1. Are there alternate optimal solutions to this problem?
It is not possible to tell if there are alternate optimal solutions to this problem as e cannot rule out Degeneracy.
Question 44 (2 points)
Exhibit 4.1
The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report.
Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week
Trucking Limits
Railway Limits
Air Cargo Limits
1
45
60
15
2
50
55
10
3
55
45
5
Costs ($ per 1000 tons)
$200
$140
$400
The following is the LP model for this logistics problem.
Let Xij = amount shipped by mode i in week j
where i = 1(Truck), 2(Rail), 3(Air)
and j = 1, 2, 3
Let WLij = weekly limit of mode i in week j (as provided in above table)
MIN: 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
Subject to:
Xij ≤ WL ij for all i and j Weekly limits by mode
X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 Total at end of three weeks
X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of two weeks
X11 + X21 + X31 ≥ 120 Total at end of first week
X11 + X12 + X13 ≥ 0.45*250 Truck mix requirement
X21 + X22 + X23 ≥ 0.40*250 Rail mix requirement
X31 + X32 + X33 ≤ 0.15*250 Air mix limit
Xij ≥ 0 for all i and j
Final
Reduced
Objective
Allowable
Allowable
Cell Name
Value
Cost
Coefficient
Increase
Decrease
$D$6 Week 1 by Truck
45
0
200
360
1E+30
$E$6 Week 1 by Rail
60
0
140
360
1E+30
$F$6 Week 1 by Air
15
0
500
1E+30
360
$D$7 Week 2 by Truck
50
0
200
0
1E+30
$E$7 Week 2 by Rail
55
0
140
0
1E+30
$F$7 Week 2 by Air
0
360
500
1E+30
360
$D$8 Week 3 by Truck
13
0
200
1E+30
0
$E$8 Week 3 by Rail
12
0
140
60
0
$F$8 Week 3 by Air
0
360
500
1E+30
360
Constraints
Final
Shadow
Constraint
Allowable
Allowable
Cell Name
Value
Price
R.H. Side
Increase
Decrease
$D$18 Week 1 by Truck
45
−360
45
13
0
$E$18 Week 1 by Rail
60
−360
60
15
0
$F$18 Week 1 by Air
15
0
15
1E+30
0
$D$19 Week 2 by Truck
50
0
50
13
25
$E$19 Week 2 by Rail
55
0
55
12
25
$F$19 Week 2 by Air
0
0
10
1E+30
10
$D$20 Week 3 by Truck
13
0
55
1E+30
42
$E$20 Week 3 by Rail
12
0
45
1E+30
33
$F$20 Week 3 by Air
0
0
5
1E+30
5
$D$9 Shipped by Truck
108
60
108
12
13
$E$9 Shipped by Rail
127
0
100
27
1E+30
$F$13 Total Shipped Tons
250
140
250
33
0
$F$9 Shipped by Air
15
0
37.5
1E+30
22.5
$G$6 Week 1 Totals
120
360
120
0
15
$G$7 Week 2 Totals
225
0
200
25
1E+30
$G$8 Week 3 Totals
250
0
250
0
1E+30
Refer to Exhibit 4.1. Should the company negotiate for additional air delivery capacity?
No. The shadow prices for each week of air delivery are zero.
Question 45 (1 point)
Almost all network problems can be viewed as special cases of the
Question 45 options:
a)
transshipment problem.
b)
shortest path problem.
c)
maximal flow problem.
d)
minimal spanning tree problem.
Question 46 (1 point)
Decision variables in network flow problems are represented by
Question 46 options:
a)
nodes.
b)
arcs.
c)
demands.
d)
supplies.
Question 47 (1 point)
The arcs in a network indicate all of the following except?
Question 47 options:
a)
routes
b)
paths
c)
constraints
d)
connections
Question 48 (2 points)
An oil company wants to produce lube oil, gasoline and diesel fuel at two refineries at the minimum cost. There are two sources of crude oil. The following network representation depicts this problem.
Write out the LP formulation for this problem.
Minimize: 15X13 + 13X14 + 9X23 + 11X24 + 4X35 + 7X36 + 8X37 + 3X45 + 9X46 + 6X47
Subject to
-X13 - X14 = -100
-X23 - X24 = -50
0.80X13 + 0.95X23 - X35 - X36 - X37 = 0
0.85X14 + 0.85X24 - X45 - X46 - X47 = 0
0.95X35 + 0.90X45 = 50
0.90X36 + 0.95X46 = 25
0.90X37 + 0.95X47 = 75
Xij >= 0
Question 49 (2 points)
An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following Excel spreadsheet shows this problem. What formula should be entered in cell E6 (and copied to cells E7:E15) in this spreadsheet?
A
B
C
D
E
F
G
H
I
J
K
L
M
1
2
3
4
Unit
Net
Supply/
5
Flow from Node
Yield
Flow into Node
Cost
Nodes
Flow
Demand
6
1
Crude A
0.90
3
Refinery 1
15
1
Crude A
−120
7 1 Crude A 0.85 4 Refinery 2 13 2 Crude B −60
8
2
Crude B
0.80
3
Refinery 1
9
3
Refinery 1
0
9
2
Crude B
0.85
4
Refinery 2
11
4
Refinery 2
0
10
3
Refinery 1
0.95
5
Lube Oil
4
5
Lube Oil
75
11
3
Refinery 1
0.90
6
Gasoline
7
6
Gasoline
50
12
3
Refinery 1
0.90
7
Diesel
8
7
Diesel
25
13
4
Refinery 2
0.90
5
Lube Oil
3
14
4
Refinery 2
0.95
6
Gasoline
9
15
4
Refinery 2
0.95
7
Diesel
6
16
17
Total cost
D6*A6, copied to E7:E15
Question 50 (2 points)
An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following Excel spreadsheet shows this problem.
What values would you enter in the Analytic Solver Platform task pane for the following Excel spreadsheet?
Objective Cell:
Variables Cells:
Constraints Cells:
A
B
C
D
E
F
G
H
I
J
K
L
M
1
2
3
4
Unit
Net
Supply/
5
Flow from Node
Yield
Flow into Node
Cost
Nodes
Flow
Demand
6
1
Crude A
0.90
3
Refinery 1
15
1
Crude A
−120
7 1 Crude A 0.85 4 Refinery 2 13 2 Crude B −60
8
2
Crude B
0.80
3
Refinery 1
9
3
Refinery 1
0
9
2
Crude B
0.85
4
Refinery 2
11
4
Refinery 2
0
10
3
Refinery 1
0.95
5
Lube Oil
4
5
Lube Oil
75
11
3
Refinery 1
0.90
6
Gasoline
7
6
Gasoline
50
12
3
Refinery 1
0.90
7
Diesel
8
7
Diesel
25
13
4
Refinery 2
0.90
5
Lube Oil
3
14
4
Refinery 2
0.95
6
Gasoline
9
15
4
Refinery 2
0.95
7
Diesel
6
16
17
Total cost
Objective Cell: H17
Variables Cells: A6:A15
Constraints Cells:
A6:A15 ≥ 0
L6:L12 ≥ M6:M12
Question 51 (1 point)
A company wants to select 1 project from a set of 4 possible projects. Which of the following constraints ensures that only 1 will be selected?
Question 51 options:
a)
X1 + X2 + X3 + X4 = 1
b)
X1 + X2 + X3 + X4 ≤ 1
c)
X1 + X2 + X3 + X4 ≥ 1
d)
X1 + X2 + X3 + X4 ≥ 0
Question 52 (1 point)
A company wants to select no more than 2 projects from a set of 4 possible projects. Which of the following constraints ensures that no more than 2 will be selected?
Question 52 options:
a)
X1 + X2 + X3 + X4 = 2
b)
X1 + X2 + X3 + X4 ≤ 2
c)
X1 + X2 + X3 + X4 ≥ 2
d)
X1 + X2 + X3 + X4 ≥ 0
Question 53 (1 point)
How is an LP problem changed into an ILP problem?
Question 53 options:
a)
by adding constraints that the decision variables be non-negative.
b)
by adding integrality conditions.
c)
by adding discontinuity constraints.
d)
by making the RHS values integer.
Question 54 (1 point)
How is an LP problem changed into an ILP problem?
Question 54 options:
a)
by adding constraints that the decision variables be non-negative.
b)
by adding integrality conditions.
c)
by adding discontinuity constraints.
d)
by making the RHS values integer.
Question 55 (1 point)
The LP relaxation of an ILP problem
Question 55 options:
a)
always encompasses all the feasible integer solutions to the original ILP problem.
b)
encompasses at least 90% of the feasible integer solutions to the original ILP problem.
c)
encompasses different set of feasible integer solutions to the original ILP problem.
d)
will not contain the feasible integer solutions to the original ILP problem.
Question 56 (1 point)
The objective function value for the ILP problem can never
Question 56 options:
a)
be as good as the optimal solution to its LP relaxation.
b)
be as poor as the optimal solution to its LP relaxation.
c)
be worse than the optimal solution to its LP relaxation.
d)
be better than the optimal solution to its LP relaxation.
