1. Many organizations must determine how to schedule employees to provide adequate service. If we assume that an
organization faces the same situation each week, this is referred to as
A meat market manager for a l
...
1. Many organizations must determine how to schedule employees to provide adequate service. If we assume that an
organization faces the same situation each week, this is referred to as
A meat market manager for a large grocery store is preparing a processing plan to stock the shelves with
sausage, ground meat, and jerky, which he can prepare from beef, pork and venison. Sausage and ground
meat can be made of any mix of the beef, pork and venison, as long at the fat contents are below 15% for
sausage and 10% for ground meat. Sausage sells for $5/pound and ground meat sells for $3/pound. Jerky,
which sells or $10/pound, is made in a drying process from beef or venison. In the drying process, there is a
50% loss in weight for jerky made from beef (e.g., one pound of beef yields 0.5 pounds of beef jerky) and a
30% loss in weight for jerky made from venison. The market can sell at most 500 pounds of sausage, 1000
pounds of ground meat, and 100 pounds of jerky before their expiration dates. There are currently 1,000
pounds of beef (10% fat content), 500 pounds of pork (8% fat content), and 200 pounds of venison (2% fat
content) available for processing.
2. What happens to the revenue when the optimal plan changes?
Nothing – it stays at $6,500 because the amount of the three products made stays that same.
3. Which of the following does not represent a broad class of applications of linear programming models?
a. Blending models
b. Financial portfolio models
c. Logistics models
d. Set covering models
e. Forecasting models
4. Many of the most successful applications of optimization in the real world have been in the areas of scheduling,
blending, logistics and aggregate planning.
a. True
b. False
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5. Many organizations must determine how to schedule employees to provide adequate service. If we assume that an
organization faces the same situation each week, this is referred to as
a. static scheduling problem
b. dynamic scheduling problem
c. transportation scheduling problem
d. All of these options
6. Workforce scheduling problems are often integer programming models, which means that they have:
a. an integer objective function
b. integer decision variables
c. integer constraints
d. all of these options
7. A common characteristic of integer programming models is that they:
a. are easy to solve graphically
b. produce the same answer and standard linear programming models
c. often produce multiple optimal solutions
d. all of these options
8. Which of the following is true regarding multiple optimal solutions?
a. All solutions have the same values for the decision variables
b. All solutions have the same value for the objective function
c. All solutions have the same shadow prices
d. All of these options
9. Rounding the solution of a linear programming to the nearest integer values provides a(n)
a. integer solution that is optimal
b. integer solution that may be neither feasible nor optimal
c. feasible solution that is not necessarily optimal
d. infeasible solution
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10. Which of the following statements are false?
a. Solver does not offer a sensitivity report for models with integer constraints
b. Solver’s sensitivity report is not suited for questions about multiple input changes
c. Solver’s sensitivity report is used primarily for questions about one-at-a time changes to input
d. None of these options
11. If refers to the number of hours employee works in week , then to indicate that the number of working hours of 4
employees in week 3 should not exceed 160 hours, we must have a constraint of the form
a.
b.
c.
d.
12. When we solve a linear programming problem with Solver, we cannot guarantee that the solution obtained is an optimal
solution.
a. True
b. False
alse
13. Multiple optimal solutions are quite common in linear programming models.
a. True
b. False
14. If an LP problem is not correctly formulated, Solver will automatically indicate that it is infeasible when trying to solve
it.
a. True
b. False
alse
15. Integer programming (IP) models are optimization models in which all of the variables must be integers.
a. True
b. False
alse
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16. Solver may be unable to solve some integer programming problems, even when they have an optimal solution.
a. True
b. False
17. If Solver fails to find an optimal solution to an integer programming problem, we might be able to find a near optimal
solution by increasing the tolerance setting.
a. True
b. False
18. The LP relaxation of an integer programming (IP) problem is the same model as the IP model except that some integer
constraints are omitted.
a. True
b. False
alse
19. The optimal solution to an LP problem was 3.69 and 1.21. If and were restricted to be integers, then
= 4 and 1 will be a feasible solution, but not necessarily an optimal solution to the IP problem.
a. True
b. False
alse
20. A good shipping plan uses as many cheap routes as possible, but ultimately is constrained by capacities and
demands.
a. True
b. False
alse
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21. A Michigan company consists of three subsidiaries. Each has the respective average payroll, unemployment reserve
fund, and estimated payroll shown in the table below (all figures are in millions of dollars). Any employer in the state of
Michigan whose reserve to average payroll ratio is less than 1 must pay 25% of its estimated payroll in unemployment
insurance premiums. Otherwise, if the ratio is at least one, the employer pays 13%. The company can aggregate its
subsidiaries and label them as separate employers. For example, if subsidiaries 1 and 2 are aggregated, they must
pay 25% of their combined payroll in unemployment insurance premiums. Determine which subsidiaries should be
aggregated.
Subsidiary Ave. Payroll Reserve Est. Payroll
1 350 450 400
2 650 560 450
3 850 650 550
ata on subsidiaries
Subsidiary Ave. Payroll Reserve
Est.
Payroll
1 350 450 400
2 650 560 450
3 850 650 550
Any employer pays A% of its estimate payroll if its ratio of reserve to average payroll is
less than 1.0. Otherwise, it pays B%.
Given percentages
A 25%
B 13%
Companies that aggregate (1 if they
do, 0 otherwise)
None 1, 2 1,3 2,3 1,2,3 Sum Required
0 1 0 0 0 1 = 1
Total paid in each case 302 248 350 302 350
Total paid 248
22. A company has daily staffing requirements for two types of jobs, cleaning and customer service persons. The minimum
numbers of workers required each day for each type of job are shown in the table below. To meet these requirements,
the company can employ three types of workers: those who clean only, those who can perform customer service only,
and those who are able to do both. In each of these three categories, the company wants to meet its daily
requirements using only full-time workers. A full-time worker must work five consecutive days with two days off.
Workers who are able to perform only one type of work (cleaning or customer service) earn $50 per day. Those who are
able to perform both types of work earn $60 per day. As a matter of policy, the company wants to ensure that at least
20% of its total hours are staffed by “swing workers”; those who can do both types of jobs. The company wants to find
a staffing policy that covers the daily worker requirements at minimum total costs per week. Use solver to formulate
and solve the company’s problem.
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Daily requirements
Mon Tue Wed Thu Fri Sat Sun
Cleaning 8 7 7 10 9 15 11
Customer service 13 13 9 14 18 20 19
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A post office requires different numbers of full-time employees on different days of the week. The number of full-time
employees required each day is given in the table below.
Mon Tue Wed Thu Fri Sat Sun
20 16 18 22 17 19 14
Union rules state that each full-time employee must work five consecutive days and then receive two days off. The post
office wants to meet its daily requirements using only full-time employees. Its objective is to minimize the number of
full-time employees that must be hired.
23. (A) Use Solver to formulate and solve the post office’s problem.
(B) Suppose the post office has 30 full-time employees and is not allowed to hire or fire any employees.
Determine a schedule that maximizes the number of weekend days off received by the employees.
(A)
Number starting their five-day shift on various
days
Mon 6
Tue 7
Wed 0
Thu 9
Fri 0
Sat 5
Sun 0
Number working on various days (along top) who started their shift on various days (along side)
Mon Tue Wed Thu Fri Sat Sun
Mon 6 6 6 6 6
Tue 7 7 7 7 7
Wed 0 0 0 0 0
Thu 9 9 9 9 9
Fri 0 0 0 0 0
Sat 5 5 5 5 5
Sun 0 0 0 0 0
Totals Available 20 18 18 22 22 21 14
Min required 20 16 18 22 17 19 14
Total employees 27
(B)
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Starting day of
5-day shift
Mon. Tue. Wed. Thu. Fri. Sat. Sun.
Total Total
employed
Number starting 11 5 2 12 0 0 0 30 = 30
Number working on each day 23 16 18 30 30 19 14
Minimal number required each day 20 16 18 22 17 19 14
Weekend days-off 22 5 0 0 0 0 0
Total weekend days-off 27
24. During each 4-hour period, the police force in a small town in Ohio requires the following number of on-duty police
officers: 8 from midnight to 4 A.M.; 7 from 4 A.M. to 8 A.M.; 6 from 8 A.M. to noon; 6 from noon to 4 P.M.; 5 from 4
P.M. to 8 P.M.; and 4 from 8 P.M. to midnight. Each police officer works two consecutive 4-hour shifts. Determine how
to minimize the number of police officers needed to meet the town’s daily requirements.
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A construction company is preparing for a nine-month project, and will need to develop a staffing plan. The
company can assign up to 30 of its own full-time employees to the project, and will hire short-term contract
employees to make up any shortage in meeting the personnel requirements. Company employees earn
$6,000 per month, while short-term contract employees make $8,600/month. Contract employees can be
assigned to the project beginning in any month, and their contract period is two months. The number of
workers required for the project by month is shown below:
25. Determine the optimal staffing plan for the project.
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26. The project manager is evaluating options to complete the project early so that the company can earn a
bonus. He has determined that the project schedule can be compressed into a six-month schedule, with the
same total number of worker-months. In that case, the staffing requirements are as shown below.
Develop an optimal staffing plan for the project under the accelerated schedule.
27. Suppose the bonus for completing the project three months early is $250,000. What would be the net bonus
to the company, after adjusting for any difference in personnel costs under the accelerated schedule?
$250,000 – ($3,436,400-$3,387,200) = $200,800
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28. The production manager believes the cost of the contract employees, who are currently in high demand,
could be somewhat higher – perhaps as high as $10,000 per month. Perform a sensitivity analysis to
determine the effect on the number of full-time employees that will be needed for the project.
Once the salary reaches 9000, all 30 available full-time employees with be assigned to the
project (only 21 are needed under the existing plan).
29. Which of the following statements is a type of constraint that is often required in blending problems?
a. Integer constraint
b. Binary constraint
c. Quality constraint
d. None of these options
30. The constraints in a blending problem can be specified in a valid way and still lead to which of the following problems?
a. Unboundedness
b. Infeasibility
c. Nonlinearity
d. None of these options
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31. To specify that must be at most 75% of the blend of we must have a constraint of the form
a.
b.
c.
d.
e.
32. In blending problems, if a quality constraint involves a quotient, then the problem will be nonlinear.
a. True
b. False
33. You have decided to enter the candy business. You are considering producing two types of candies: A and
B, both of which consist solely of sugar, nuts, and chocolate. At present you have in stock 12,000 ounces of
sugar, 3000 ounces of nuts, and 3000 ounces of chocolate. The mixture used to make candy A must contain
at least 10% nuts and 10% chocolate. The mixture used to make candy B must contain at least 20% nuts.
Each ounce of candy A can be sold for $0.40 and each ounce of candy for $0.50. Determine how you can
maximize your revenues from candy sales.
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34. A company blends silicon and nitrogen to produce two types of fertilizers. Fertilizer 1 must be at least 40%
nitrogen and sells for $75 per pound. Fertilizer 2 must be at least 70% silicon and sells for $45 per pound.
The company can purchase up to 9000 pounds of nitrogen at $16 per pound and up to 12,000 pounds of
silicon at $12 per pound. Assuming that all fertilizer produced can be sold, determine how the company can
maximize its profit.
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A meat market manager for a large grocery store is preparing a processing plan to stock the shelves with
sausage, ground meat, and jerky, which he can prepare from beef, pork and venison. Sausage and ground
meat can be made of any mix of the beef, pork and venison, as long at the fat contents are below 15% for
sausage and 10% for ground meat. Sausage sells for $5/pound and ground meat sells for $3/pound. Jerky,
which sells or $10/pound, is made in a drying process from beef or venison. In the drying process, there is a
50% loss in weight for jerky made from beef (e.g., one pound of beef yields 0.5 pounds of beef jerky) and a
30% loss in weight for jerky made from venison. The market can sell at most 500 pounds of sausage, 1000
pounds of ground meat, and 100 pounds of jerky before their expiration dates. There are currently 1,000
pounds of beef (10% fat content), 500 pounds of pork (8% fat content), and 200 pounds of venison (2% fat
content) available for processing.
35. Determine the optimal processing plan for the meat market.
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36. Suppose that later in the year, venison will be out of season, but the market will be able to obtain an
additional 300 pounds of pork for the same costs. Develop a processing plan in that case. How does the
solution change?
The maximum amounts of each product can still be made. The jerky will be made from beef,
instead of venison, and the additional pork will be used in making ground meat, since less beef
will be available for that purpose.
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37. The market manager is concerned about variability in the fat content of beef, noting that it actually can be
as high as 20% and as low as 5%. Perform a sensitivity analysis to determine the effect, first on the amount
of beef used, and then on the revenue. What do the results indicate? Should the manager be concerned?
There’s nothing to worry about if the fat content is lower than 10%; however, if it increases above
10%, then less beef will be used in the processing plan, and if it increases above 15%, the
revenue will start to decrease, since the amounts of the products made will start to become
limited by their specifications.
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38. The problem which deals with the direct distribution of products from supply locations to demand locations is called
a(n)
a. transportation problem
b. assignment problem
c. network problem
d. transshipment problem
39. The objective in transportation problems is typically to:
a. maximize profits
b. maximize revenue
c. minimize costs
d. maximize feasibility
40. A typical transportation problem requires which of the following sets of input numbers:
a. Capacities, demands and flows
b. Capacities, demands and unit shipping costs
c. Supplies, demands and flows
d. Supplies, demands and arcs
41. Which of the following is not a required input for a typical transportation problem?
a. Capacities (or supplies)
b. Demands
c. Unit shipping (and possibly production) costs
d. Distance from origins to destinations
42. The decision variables in transportation problems are:
a. profits
b. costs
c. flows
d. capacities
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43. In a network representation of a transportation problem, the nodes generally represent:
a. warehouses
b. geographic locations
c. flows
d. capacities
44. In a network representation of a transportation problem, the arcs generally represent:
a. warehouses
b. geographic locations
c. flows
d. capacities
45. In formulating a transportation problem as linear programming model, which of the following statements are correct?
a. There is one constraint for each supply location
b. There is one constraint for each demand location
c. The sum of decision variables out of a supply location is constrained by the supply at that location
d. The sum of decision variables out of all supply locations to a specific demand location is constrained by the
demand at that location
e. All of these options
46. In a transshipment problem, shipments
a. can occur between any two nodes (suppliers, demanders, and transshipment locations)
b. cannot occur between two supply locations
c. cannot occur between two demand locations
d. cannot occur between a transshipment location and a demand location
e. cannot occur between a supply location and a demand location
47. Transportation and transshipment problems are both considered special cases of a class of linear programming
problems called
a. minimum cost problems
b. minimum cost network flow problems
c. supply locations network problems
d. demand locations network problems
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48. In a typical minimum cost network flow model, the nodes indicate
a. roads
b. rail lines
c. geographic locations
d. rivers
49. The flow balance constraint for each transshipment node, in a minimum cost network flow model, takes the form
a. Flow in Flow out + Net supply
b. Flow out Flow in + Net supply
c. Flow in = Flow out
d. Flow out Flow in + Net supply
e. Flow in Flow out + Net demand
50. In a minimum cost network flow model, the flow balance constraint for each supply node takes the form
a. Flow in Flow out + Net supply
b. Flow out Flow in + Net demand
c. Flow in = Flow out
d. Flow out Flow in + Net supply
e. Flow in Flow out + Net demand
51. In a minimum cost network flow model, the flow balance constraint for each demand node takes the form
a. Flow out Flow in + Net supply
b. Flow in Flow out + Net demand
c. Flow in = Flow out
d. Flow in Flow out + Net demand
e. Flow out Flow in + Net demand
52. Logistics problems are problems of finding the least expensive way to transport products from their origin to
their destination.
a. True
b. False
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53. If all the supplies and demands for a transportation model are integers, then the optimal Solver solution may or may
not have integer-valued shipments.
a. True
b. False
alse
54. In transportation problems, shipping costs are often nonlinear due to quantity discounts.
a. True
b. False
55. In a transportation problem, if it costs $4 per item to ship up to 200 items between cities, and $2 per item for each
additional item, the proportionality assumption of LP is satisfied.
a. True
b. False
alse
56. The transportation model is a special case of the minimum cost network flow model (MCNFM).
a. True
b. False
57. In transportation problems, shipments between supply points or between demand points are possible.
a. True
b. False
alse
58. In transportation problems, the three sets of input numbers that are required are capacities, demands and flows.
a. True
b. False
alse
59. In network models of transportation problems, arcs represent the routes for getting a product from one node to another.
a. True
b. False
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60. Transshipment points are locations where goods neither originate nor end up, but goods are allowed to enter such
points to be shipped out to their eventual destinations.
a. True
b. False
61. The flows in a general minimum cost network flow model (MCNFM) do all necessarily have to be from “left to right”;
that is, from supply points to demand points.
a. True
b. False
alse
62. In an optimized network flow model (MCNFM), all the available capacity will be used.
a. True
b. False
alse
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63. An auto company must meet (on time) the following demands for cars: 5000 in quarter 1; 3000 in quarter 2; 6000 in
quarter 3; 2000 in quarter 4. At the beginning of quarter 1, there are 500 autos in stock. The company has the capacity
to produce at most 3600 cars per quarter. At the beginning of each quarter, the company can change production
capacity. It costs $125 to increase quarterly production capacity by one unit. It also costs $60 per quarter to maintain
each unit of production capacity (even if it is unused during the current quarter). The variable cost of producing a car is
$2400. A holding cost of $200 per car is assessed against each quarter’s ending inventory. It is required that at the
end of quarter 4, plant capacity must be at least 5000 cars. Determine how to minimize the total cost incurred during
the next four quarters.
Initial inventory 500
Initial capacity per quarter 3600
Cost to increase capacity by 1 unit $125
Cost to maintain 1 unit of capacity $60
Variable cost of producing a car $2,400
Holding cost per car in inventory $200
Quarter 1 2 3 4
Increase in capacity 900 0 500 0
Production 4500 4000 5000 2000
Capacity 4500 4500 5000 5000
Required final capacity 5000
On hand after production 5000 4000 6000 2000
Demand 5000 3000 6000 2000
End inventory 0 1000 0 0
Production cost $37,200,000
Holding cost $200,000
Cost of increasing capacity $175,000
Cost of maintaining capacity $1,140,000
Total cost $38,715,000
64. An electronic company is considering opening warehouses in New York, Los Angeles, Madison, and Tampa. Each
warehouse can ship 125 units per week. The weekly fixed cost for keeping each warehouse open is $500 for New
York, $600 for Los Angeles, $400 for Madison, and $200 for Tampa. Region 1 of the country requires 90 units per
week, region 2 requires 80 units per week, and region 3 requires 50 units per week. The costs (including production
and shipping costs) of sending one unit from a plant to a region are shown in the table below.
To
From Region 1 Region 2 Region 3
NY $25 $45 $55
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LA $52 $19 $30
Madison $29 $38 $21
Tampa $28 $54 $39
Show how the company can meet weekly demands at a minimum cost, subject to the above information and the
following restrictions:
If the New York warehouse is opened, then Los Angeles must be opened.
At most two warehouses can be opened.
Either the Tampa or the Los Angeles warehouses must be opened.
Unit production and
shipping costs
To
Region 1 Region 2
Region
3
Fixed
cost Capacity
NY $25 $45 $55 $500 125
From LA $52 $19 $30 $600 125
Madison $29 $38 $21 $400 125
Tampa $28 $54 $39 $200 125
Warehouses to open (1 if
open, 0 if not)
NY 0
LA 1
Madison 1
Tampa 0
Amount produced and
shipped To
Region 1 Region 2
Region
3 Total Capacity
NY 0 0 0 0 0
From LA 0 80 15 95 125
Madison 90 0 35 125 125
Tampa 0 0 0 0 0
Total 90 80 50
Required 90 80 50
If NY, then LA constraint
NY LA
0 1
At most two open
warehouses constraint
# open Max
2 2
Either Tampa or LA
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constraint
# open Min
1 1
Cost of opening
warehouses $1,000
Cost of production and
shipping $5,315
Total cost $6,315
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A company supplies goods to three customers, each of whom requires 50 units. The company has two warehouses. In
warehouse 1, 75 units are available, and in warehouse 2, 55 units are available. The costs of shipping one unit from
each warehouse to each customer are shown in the table below.
To
From Customer 1 Customer 2 Customer 3
Warehouse 1 $20 $40 $30
Warehouse 2 $15 $35 $45
Not shipped (shortage) $100 $90 $115
There is a penalty for each unsatisfied customer unit of demand – with customer 1, a penalty cost of $100 is incurred;
with customer 2, $90; and with customer 3, $115.
65. Suppose that the company can purchase and ship extra units to either warehouse for a total cost of $125 per unit and
that all customer demand must be met. Determine how to minimize the sum of purchasing and shipping costs.
Unit shipping costs To
From Customer 1
Customer
2
Customer
3
Unit
purchase
cost
Warehouse 1 $20 $40 $30 $125
Warehouse 2 $15 $35 $45 $125
Units purchased for: Originally available
Warehouse 1 0 75
Warehouse 2 20 55
Shipments To
From Customer 1
Customer
2
Customer
3 Total Available
Warehouse 1 0 25 50 75 75
Warehouse 2 50 25 0 75 75
Total 50 50 50
Required 50 50 50
Costs
Total purchase cost $2,500
Total shipping cost $4,125
Total cost $6,625
An auto company produces cars at Los Angeles and Detroit, and has a warehouse in Atlanta. The company supplies
cars to dealers in Dallas and Orlando. The costs of shipping a car between various points are shown in the table below,
where “NA” means that a shipment is not allowed. Los Angeles can produce up to 1400 cars, and Detroit can produce
up to 3200 cars. Dallas must receive 2800 cars, and Orlando must receive 1800 cars.
To
From LA Detroit Atlanta Dallas Orlando
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LA $12,000 $190 $150 $140 $275
Detroit $195 $12,000 $160 $160 $160
Atlanta $155 $165 $12,000 $160 $130
Dallas $140 $160 $170 $12,000 $12,000
Orlando $260 $170 $130 $12,000 $12,000
66. (A) Determine how to minimize the cost of meeting demands at Dallas and Orlando.
(B) Modify the answer to (A) if shipments between Los Angeles and Detroit are not allowed.
(A)
Unit shipping costs
To
LA Detroit Atlanta Dallas Orlando
LA $12,000 $190 $150 $140 $275
Detroit $195 $12,000 $160 $160 $160
From Atlanta $155 $165 $12,000 $160 $130
Dallas $140 $160 $170 $12,000 $12,000
Orlando $260 $170 $130 $12,000 $12,000
Shipments
To
LA Detroit Atlanta Dallas Orlando
Total
outflow
LA 0 0 0 1400 0 1400
Detroit 0 0 0 1400 1800 3200
From Atlanta 0 0 0 0 0 0
Dallas 0 0 0 0 0 0
Orlando 0 0 0 0 0 0
Total inflow 0 0 0 2800 1800
Total outflow 1400 3200 0 0 0
Net flow out or in 1400 3200 0 2800 1800
= = =
Required 1400 3200 0 2800 1800
Total cost $708,000
(B)
To disallow shipments between LA and Detroit, enter large unit costs in LA / Detroit routes.
However, since no cars were shipped between LA and Detroit anyway, the optimal shipments
are still the same.
Each year, a computer company produces up to 600 computers in New York and up to 500 computers in Memphis.
Los Angeles customers must receive 600 computers, and 500 computers must be supplies to Oklahoma City
customers. Producing a computer costs $850 in New York and $950 in Memphis. Computers are transported by plane
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and can be sent through Chicago. The costs of shipping a computer between cities are shown below.
To
From Chicago Oklahoma City L.A.
New York $95 $245 $295
Memphis $115 $155 $195
Chicago NA $60 $65
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67. (A) Determine how to minimize the total (production plus distribution) costs of meeting the company’s annual demand.
(B) How would you modify the model in (A) if at most 300 units can be shipped through Chicago?
(A)
Unit ship costs To
Chicago
Oklahoma
City L.A.
New York $95 $245 $295
From Memphis $115 $155 $195
Chicago NA $60 $65
Unit production
costs
New York $850
Memphis $950
To
From New York Memphis Chicago
Oklahoma
City LA
New York $55,000 $55,000 $945 $1,095 $1,145
Memphis $55,000 $55,000 $1,065 $1,105 $1,145
Chicago $55,000 $55,000 $55,000 $40 $50
Oklahoma
City $55,000 $55,000 $55,000 $55,000 $55,000
LA $55,000 $55,000 $55,000 $55,000 $55,000
(B)
Shipments
To
From
New
York Memphis Chicago
Oklahoma
City LA
Total
outflow Capacity
New York 0 0 300 300 0 600
Memphis 0 0 0 200 300 500
Chicago 0 0 0 0 300 300 300
Oklahoma City 0 0 0 0 0 0
LA 0 0 0 0 0 0
Total inflow 0 0 300 500 600
Total outflow 600 500 300 0 0
Net flow in or out600 500 0 500 600
= = =
Required 600 500 0 500 600
Total cost =
$1,191,500
An oil company produces oil at two wells. Well 1 can produce up to 150,000 barrels per day, and well 2 can produce
up to 200,000 barrels per day. It is possible to ship oil directly from the wells to customers in Los Angeles and New
York. Alternatively, the company could transport oil to the ports of Mobile and Galveston and then ship it by tanker to
New York or Los Angeles. Los Angeles requires 160,000 barrels per day, and New York requires 140,000 barrels per
day. The costs (in dollars) of shipping 1000 barrels between various locations are shown below:
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To
From Well 1 Well 2 Mobile Galveston New York Los Angeles
Well 1 $10,000 $10,000 $10 $13 $25 $28
Well 2 $10,000 $10,000 $15 $12 $26 $25
Mobile $10,000 $10,000 $10,000 $6 $16 $17
Galveston $10,000 $10,000 $6 $10,000 $14 $16
New York $10,000 $10,000 $10,000 $10,000 $10,000 $15
Los Angeles $10,000 $10,000 $10,000 $10,000 $15 $10,000
68. (A) Assume that before being shipped to Los Angeles or New York, all oil produced at the wells must be refined at
either Mobile or Galveston. To refine 10000 barrels of oil costs $12 at Mobile and $10 at Galveston. Assuming that both
Mobile and Galveston have infinite refinery capacity, determine how to minimize the daily cost of transporting and
refining the oil requirements of Los Angeles and New York.
(B) Rework (A) under the assumption that Galveston has a refinery capacity of 150,000 barrels per day, and Mobile
has a refinery capacity of 180,000 barrels per day.
(A)
Unit shipping
costs (per 1000
barrels)
To
From Well 1 Well 2 Mobile Galveston NY LA
Well 1 $10,000 $10,000 $10 $13 $10,000 $10,000
Well 2 $10,000 $10,000 $15 $12 $10,000 $10,000
Mobile $10,000 $10,000 $10,000 $6 $16 $17
Galveston $10,000 $10,000 $6 $10,000 $14 $16
NY $10,000 $10,000 $10,000 $10,000 $10,000 $15
LA $10,000 $10,000 $10,000 $10,000 $15 $10,000
Refinery cost (per
1000 barrels)
Mobile $12
Effectively requires all oil
from the wells to go to
Mobile or Galveston first,
where it can be refined.
Galveston $10
Shipments (in
1000s of barrels)
To
From Well 1 Well 2 Mobile Galveston NY LA
Total
outflow
Well 1 0 0 100 0 0 0 100
Well 2 0 0 0 200 0 0 200
Mobile 0 0 0 0 0 100 100
Galveston 0 0 0 0 140 60 200
NY 0 0 0 0 0 0 0
LA 0 0 0 0 0 0 0
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Total inflow 0 0 100 200 140 160
Total outflow 100 200 100 200 0 0
Net flow in or out 100 200 0 0 140 160
<= <= = = = =
Required 150 200 0 0 140 160
Total Cost =
$11,220,000
(B)
Unit shipping
costs (per
1000 barrels)
To
From Well 1 Well 2 Mobile Galveston NY LA
Well 1 $10,000 $10,000 $10 $13 $10,000 $10,000
Well 2 $10,000 $10,000 $15 $12 $10,000 $10,000
Mobile $10,000 $10,000 $10,000 $6 $16 $17
Galveston $10,000 $10,000 $6 $10,000 $14 $16
NY $10,000 $10,000 $10,000 $10,000 $10,000 $15
LA $10,000 $10,000 $10,000 $10,000 $15 $10,000
Refinery cost
(per 1000
barrels)
Mobile $12
Galveston $10
Shipments (in
1000s of
barrels)
To Total
From Well 1 Well 2 Mobile Galveston NY LA outflow Capacity
Well 1 0 0 150 0 0 0 150
Well 2 0 0 0 150 0 0 150
Mobile 0 0 0 0 0 150 150 <= 180
Galveston 0 0 0 0 140 10 150 <= 150
NY 0 0 0 0 0 0 0
LA 0 0 0 0 0 0 0
Total inflow 0 0 150 150 140 160
Total outflow 150 150 150 150 0 0
Net flow in or
out 150 150 0 0 140 160
<= <= = = = =
Required 150 200 0 0 140 160
Total Cost =
$11,270,000
An oil company has oil fields in San Diego and Los Angeles. The San Diego field can produce up to 500,000 barrels
per day, and the Los Angeles field can produce up to 400,000 barrels per day. Oil is sent from the fields to a refinery,
either in Dallas or in Houston. Assume that each refinery has unlimited capacity. To refine 100,000 barrels costs $725
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at Dallas and $950 at Houston. Refined oil is shipped to customers in Chicago and New York. Chicago customers
require 400,000 barrels per day, and New York customers require 300,000 barrels per day. The costs of shipping
100,000 barrels of oil (refined or unrefined) between cities are shown in the table below:
69. (A) Determine how to minimize the total cost of meeting all demands.
(B) If each refinery had a capacity of 380,000 barrels per day, how would you modify the model in (A)?
(A)
(B)
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70. During the next 4 quarters, an automobile company must meet (on time) the following demands for cars: 4000 in
quarter 1; 2000 in quarter 2; 5000 in quarter 3; 1000 in quarter 4. At the beginning of quarter 1, there are 300 autos in
stock. The company has the capacity to produce at most 3000 cars per quarter. At the beginning of each quarter, the
company can change production capacity. It costs $100 to increase quarterly production capacity by 1 unit. For
example, it would cost $20,000 to increase capacity from 3000 to 3200. It also costs $60 per quarter to maintain each
unit of production capacity (even if it is unused during the current quarter). The variable cost of producing a car is
$2200. A holding cost of $160 per car is assessed against each quarter’s ending inventory. It is required that at the
end of quarter 4, plant capacity must be at least 4000 cars. Determine how to minimize the total cost incurred during
the next 4 quarters.
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71. A large accounting firm has three auditors. Each can work up to 180 hours during the next month, during
which time three projects must be completed. Project 1 takes 140 hours, project 2 takes 150 hours, and
project 3 takes 170 hours. The amount per hour that can be billed for assigning each auditor to each project
is given in the table below:
Determine how to maximize total billings during the next month by formulating the company’s problem as
a transportation model.
72. In aggregate planning models, which of the following statements are correct?
a. The number of workers available influences the possible production levels
b. We allow the workforce level to be modified each month through the hiring and firing of workers
c. We eventually allow demand to be backlogged; that is, demand need not be met on time
d. All of these options
73. In aggregate planning models, the number of workers available influences the possible production levels.
a. True
b. False
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74. Aggregate planning models are usually implemented through a rolling planning horizon.
a. True
b. False
75. In aggregate planning models, we can model backlogging of demand by allowing a month’s inventory to be negative.
a. True
b. False
76. An oil delivery truck contains five compartments, holding up to 2800, 2900, 1200, 1800, and 3200 gallons of fuel,
respectively. The company must deliver three types of fuel (super, regular, and unleaded) to a customer. The demands,
penalty per gallon short, and the maximum allowed shortage are shown in the table below. Each compartment of the
truck can carry only one type of gasoline. Determine how to load the truck in a way that minimizes shortage costs.
Demand Cost (Gallon short) Maximum shortage allowed
Super 3000 $9 400
Regular 4000 $7 400
Unleaded 5000 $5 400
ata on customer demands
Cost/gallon
short
Max
shortage
(in 100s)
Super $9 4
Regular $7 4
Unleaded $5 4
Compartments used for various types of
gasoline (1 if used, 0 if not)
Compt 1 Compt 2 Compt 3 Compt 4 Compt 5
Super 0 1 0 0 0
Regular 1 0 1 0 0
Unleaded 0 0 0 1 1
Sum 1 1 1 1 1
Max 1 1 1 1 1
Amounts stored in various
compartments (in 100s of
gallons)
Compt 1 Compt 2 Compt 3 Compt 4 Compt 5Total Demand
Super 0 29 0 0 0 29 30
Regular 28 0 12 0 0 40 40
Unleaded 0 0 0 18 32 50 50
Total 28 29 12 18 32
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Capacity 28 29 12 18 32
Max shortage constraints (in
100s of gallons)
Shortage Max
Super 1 4
Regular 0 4
Unleaded 0 4
Logical constraints (can't ship gas type in container unless 0-1 variable is 1; then upper limit
is container capacity)
Sent 0 29 0 0 0
28 0 12 0 0
0 0 0 18 32
Logical bounds 0 29 0 0 0
28 0 12 0 0
0 0 0 18 32
Shortage cost $900
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A company supplies goods to three customers, each of whom requires 50 units. The company has two warehouses. In
warehouse 1, 75 units are available, and in warehouse 2, 55 units are available. The costs of shipping one unit from
each warehouse to each customer are shown in the table below.
To
From Customer 1 Customer 2 Customer 3
Warehouse 1 $20 $40 $30
Warehouse 2 $15 $35 $45
Not shipped (shortage) $100 $90 $115
There is a penalty for each unsatisfied customer unit of demand – with customer 1, a penalty cost of $100 is incurred;
with customer 2, $90; and with customer 3, $115.
77. Determine how to minimize the sum of shortage and shipping costs.
Unit
shipping
costs To
From Customer 1 Customer 2 Customer 3
Warehouse 1 $20 $40 $30
Warehouse 2 $15 $35 $45
Not shipped
(shortage) $100 $90 $115
Shipments To
From Customer 1 Customer 2 Customer 3 Total Available
Warehouse 1 0 25 50 75 = 75
Warehouse 2 50 5 0 55 = 55
Not shipped
(shortage) 0 20 0 20 = 20
Total 50 50 50
= = =
Required 50 50 50
Costs
Total
shipping
cost $3,425
Total
shortage
cost $1,800
Total cost $5,225
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78. The cost per day running a hotel is 200,000 + 0.002 dollars, where x is the number of customers served per day.
What number of customers served per day minimizes the cost per customer of running the hotel?
ost per day of serving x customers is a + bx2, where
a 200000
b 0.002
Number of customers served 10000.02
Cost per customer per day $40
79. An oil company controls two oil fields. Field 1 can produce up to 45 million barrels of oil per day, and field 2 can
produce up to 55 million barrels of oil per day. At field1, it costs $3 to extract and refine a barrel of oil; at field 2, the
cost is $2. The company sells oil to two countries: France and Japan. The shipping costs per barrel are shown below.
Each day, France is willing to buy up to 45 million barrels (at $6 per barrel), and Japan is willing to buy up to 35 million
barrels (at $6.50 per barrel). Determine how to maximize the company’s profit.
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80. The cost per day of running a hospital is 250,000 + dollars, where x is the number of patients served
per day. What number of patients served per day minimizes the cost per patient of running the hospital?
81. A company manufactures two products. If it charges price for product I, it can sell units of product I, where
. It costs $25 to produce a unit of product 1 and $72 to produce a unit of
product 2. How many units of each product should the company produce, and what prices should it charge, to
maximize its profit?
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A motorcycle company is determining its production schedule for the next four quarters. Demands for
motorcycles are forecasted to be 40 in quarter 1; 70 in quarter 2; 50 in quarter 3; 20 in quarter 4. The
company incurs four types of costs:
∙ It costs the company $400 to manufacture each motorcycle
∙ A holding cost of $100 per motorcycle left in inventory is incurred at the end of each quarter.
∙ Increasing production from one quarter to the next incurs costs for training employees. It is estimated that
a cost of $700 per motorcycle is incurred if production is increased from one quarter to the next.
∙ Decreasing production from one quarter to the next incurs costs for severance pay, decreasing morale, and
so forth. It is estimated that a cost of $600 per motorcycle is incurred if production is decreased from one
quarter to the next.
All demands must be met on time, and a quarter’s production can be used to meet demand for the current
quarter (as well as future quarters). During the quarter immediately preceding quarter1, 50 motorcycles
were produced. Assume that at the beginning of quarter 1, no motorcycles are in inventory.
82. Determine how to minimize the company’s total cost during the next four quarters.
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83. Discuss how the company’s optimal production schedule would be affected by a change in the cost of
increasing production from one quarter to the next.
One way to answer this question is to use SolverTable to see how the previous month’s
production quantities (namely; 50, 55, 55, and 50) and total cost vary as costs of decreasing
production change.
When cost of decreasing production changes, total cost increases, but the production schedule stays the
same except for production in quarter 4. It is less when this cost is low.
84. Discuss how the company’s optimal production schedule would be affected by a change in the cost of
decreasing production from one quarter to the next.
One way to answer this question is to use SolverTable to see how the previous month’s
production quantities (namely; 50, 55, 55, and 50) and total cost vary as costs of decreasing
production change.
When cost of decreasing production changes, total cost increases, but the production schedule stays the
same except for production in quarter 4. It is less when this cost is low.
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85. A total of 160 hours of labor are available each week at $15 per hour. Additional labor can be purchased at
$25 per hour. Capital can be purchased in unlimited quantities at a cost of $45 per unit. If K units of capital
and L units of labor are available during a week, then machines can be produced. Each machine
sells for $270. How can the firm maximize its weekly profit?
86. A statistician is currently trying to maximize his profit in the bond market. Four bonds are available for purchase and
sale at the bid and ask prices shown in the table below. The statistician can buy up to 1300 units of each bond at the
ask price or sell up to 1300 units of each bond at the bid price. During each of the next three years the person who
sells a bond will pay the owner of the bond the cash payments that are also shown in the table below. The
statistician’s goal is to maximize his revenue from selling bonds less his payments for buying bonds, subject to the
constraint that after each year’s payments are received, his current cash position (due only to cash payments from
bonds and not purchases or sales of bonds) is nonnegative. His current cash position can depend on past coupons
and that cash accumulated at the end of each year earns 12% annual interest. Determine how to maximize net profit
from buying and selling bonds, subject to the constraints previously described.
Bid (for selling) and ask (for buying) prices of bonds
Bond 1 Bond 2 Bond 3 Bond 4
Bid $1,000 $99 $980 $960
Ask $1,020 $1,015 $1,002 $184
Cash payments from seller to buyer
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Bond 1 Bond 2 Bond 3 Bond 4
Year 1 $120 $100 $90 $70
Year 2 $140 $130 $110 $90
Year 3 $1,300 $1,320 $1,290 $1,310
ond 1 Bond 2 Bond 3 Bond 4
Bid $1,000 $99 $980 $960
Ask $1,020 $1,015 $1,002 $184
Bond 1 Bond 2 Bond 3 Bond 4
Year 1 $120 $100 $90 $70
Year 2 $140 $130 $110 $90
Year 3 $1,300 $1,320 $1,290 $1,310
Interest rate for cash 12.000%
Bond 1 Bond 2 Bond 3 Bond 4
Buys 1300 943 0 1300
Maximum 1300 1300 1300 1300
Sells 0 0 1300 1300
Maximum 1300 1300 1300 1300
Cash in Cash out Net cash
Accumulated cash
with interest
Year 0 (right after buy,
sells) $2,522,000 $2,522,000 $0 $0
Year 1 $341,266 $208,000 $133,266 $133,266
Year 2 $421,546 $260,000 $161,546 $310,804
Year 3 $4,637,311 $3,380,000 $1,257,311 $1,605,412
Cash at end of year 3 $1,605,412
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87. Assume that you are given the following means, standard deviations, and correlations for the annual return on three
stocks.
Stock 1 Stock 2 Stock 3
Mean return 0.15 0.18 0.23
Stdev. of return 0.18 0.28 0.38
Correlation matrix
Stock 1 Stock 2 Stock 3
Stock 1 1.00 0.62 0.72
Stock 2 0.62 1.00 0.39
Stock 3 0.72 0.39 1.00
The correlation between stocks 1 and 2 is 0.62, between stocks 1 and 3 is 0.72, and between stocks 2 and 3 is 0.39.
You have $12,000 to invest and can invest no more than 55% of your money in any single stock. Determine the
minimum variance portfolio that yields an expected annual return of at least 0.15.
Stock 1 Stock 2 Stock 3
Mean return 0.15 0.18 0.23
Stdev of return 0.18 0.28 0.38
Correlation matrix
Stock 1 Stock 2 Stock 3
Stock 1 1 0.62 0.72
Stock 2 0.62 1 0.39
Stock 3 0.72 0.39 1
Stocks Stock 1 Stock 2 Stock 3 Total Available
Dollars $6,600 $4,824 $576 $12,000 = $12,000
Fraction 0.55 0.402028 0.047972
Maximum 0.55 0.55 0.55
Actual Required
0.165899 0.15
Standard deviations times fractions invested
Stock 1 Stock 2 Stock 3
0.099 0.112568 0.018229
Portfolio variance 0.040823
Portfolio Stdev. 0.202047
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88. At time 0, you have $10,000. Investments A and B are available; their cash flows are shown in the table below:
Assume that any money not invested in A or B earns interest at an annual rate of 8%. Determine how to maximize
your cash on hand at time 3.
Investment Time 0 Time 1 Time 2 Time 3
A -$1.00 $0.20 $1.50 $0.00
B $0.00 -$1.00 $0.00 $1.90
ash flows per dollar invested
Time 0 Time 1 Time 2 Time 3
Chapter 14
89. You are given the following means, standard deviations, and correlations for the annual return on three
stocks. The means are 0.12, 0.15, and 0.20. The standard deviations are 0.20, 0.30, and 0.40. The correlation
between stocks 1 and 2 is 0.65, between stocks 1 and 3 is 0.75, and between stocks 2 and 3 is 0.41. You have
$13,000 to invest and can invest no more than half of your money in any single stock. Determine the
minimum variance portfolio that yields an expected annual return of at least 0.14.
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90. Laila, an Egyptian Broker, is currently trying to maximize her profit in the bond market. Four bonds are available for
purchase and sale at the bid and ask prices shown in the table be below:
Laila can buy up to 1000 units of each bond at the ask price or sell up to 1000 units of each bond at the bid price.
During each of the next 3 years, the person who sells a bond will pay the owner of the bond the cash payments shown
in the table below:
Laila’s goal is to maximize her revenue from selling bonds less her payment for buying bonds, subject to the constraint
that after each year’s payments are received, her current cash position (due only to cash payments from bonds and
not purchases or sales of bonds) is nonnegative. Note that her current cash position can depend on past coupons and
that cash accumulated at the end of each year earns 13% annual interest. Determine how to maximize net profit from
buying and selling bonds, subject to the constraints previously described. Why do you think we limit the number of
units of each bond that can be bought or sold?
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91. Any integer program involving 0 – 1 variables with constraint(s) is called a knapsack problem.
a. three
b. two
c. one
d. zero
92. The binary variables in the fixed cost models correspond to:
a. the number of units or products produced
b. the total profit
c. the amount of labor hours
d. a process for which a fixed cost occurs
93. A 0-1 variable, also called a binary variable, is a variable that must equal 0 or 1.
a. True
b. False
alse
94. Any integer programming problem involving 0-1 variables with only one constraint is called a knapsack problem.
a. True
b. False
95. In a set-covering model, each member of a given set (set 1) must be “covered” by an acceptable member of another
set (set 2). The objective of such problems is to minimize the number of elements in set 2 that are needed to cover all
the elements in set 1.
a. True
b. False
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96. Suppose that on Monday morning you have $5000 in cash on hand. For the following seven days, the following cash
requirements must be met: Monday, $6,000; Tuesday, $7,000; Wednesday, $10,000; Thursday, $3,000; Friday,
$8,000; Saturday, $3,000; Sunday, $4,000. At the beginning of each day, you must decide how much money (if any) to
withdraw from the bank. It costs $8 to make a withdrawal of any size. You believe that the opportunity cost of having
$1 of cash on hand for a year is $0.25. Assume that opportunity costs are incurred on each day’s ending balance.
Determine how much money you should withdraw from the bank during each of the next 7 days.
Initial cash (in $1000s) $5
Fixed cost of withdrawal $8
Opportunity cost/$1/day $0.25
Days to withdraw (1 if a withdrawal, 0 if
not)
Mon Tue Wed Thur Fri Sat Sun
1 0 0 0 0 0 0
Amount withdrawn (in $1000s) $36 $0 $0 $0 $0 $0 $0
Logical upper bound $36 $0 $0 $0 $0 $0 $0
Cash requirement (in $1000s) $6 $7 $10 $3 $8 $3 $4
Cash on hand at end of day $35 $28 $18 $15 $7 $4 $0
Meet requirements on time 0 0 0 0 0 0 0
Summary of costs (in dollars)
Withdrawal cost $8.0
Opportunity cost $24.7
Total cost $32.7
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97. A manufacturer can sell product 1 at a profit of $2 per unit and product 2 at a profit of $6 per unit. Three units of raw
material are needed to manufacture one unit of product 1, and 6 units of raw material are needed to manufacture unit of
product 2. A total of 120 units of raw material are available. If any of product 1 is produced, a setup cost of $10 is
incurred, and if any of product 2 is produced, a setup cost of $20 is incurred. Determine how to maximize the
manufacturer’s profit.
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98. The financial CEO is given a group of possible investments projects for his company’s capital. For each
project, he is given the NPV that each project would add to the firm, as well as the cash outflow required by
each project during each year as shown in the table below:
Determine the investments that maximize the firm’s NPV. The firm has 30 million dollars available during
each of the next 5 years. All numbers are in millions of dollars.
99. A company produces three types of glue on two different production lines. Each line can be utilized by up to
six workers at a time. Workers are paid $550 per week on production line 1 and $950 per week on
production line 2. For a week of production it costs $1,050 to set up production line 1 and $2,050 to set up
production line 2. During a week on a production line each worker produces the number of units of glue
shown in the table below:
Each week at least 120 units of glue 1, at least 150 units of glue 2, and at least 200 units of glue 3 must be
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produced. Determine how to minimize the total cost of meeting weekly demands.
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100. A product can be produced on four different machines. Each machine has a fixed setup cost, variable
production cost per unit processed, and a production capacity, as shown below:
A total of 2000 units of the product must be produced. Determine how to minimize the total cost.
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101. My friend Lee Meadow moved from Ferris State University in Michigan to Indiana University East in Indiana and has
rented a truck that can haul up to 1100 cubic feet of furniture. The volume and value of each item he considered moving
on the truck are giving below.
Which items did he take with him to Indiana? What is the value of these items?
102. In nonlinear models, which of the following statements are correct?
a. Only the objective function is not a linear function of the decision variables
b. Only the constraints are not linear functions of the decision variables
c. The objective function and/or the constraints are not linear functions of the decision variables
d. All of these options
103. A nonlinear programming problem (NLP) is an optimization problem in which the objective function and/or the
constraints are not linear functions of the decision variables.
a. True
b. False
104. When we solve a nonlinear programming problem (NLP), it is very possible that Solver will obtain the wrong answer.
a. True
b. False
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105. For some types of integer programming problems, their LP relaxation solutions are optimal.
a. True
b. False
106. A local optimal solution is better than all nearby solutions, but a solution far away might be better than it.
a. True
b. False
107. A global optimal solution is not necessarily the best solution overall.
a. True
b. False
alse
108. Chemical Bank is attempting to determine where its assets should be invested during the current year. At present,
$800,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual rate of return on
each type of investment is known to be the following: bonds, 12%, home loans, 18%, auto loans, 15%, personal loans,
22%. To ensure that the bank's portfolio is not too risky, the bank’s investment manager has placed the following
restrictions on the bank portfolio:
∙ No more than 30% of the total amount invested may be in personal loans
∙ The amount invested in home loans cannot exceed the amount invested in auto loans
∙ The amount invested in personal loans cannot exceed the amount invested in bonds.
Determine how the bank can maximize the annual return on its investment portfolio.
nnual rates
Bonds Home loans Auto loans Personal loans
12% 18% 15% 24%
Maximum percent in personal loans 30%
Amounts invested
Bonds Home loans Auto loans Personal loans
$240,000 $160,000 $160,000 $240,000
$240,000 $160,000 $240,000
Annual return $139,200
109. A company is considering investing a total amount of $2.50 million in four bonds. The expected annual return, the
worst-case annual return on each bond, and the “duration” of each bond are given in the table below.
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Bond 1 Bond 2 Bond 3 Bond 4
Expected 16% 11% 14% 19%
Worst case 7% 9% 11% 10%
Duration 4 5 8 10
The duration of a bond is a measure of the bond’s sensitively to interest rates. The company wants to maximize the
expected return from its bond investments, subject to the following constraints:
∙ The worst-case return of the bond portfolio must be at least 90%.
∙ The average duration of the portfolio must be at most 7
∙ Because of diversification requirements, at most 35% of the total amount invested in a single bond.
Determine how the company can maximize the expected return on its investment.
Chapter 14
110. The risk index of an investment can be obtained by taking the absolute values of percentage changes in the
value of the investment for each year and averaging them. Suppose you are trying to determine what
percentage of your money you should invest in T-bills, gold, and stocks. The table below lists the annual
returns (percentage changes in value) for these investments for the years 1968–1988.
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Chapter 14
Let the risk index of a portfolio be the weighted average of the risk indexes of these investments, where the
weights are the fractions of your money assigned to the investments. Suppose that the amount of each
investment must be between 20% and 50% of the total invested. You would like the risk index of your
portfolio to equal 0.15, and your goal is to maximize the expected return on your portfolio. Determine the
maximum expected return on your portfolio, subject to the stated constraints. Use the average return
earned by each investment during the years 1968-1988 as your estimate of expected return.
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Chapter 14
Determine the cheapest way of producing today’s batch of this drug.
112. A minimum cost network flow model (MCNFM) has the following advantage relative to the special case of a simple
transportation model:
a. a MCNFM does not require capacity restrictions on the arcs of the network
b. the flows in a general MCNFM don’t all necessarily have to be from supply locations to demand locations
c. a MCNFM is generally easier to formulate and solve
d. All of these options
Chapter 14
An oil company produces oil at two wells. Well 1 can produce up to 150,000 barrels per day, and well 2 can produce
up to 200,000 barrels per day. It is possible to ship oil directly from the wells to customers in Los Angeles and New
York. Alternatively, the company could transport oil to the ports of Mobile and Galveston and then ship it by tanker to
New York or Los Angeles. Los Angeles requires 160,000 barrels per day, and New York requires 140,000 barrels per
day. The costs (in dollars) of shipping 1000 barrels between various locations are shown below:
To
From Well 1 Well 2 Mobile Galveston New York Los Angeles
Well 1 $10,000 $10,000 $10 $13 $25 $28
Well 2 $10,000 $10,000 $15 $12 $26 $25
Mobile $10,000 $10,000 $10,000 $6 $16 $17
Galveston $10,000 $10,000 $6 $10,000 $14 $16
New York $10,000 $10,000 $10,000 $10,000 $10,000 $15
Los Angeles $10,000 $10,000 $10,000 $10,000 $15 $10,000
113. Determine how to minimize the transportation cost in meeting the oil demands of Los Angeles and New York.
114. A company manufactures two products. If it charges price for product , it can sell units of product , where
It costs the company $20 to produce a unit of product 1 and $65 to
produce a unit of product 2. How many units of each product should the company produce, and what prices should it
charge, to maximize its profit?115. Linear programming models are used by many financial firms to select a desirable bond portfolio. The following is a
simplified version of such a model. Abby is considering investing in four bonds; $1.5 million is available for investment.
The expected annual return, the worst-case annual return on each bond, and the “duration” of each bond are given
below (The duration of a bond is a measure of the bond’s sensitivity to interest rates.)
Abby wants to maximize the expected return from its bond investments, subject to the following three constraints:
∙ The worst-case return of the bond portfolio must be at least 8%.
∙ The average duration of the portfolio must be at most 6. For example, a portfolio that invests $600,000 in bond 1 and
$400,000 in bond 4 has an average duration of [600,000(3) + 400,000 (9)]/1,000,000 = 5.4
∙ Because of diversification requirements, at most 40% of the total amount invested can be invested in a single bond.
Determine how Abby can maximize the expected return on her investment.
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