1.

Given a transportation problem with the following costs, supply, and demand, find the initial
solution using the minimum cell cost method and Vogel’s approximation model. Is the VAM solution
optimal?

To

From 1 2 3 Supply

A $ 6 $ 7 $ 4 100
B 5 3 6 180
C 8 5 7 200

Demand 135 175 170

2. Consider the following transportation problem.

To

From 1 2 3 Supply

A $ 6 $ 9 $ 7 130
B 12 3 5 70
C 4 11 11 100

Demand 80 110 60

a. Find the initial solution using the minimum cell cost method.
b. Solve using the stepping-stone method.
3. Steel mills in three cities produce the following amounts of steel.
Location Weekly Production (tons)
A. Bethlehem 150
B. Birmingham 210
C. Gary 320
680
These mills supply steel to four cities where manufacturing plants have the following
demand.
Location Weekly Demand (tons)
1. Detroit 130
2. St. Louis 70
3. Chicago 180
4. Norfolk 240
620
Shipping costs per ton of steel are as follows.

To

From 1 2 3 4

A $14 9 16 18
B 11 8 7 16
C 16 12 10 22

Because of a truckers’ strike, shipments are at present prohibited from Birmingham to Chicago.
a. Set up a transportation tableau for this problem and determine the initial solution. Identify
the method used to find the initial solution.
b. Solve this problem using MODI.
c. Are there multiple optimal solutions? Explain. If so, identify them.

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 d. Formulate this problem as a general linear programming model.
4. Coal is mined and processed at the following four mines in Kentucky, West Virginia, and Virginia.

Location Capacity (tons)

A. Cabin Creek 90
B. Surry 50
C. Old Fort 80
D. McCoy 60
280

These mines supply the following amount of coal to utility power plants in three cities.

Plant Demand (tons)

1. Richmond 120
2. Winston-Salem 100
3. Durham 110
330
The railroad shipping costs ($1,000s) per ton of coal are shown in the following table.
Because of
railroad construction, shipments are now prohibited from Cabin Creek to Richmond.

To
From 1 2 3
A $7 $10 $5
B 12 9 4
C 7 3 11
D 9 5 7

a. Set up the transportation tableau for this problem, determine the initial solution using
VAM,
and compute total cost.
b. Solve using MODI.
c. Are there multiple optimal solutions? Explain, If there are alternative solutions, identify
them.
d. Formulate this problem as a linear programming model.
5. Oranges are grown, picked, and then stored in warehouses in Tampa, Miami, and Fresno.
These
warehouses supply oranges to markets in New York, Philadelphia, Chicago, and Boston. The
fol-
lowing table shows the shipping costs per truckload ($100s), supply, and demand. Because of
an
agreement between distributors, shipments are prohibited from Miami to Chicago.
To
From New York Philadelphia Chicago Boston
Supply

Tampa $ 9 $ 14 $ 12 $17
200

Miami 11 10 6 10
200

Fresno 12 8 15 7
200

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 Demand 130 170 100 150

a. Set up the transportation tableau for this problem and determine the initial solution
using the
minimum cell cost method.
b. Solve using MODI.
c. Are there multiple optimal solutions? Explain. If so, identify them.

6. A manufacturing firm produces diesel engines in four cities—Phoenix, Seattle, St. Louis,
and Detroit. The company is able to produce the following numbers of engines per
month.
Plant Production
1. Phoenix 5
2. Seattle 25
3. St. Louis 20
4. Detroit 25
Three trucking firms purchase the following numbers of engines for their plants in three cities.
Firm Demand
A. Greensboro 10
B. Charlotte 20
C. Louisville 15
The transportation costs per engine ($100s) from sources to destinations are shown in the
following table. However, the Charlotte firm will not accept engines made in Seattle, and the
Louisville firm will not accept engines from Detroit; therefore, these routes are prohibited.

To

From A B C

1 $ 7 $ 8 $ 5
2 6 10 6
3 10 4 5
4 3 9 11

a. Set up the transportation tableau for this problem. Find the initial solution using VAM.
b. Solve for the optimal solution using the stepping-stone method. Compute the total
minimum
cost.
7. The Interstate Truck Rental firm has accumulated extra trucks at three of its truck leasing outlets, as
shown in the following table.

Extra
Leasing Outlet Trucks
1. Atlanta 70
2. St. Louis 115
3. Greensboro 60
Total 245

The firm also has four outlets with shortages of rental trucks, as follows.

Trucks
Leasing Outlet Shortage
A. New Orleans 80

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 B. Cincinnati 50
C. Louisville 90
D. Pittsburgh 25
Total 245

The firm wants to transfer trucks from those outlets with extras to those with shortages at the minimum
total cost. The following costs of transporting these trucks from city to city have been determined.

To

From A B C D

1 $ 70 80 45 90
2 120 40 30 75
3 110 60 70 80

a. Find the initial solution using the minimum cell cost method.
b. Solve using the stepping-stone method.
8. A large manufacturing company is closing three of its existing plants and intends to
transfer some of its more skilled employees to three plants that will remain open. The
number of employees available for transfer from each closing plant is as follows.

Closing Plant Transferable Employees

1 60
2 105
3 70
Total 235

The following number of employees can be accommodated at the three plants remaining open.

Open Plants Employees Demanded

A 45
B 90
C 35
Total 170
Each transferred employee will increase product output per day at each plant as shown in the
folowing table. The company wants to transfer employees so as to ensure the maximum increase in
product output.

To

From A B C

1 5 8 6
2 10 9 12
3 7 6 8

a. Find the initial solution using VAM.

b. Solve using MODI.
9. A plant has four operators to be assigned to four machines. The time (minutes) required by
each worker to produce a product on each machine is shown in the following table.

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 Determine the optimal assignment and compute total minimum time.

Machine

Operator A B C D

1 10 12 9 11
2 5 10 7 8
3 12 14 13 11
4 8 15 11 9

10. A shop has four machinists to be assigned to four machines. The hourly cost of having
each machine operated by each machinist is as follows.

Machine

Machinist A B C D

1 $12 11 8 14
2 10 9 10 8
3 14 8 7 11
4 6 8 10 9

However, because he does not have enough experience, machinist 3 cannot operate machine B.
a. Determine the optimal assignment and compute total minimum cost.
b. Formulate this problem as a general linear programming model.
11. The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales
regions. Given their various previous contacts, the salespersons are able to cover the regions in
different amounts of time. The amount of time (days) required by each salesperson to cover each
city is shown in the following table. Which salesperson should be assigned to each region to mini-
mize total time? Identify the optimal assignments and compute total minimum time.

Region

Salesperson A B C D E

1 17 10 15 16 20
2 12 9 16 9 14
3 11 16 14 15 12
4 14 10 10 18 17
5 13 12 9 15 11

12. The Bunker Manufacturing firm has five employees and six machines and wants to assign
the
employees to the machines to minimize cost. A cost table showing the cost incurred by
each
employee on each machine follows. Because of union rules regarding departmental
transfers,
employee 3 cannot be assigned to machine E and employee 4 cannot be assigned to machine
B.
Solve this problem, indicate the optimal assignment, and compute total minimum cost.

Machine

Employee A B C D E F

1 $12 $ 7 $20 $14 $ 8 $10

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 2 10 14 13 20 9 11
3 5 3 6 9 7 10
4 9 11 7 16 9 10
5 10 6 14 8 10 12

13. Given the following cost table for an assignment problem, determine the optimal
assignment
and compute total minimum cost. Identify all alternative solutions if there are multiple
optimal
solutions.
Machine

Operator A B C D

1 $10 $2 $8 $6
2 9 5 11 9
3 12 7 14 14
4 3 1 4 2

14. An electronics firm produces electronic components, which it supplies to various electrical
manufacturers. Quality control records indicate that different employees produce different numbers of
defective items. The average number of defects produced by each employee for each of six
components is given in the following table. Determine the optimal assignment that will minimize the total
average number of defects produced by the firm per month.

Component

Employee A B C D E
F

1 30 24 16 26 30
22
2 22 28 14 30 20
13
3 18 16 25 14 12
22
4 14 22 18 23 21
30
5 25 18 14 16 16
28
6 32 14 10 14 18
20

15. State University has planned six special catered events for the November Saturday of its homecoming
football game. The events include an alumni brunch, a parent’s brunch, a booster club luncheon,
a postgame party for season ticket holders, a lettermen’s dinner, and a fund-raising dinner for
major contributors. The university wants to use local catering firms as well as the university cater-
ing service to cater these events and it has asked the caterers to bid on each event. The bids
(in $1,000s) based on menu guidelines for the events prepared by the university are shown in the
following table.

Event

Alumni Parent’s Booster Postgame Lettermen’s Contributor’s

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 Caterer Brunch Brunch Club Lunch Party Dinner Dinner

Al’s $12.6 $10.3 $14.0 $19.5 $25.0 $30.0

Bon Apetít 14.5 13.0 16.5 17.0 22.5 32.0
Custom 13.0 14.0 17.6 21.5 23.0 35.0
Divine 11.5 12.6 13.0 18.7 26.2 33.5
Epicurean 10.8 11.9 12.9 17.5 21.9 28.5
Fouchés 13.5 13.5 15.5 22.3 24.5 36.0
University 12.5 14.3 16.0 22.0 26.7 34.0

The Bon Apetít, Custom, and University caterers can handle two events, whereas the other four caterers can handle only
one. The university is confident all the caterers will do a high-quality job, so it wants to select the caterers for the events that
will result in the lowest total cost. Determine the optimal selection of caterers that will minimize total cost.
16. A university department head has five instructors to be assigned to four different courses. All of the instructors have taught
the courses in the past and have been evaluated by the students. The rating for each instructor for each course is given in
the following table (a perfect score is 100). The department head wants to know the optimal assignment of
instructors to courses that will maximize the overall average evaluation. The instructor who is not assigned to teach a
course will be assigned to grade exams. Solve this problem using the assignment method.

Course

Instructor A B C D

1 80 75 90 85
2 95 90 90 97
3 85 95 88 91
4 93 91 80 84
5 91 92 93 88

17. The coach of the women’s swim team at State University is preparing for the conference swim
meet and must choose the four swimmers she will assign to the 800-meter medley relay team. The
medley relay consists of four strokes—the backstroke, breaststroke, butterfly, and freestyle. The
coach has computed the average times (in minutes) each of her top six swimmers has achieved in
each of the four strokes for 200 meters in previous swim meets during the season as follows.

Stroke (min)

Swimmer Backstroke Breaststroke Butterfly Freestyle

Annie 2.56 3.07 2.90 2.26

Beth 2.63 3.01 3.12 2.35
Carla 2.71 2.95 2.96 2.29
Debbie 2.60 2.87 3.08 2.41
Erin 2.68 2.97 3.16 2.25
Fay 2.75 3.10 2.93 2.38

Determine the medley relay team and its total expected relay time for the coach.

18. The Vanguard Publishing Company has eight college students it hires as salespeople to sell encyclopedias
during the summer. The company desires to allocate them to three sales territories. Territory 1 requires
threesalespeople, and territories 2 and 3 require two salespeople each. It is estimated that each salesperson
will be able to generate the amounts of dollar sales per day in each of the three territories as given in the
following table. The company desires to allocate the salespeople to the three territories so that sales will
be maximized. Solve this problem using any method to determine the initial solution and any
solution method. Compute the maximum total salesperday.
Territory

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 Salesperson 1 2 3

A $110 $150 $130

B 90 120 80
C 205 160 175
D 125 100 115
E 140 105 150
F 100 140 120
G 180 210 160
H 110 120 90

19. You are given the following payoff table (in units of thousands of dollars) for a decision analysis
problem:

State of Nature
Alternative S1 S2 S3
A1 220 170 110
A2 200 180 150
(a) Which alternative should be chosen under the maximin payoff criterion?
(b) Which alternative should be chosen under the maximum likelihood criterion?
(c) Which alternative should be chosen under minimax criterion?
(d) Which alternative should be chosen under equally likely criterion?
(e) Which alternative should be chosen under realism criterion? (Assuming 0.75 for optimistic)

20. Betsy Pitzer makes decisions according to Bayes’ decision rule. For her current problem,
Betsy has constructed the following payoff table (in units of dollars):
State of Nature
Alternative S1 S2 S3
A1 50 100 100
A2 20 210 0 10
A3 20 240 2 40
Prior probability 0.5 0.3 0.2

(a) Find EMV

(b) Find EVPI.
(c) What is the most that Betsy should consider paying to obtain more information about which
state of nature will occur?

21. Suppose that the demand for a product is 30 units per month and the items are
withdrawn at a constant rate. The setup cost each time a production run is
undertaken to replenish inventory is $15. The production cost is $1 per item, and
the inventory holding cost is $0.30 per item per month.
(a) Assuming shortages are not allowed, determine how often to make a
production run and what size it should be.
(b) If shortages are allowed but cost $3 per item per month, determine how often
to make a production run and what size it should be.
(c) What is the Total optimal cost?
22. The demand for a product is 600 units per week, and the items are withdrawn at a
constant rate. The setup cost for placing an order to replenish inventory is $25. The
unit cost of each item is $3, and the inventory holding cost is $0.05 per item
perweek.
(a) Assuming shortages are not allowed, determine how often to order and what
size the order should be.
(b) If shortages are allowed but cost $2 per item per week, determine how often to

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 order and what size the order should be.
(c) What is the Total optimal cost?

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