DISCUSSION QUESTIONS AND PROBLEMS 381
8. In solving a facility location problem in which there are two possible locations being considered, the transportation algorithm may be used. In doing this, a. two rows (sources) would be added to the existing
rows and the enlarged problem would be solved. b. two separate transportation problems would be solved. c. costs of zero would be used for each of the new facilities. d. the problem would be a transshipment problem.
9. The Hungarian method is a. a way to develop an initial solution to a transportation
problem. b. used to solve assignment problems. c. also called Vogel’s approximation method. d. only used for problems in which the objective is to
10. In an assignment problem, it may be necessary to add more than one row to the table. a. True b. False
11. When using the Hungarian method, an optimal assignment can always be made when every row and every column has at least one zero. a. True b. False
12. An assignment problem can be viewed as a special type of transportation problem with which of the following features? a. the capacity for each source and the demand for each
destination is equal to one b. the number of rows is equal to the number of columns c. the cost for each shipping route is equal to one d. all of the above
Discussion Questions and Problems
Discussion Questions 9-1 Is the transportation model an example of decision
making under certainty or decision making under uncertainty? Why?
9-2 Explain how to determine the number of variables and constraints that would be in a transportation problem simply by knowing the number of sources and the number of destinations.
9-3 What is a balanced transportation problem? Describe the approach you would use to solve an unbalanced problem.
9-4 The stepping-stone method is being used to solve a transportation problem. The smallest quantity in a cell with a minus sign is 35, but two different cells with minus signs have 35 units in them. What prob- lem will this cause, and how should this difficulty be resolved?
9-5 The stepping-stone method is being used to solve a transportation problem. There is only one empty cell having a negative improvement index, and this index is The stepping-stone path for this cell indicates that the smallest quantity for the cells with minus signs is 80 units. If the total cost for the current solution is $900, what will the total cost be for the im- proved solution? What can you conclude about how much the total cost will decrease when developing each new solution for any transportation problem?
9-6 Explain what happens when the solution to a trans- portation problem does not have occu- pied squares (where number of rows in the table and number of columns in the table).
9-7 What is the enumeration approach to solving assign- ment problems? Is it a practical way to solve a 5 row 5 column problem? a problem? Why?7 * 7
n = m =
m + n – 1
9-8 How could an assignment problem be solved using the transportation approach? What condition will make the solution of this problem difficult?
9-9 You are the plant supervisor and are responsible for scheduling workers to jobs on hand. After estimat- ing the cost of assigning each of five available work- ers in your plant to five projects that must be completed immediately, you solve the problem us- ing the Hungarian method. The following solution is reached and you post these job assignments:
Jones to project A
Smith to project B
Thomas to project C
Gibbs to project D
Heldman to project E
The optimal cost was found to be $492 for these as- signments. The plant general manager inspects your original cost estimates and informs you that in- creased employee benefits mean that each of the 25 numbers in your cost table is too low by $5. He suggests that you immediately rework the problem and post the new assignments. Is this necessary? Why? What will the new optimal cost be?
9-10 Sue Simmons’s marketing research firm has local representatives in all but five states. She decides to expand to cover the whole United States by transfer- ring five experienced volunteers from their current locations to new offices in each of the five states. Simmons’s goal is to relocate the five representa- tives at the least total cost. Consequently, she sets up a relocation cost table and prepares to solve it for the best assignments by use of the Hungarian
5 * 5
382 CHAPTER 9 • TRANSPORTATION AND ASSIGNMENT MODELS
method. At the last moment, Simmons recalls that although the first four volunteers did not pose any objections to being placed in any of the five new cities, the fifth volunteer did make one restriction. That person absolutely refused to be assigned to the new office in Tallahassee, Florida—fear of southern roaches, the representative claimed! How should Sue alter the cost matrix to ensure that this assign- ment is not included in the optimal solution?
Problems* 9-11 The management of the Executive Furniture Corpo-
ration decided to expand the production capacity at its Des Moines factory and to cut back production at its other factories. It also recognizes a shifting mar- ket for its desks and revises the requirements at its three warehouses. (a) Use the northwest corner rule to establish an ini-
tial feasible shipping schedule and calculate its cost.
(b) Use the stepping-stone method to test whether an improved solution is possible.
(c) Explain the meaning and implications of an improvement index that is equal to 0. What deci- sions might management make with this informa- tion? Exactly how is the final solution affected?
9-12 Formulate the transportation problem in Problem 9-11 as a linear program and solve using computer soft- ware.
9-13 The Hardrock Concrete Company has plants in three locations and is currently working on three major construction projects, each located at a different site. The shipping cost per truckload of concrete, daily plant capacities, and daily project requirements are provided in the table below. (a) Formulate an initial feasible solution to Hard-
rock’s transportation problem using the north- west corner rule. Then evaluate each unused shipping route by computing all improvement indices. Is this solution optimal? Why?
(b) Is there more than one optimal solution to this problem? Why?
NEW WAREHOUSE REQUIREMENTS NEW FACTORY CAPACITIES
Albuquerque (A) 200 desks Des Moines (D) 300 desks
Boston (B) 200 desks Evansville (E) 150 desks
Cleveland (C) 300 desks Fort Lauderdale (F) 250 desks
Data for Problem 9-11
TO FROM ALBUQUERQUE BOSTON CLEVELAND
DES MOINES 5 4 3
EVANSVILLE 8 4 3
FORT LAUDERDALE 9 7 5
Table for Problem 9-11
TO FROM PROJECT A PROJECT B PROJECT C PLANT CAPACITIES
PLANT 1 $10 $4 $11 70
PLANT 2 12 5 8 50
PLANT 3 9 7 6 30
PROJECT REQUIREMENTS 40 50 60 150
Data for Problem 9-13
Note: means the problem may be solved with QM for Windows; means the problem may be
solved with Excel QM; and means the problem may be solved with QM for Windows and/or Excel QM.
DISCUSSION QUESTIONS AND PROBLEMS 383
9-14 Hardrock Concrete’s owner has decided to increase the capacity at his smallest plant (see Problem 9-13). In- stead of producing 30 loads of concrete per day at plant 3, that plant’s capacity is doubled to 60 loads. Find the new optimal solution using the northwest corner rule and stepping-stone method. How has changing the third plant’s capacity altered the optimal shipping as- signment? Discuss the concepts of degeneracy and multiple optimal solutions with regard to this problem.
9-15 Formulate the Hardrock Concrete Company trans- portation problem in Problem 9-13 as a linear pro- gram and solve using computer software. What would change in the linear program if the change in Problem 9-14 were implemented?
9-16 The Saussy Lumber Company ships pine flooring to three building supply houses from its mills in Pineville, Oak Ridge, and Mapletown. Determine the best transportation schedule for the data given in the table. Use the northwest corner rule and the step- ping-stone method.
9-17 The Krampf Lines Railway Company specializes in coal handling. On Friday, April 13, Krampf had empty cars at the following towns in the quantities indicated:
TOWN SUPPLY OF CARS
TO SUPPLY SUPPLY SUPPLY MILL FROM HOUSE 1 HOUSE 2 HOUSE 3 CAPACITY (TONS)
PINEVILLE $3 $3 $2
OAK RIDGE 4 2 3
MAPLETOWN 3 2 3
SUPPLY HOUSE DEMAND (TONS) 30 30 35 95
Table for Problem 9-16
TOWN DEMAND FOR CARS
Coal Valley 30
Coal Junction 25
By Monday, April 16, the following towns will need coal cars as follows:
TO FROM COAL VALLEY COALTOWN COAL JUNCTION COALSBURG
MORGANTOWN 50 30 60 70
YOUNGSTOWN 20 80 10 90
PITTSBURGH 100 40 80 30
Table for Problem 9-17
Using a railway city-to-city distance chart, the dis- patcher constructs a mileage table for the preceding towns. The result is shown in the table below. Mini- mizing total miles over which cars are moved to new locations, compute the best shipment of coal cars.
9-18 Formulate the Krampf Lines Railway Company sit- uation (Problem 9-17) as a linear program and solve using computer software.
9-19 An air conditioning manufacturer produces room air conditioners at plants in Houston, Phoenix, and Memphis. These are sent to regional distributors in Dallas, Atlanta, and Denver. The shipping costs vary, and the company would like to find the least-cost way to meet the demands at each of the distribution centers. Dallas needs to receive 800 air conditioners per month, Atlanta needs 600, and Denver needs 200. Houston has 850 air conditioners available each month, Phoenix has 650, and Memphis has 300. The shipping cost per unit from Houston to Dallas is $8,
384 CHAPTER 9 • TRANSPORTATION AND ASSIGNMENT MODELS
to Atlanta is $12, and to Denver is $10. The cost per unit from Phoenix to Dallas is $10, to Atlanta is $14, and to Denver is $9. The cost per unit from Memphis to Dallas is $11, to Atlanta is $8, and to Denver is $12. How many units should be shipped from each plant to each regional distribution center? What is the total cost for this?
9-20 Formulate the air conditioning situation present in Problem 9-18 as a linear program and solve using computer software.
9-21 Finnish Furniture manufactures tables in facilities located in three cities—Reno, Denver, and Pitts- burgh. The tables are then shipped to three retail stores located in Phoenix, Cleveland, and Chicago. Management wishes to develop a distribution sched- ule that will meet the demands at the lowest possible cost. The shipping cost per unit from each of the sources to each of the destinations is shown in the following table:
TO FROM PHOENIX CLEVELAND CHICAGO
RENO 10 16 19
DENVER 12 14 13
PITTSBURGH 18 12 12
The available supplies are 120 units from Reno, 200 from Denver, and 160 from Pittsburgh. Phoenix has a demand of 140 units, Cleveland has a demand of 160 units, and Chicago has a demand of 180 units. How many units should be shipped from each man- ufacturing facility to each of the retail stores if cost is to be minimized? What is the total cost?
9-22 Finnish Furniture has experienced a decrease in the demand for tables in Chicago; the demand has fallen
TO EXCESS FROM W X Y Z SUPPLY
A 12¢ 4¢ 9¢ 5¢ 55
B 8¢ 1¢ 6¢ 6¢ 45
C 1¢ 12¢ 4¢ 7¢ 30
UNFILLED POWER DEMAND 40 20 50 20
Find an initial transmission assignment of the excess power supply. Then find the least-cost distribution system.
9-25 Consider the transportation table given below. Find an initial solution using the northwest corner rule. What special condition exists? Explain how you will proceed to solve the problem.
TO DESTINATION DESTINATION DESTINATION FROM A B C SUPPLY
SOURCE 1 $8 $9 $4
SOURCE 2 5 6 8
SOURCE 3 7 9 6
SOURCE 4 5 3 7
DEMAND 110 34 31 175
Table for Problem 9-25
to 150 units (see Problem 9-21). What special condition would exist? What is the minimum-cost solution? Will there be any units remaining at any of the manufacturing facilities?
9-23 Formulate the Finnish Furniture situation (Problem 9-21) as a linear program and solve using computer software.
9-24 The state of Missouri has three major power-generating companies (A, B, and C). During the months of peak demand, the Missouri Power Authority authorizes these companies to pool their excess supply and to distribute it to smaller independent power companies that do not have generators large enough to handle the demand. Excess supply is distributed on the ba- sis of cost per kilowatt hour transmitted. The follow- ing table shows the demand and supply in millions of kilowatt hours and the cost per kilowatt hour of transmitting electric power to four small companies in cities W, X, Y, and Z:
DISCUSSION QUESTIONS AND PROBLEMS 385
TO HOSPITAL HOSPITAL HOSPITAL HOSPITAL FROM 1 2 3 4 SUPPLY
BANK 1 $8 $9 $11 $16
BANK 2 12 7 5 8
BANK 3 14 10 6 7
DEMAND 90 70 40 50 250
Table for Problem 9-26
DEMAND FOR MONTH STAINLESS STEEL SINKS
PROPERTY (INTEREST RATES) (%)
SAVINGS AND DRURY MAXIMUM LOAN COMPANY HILL ST. BANKS ST. PARK AVE. LANE CREDIT LINE ($)
FIRST HOMESTEAD 8 8 10 11 80,000
COMMONWEALTH 9 10 12 10 100,000
WASHINGTON FEDERAL 9 11 10 9 120,000
LOAN REQUIRED TO PURCHASE BUILDING $60,000 $40,000 $130,000 $70,000
Table for Problem 9-28
9-26 The three blood banks in Franklin County are coordi- nated through a central office that facilitates blood de- livery to four hospitals in the region. The cost to ship a standard container of blood from each bank to each hospital is shown in the table below. Also given are the biweekly number of containers available at each bank and the biweekly number of containers of blood needed at each hospital. How many shipments should be made biweekly from each blood bank to each hos- pital so that total shipment costs are minimized?
9-27 Formulate the Franklin County Blood Bank situation (Problem 9-26) as a linear program and solve using computer software.
9-28 The B. Hall Real Estate Investment Corporation has identified four small apartment buildings in which it would like to invest. Mrs. Hall has approached three savings and loan companies regarding financing. Because Hall has been a good client in the past and has maintained a high credit rating in the commu- nity, each savings and loan company is willing to consider providing all or part of the mortgage loan needed on each property. Each loan officer has set differing interest rates on each property (rates are af- fected by the neighborhood of the apartment build- ing, condition of the property, and desire by the individual savings and loan to finance various-size buildings), and each loan company has placed a
maximum credit ceiling on how much it will lend Hall in total. This information is summarized in the table on this page.
Each apartment building is equally attractive as an investment to Hall, so she has decided to pur- chase all buildings possible at the lowest total pay- ment of interest. From which savings and loan companies should she borrow to purchase which buildings? More than one savings and loan can fi- nance the same property.
9-29 Formulate the B. Hall Real Estate Investment Cor- poration problem (Problem 9-28) as a linear pro- gram and solve using computer software.
9-30 The J. Mehta Company’s production manager is planning for a series of 1-month production periods for stainless steel sinks. The demand for the next 4 months is as follows:
386 CHAPTER 9 • TRANSPORTATION AND ASSIGNMENT MODELS
The Mehta firm can normally produce 100 stainless steel sinks in a month. This is done during regular production hours at a cost of $100 per sink. If de- mand in any 1 month cannot be satisfied by regular production, the production manager has three other choices: (1) He can produce up to 50 more sinks per month in overtime but at a cost of $130 per sink; (2) he can purchase a limited number of sinks from a friendly competitor for resale (the maximum num- ber of outside purchases over the 4-month period is 450 sinks, at a cost of $150 each); or (3) he can fill the demand from his on-hand inventory. The inven- tory carrying cost is $10 per sink per month. Back orders are not permitted. Inventory on hand at the beginning of month 1 is 40 sinks. Set up this “pro- duction smoothing” problem as a transportation problem to minimize cost. Use the northwest corner rule to find an initial level for production and outside purchases over the 4-month period.
9-31 Formulate the J. Mehta production problem (See Problem 9-30) as a linear program and solve using computer software.
9-32 Ashley’s Auto Top Carriers currently maintains plants in Atlanta and Tulsa that supply major distri- bution centers in Los Angeles and New York. Be- cause of an expanding demand, Ashley has decided to open a third plant and has narrowed the choice to one of two cities—New Orleans or Houston. The pertinent production and distribution costs, as well as the plant capacities and distribution demands, are shown in the table below.
Which of the new possible plants should be opened?
9-33 Formulate and solve linear programs to help Ashley’s Auto Top Carriers (See Problem 9-32) determine
where to open the new plant. How much difference in the costs for the two locations?
9-34 Marc Smith, vice president for operations of HHN, Inc., a manufacturer of cabinets for telephone switches, is constrained from meeting the 5-year forecast by limited capacity at the existing three plants. These three plants are Waterloo, Pusan, and Bogota. You, as his able assistant, have been told that because of existing capacity constraints and the expanding world market for HHN cabinets, a new plant is to be added to the existing three plants. The real estate department has advised Marc that two sites seem particularly good because of a stable po- litical situation and tolerable exchange rate: Dublin, Ireland, and Fontainebleau, France. Marc suggests that you should be able to take the data on the next page and determine where the fourth plant should be located on the basis of production costs and trans- portation costs. Which location is better?
9-35 Don Levine Corporation is considering adding an additional plant to its three existing facilities in Decatur, Minneapolis, and Carbondale. Both St. Louis and East St. Louis are being considered. Evaluating only the transportation costs per unit as shown in the tables below and on the next page, which site is best?
Indicates distribution cost (shipping, handling, storage) will be $6 per carrier if sent from Houston to New York
TO DISTRIBUTION CENTERS
Data for Problem 9-32
FROM EXISTING PLANTS
TO DECATUR MINNEAPOLIS CARBONDALE DEMAND
Blue Earth $20 $17 $21 250
Ciro 25 27 20 200
Des Moines 22 25 22 350
Capacity 300 200 150
DISCUSSION QUESTIONS AND PROBLEMS 387
9-36 Using the data from Problem 9-35 plus the unit pro- duction costs shown in the following table, which lo- cations yield the lowest cost?
9-38 Four automobiles have entered Bubba’s Repair Shop for various types of work, ranging from a transmis- sion overhaul to a brake job. The experience level of the mechanics is quite varied, and Bubba would like to minimize the time required to complete all of the jobs. He has estimated the time in minutes for each mechanic to complete each job. Billy can complete job 1 in 400 minutes, job 2 in 90 minutes, job 3 in 60 minutes, and job 4 in 120 minutes. Taylor will finish job 1 in 650 minutes, job 2 in 120 minutes, job 3 in 90 minutes, and job 4 in 180 minutes. Mark will finish job 1 in 480 minutes, job 2 in 120 minutes, job 3 in 80 minutes, and job 4 in 180 minutes. John will complete job 1 in 500 minutes, job 2 in 110 minutes, job 3 in 90 minutes, and job 4 in 150 minutes. Each mechanic should be assigned to just one of these jobs. What is the minimum total time required to finish the four jobs? Who should be assigned to each job?
MARKET AREA WATERLOO PUSAN BOGOTA FONTAINEBLEAU DUBLIN
Production cost $50 $30 $40 $50 $45
Transportation cost 10 25 20 25 25
Production cost 50 30 40 50 45
Transportation cost 20 25 10 30 30
Production cost 50 30 40 50 45
Transportation cost 25 10 25 40 40
Production cost 50 30 40 50 45
Transportation cost 25 40 30 10 20
Capacity 8,000 2,000 5,000 9,000 9,000
Data for Problem 9-34
FROM PROPOSED PLANTS
TO EAST ST. LOUIS ST. LOUIS
Blue Earth $29 $27
Ciro 30 28
Des Moines 30 31
Capacity 150 150
LOCATION PRODUCTION COSTS
East St. Louis 40
St. Louis 50
9-37 In a job shop operation, four jobs may be performed on any of four machines. The hours required for each job on each machine are presented in the fol- lowing table. The plant supervisor would like to as- sign jobs so that total time is minimized. Find the best solution.
JOB W X Y Z
A12 10 14 16 13
A15 12 13 15 12
B2 9 12 12 11
B9 14 16 18 16
388 CHAPTER 9 • TRANSPORTATION AND ASSIGNMENT MODELS
9-39 Baseball umpiring crews are currently in four cities where three-game series are beginning. When these are finished, the crews are needed to work games in four different cities. The distances (miles) from each of the cities where the crews are currently working to the cities where the new games will begin are shown in the following table:
Bardot, and Hoolihan. Believing in the quantitative analysis approach to problem solving, the adminis- trator has interviewed each nurse, considered his or her background, personality, and talents, and devel- oped a cost scale ranging from 0 to 100 to be used in the assignment. A 0 for Nurse Bardot being assigned to the cardiology unit implies that she would be per- fectly suited to that task. A value close to 100, on the other hand, would imply that she is not at all suited to head that unit. The accompanying table gives the complete set of cost figures that the hospital admin- istrator felt represented all possible assignments. Which nurse should be assigned to which unit?
9-43 The Gleaming Company has just developed a new dishwashing liquid and is preparing for a national tel- evision promotional campaign. The firm has decided to schedule a series of 1-minute commercials during the peak homemaker audience viewing hours of 1 to 5 p.m. To reach the widest possible audience, Gleam- ing wants to schedule one commercial on each of four networks and to have one commercial appear during each of the four 1-hour time blocks. The exposure rat- ings for each hour, which represent the number of viewers per $1,000 spent, are presented in the follow- ing table. Which network should be scheduled each hour to provide the maximum audience exposure?
FROM KANSAS CITY CHICAGO DETROIT TORONTO
Seattle 1,500 1,730 1,940 2,070
Arlington 460 810 1,020 1,270
Oakland 1,500 1,850 2,080 X
Baltimore 960 610 400 330
PROFESSOR STATISTICS MANAGEMENT FINANCE ECONOMICS
Anderson 90 65 95 40
Sweeney 70 60 80 75
Williams 85 40 80 60
McKinney 55 80 65 55
NURSE UROLOGY CARDIOLOGY ORTHOPEDICS OBSTETRICS
Hawkins 28 18 15 75
Condriac 32 48 23 38
Bardot 51 36 24 36
Hoolihan 25 38 55 12
The X indicates that the crew in Oakland cannot be sent to Toronto. Determine which crew should be sent to each city to minimize the total distance traveled. How many miles will be traveled if these assignments are made?
9-40 In Problem 9-39, the minimum travel distance was found. To see how much better this solution is than the assignments that might have been made, find the assignments that would give the maximum distance traveled. Compare this total distance with the dis- tance found in Problem 9-39.
9-41 Roscoe Davis, chairman of a college’s business department, has decided to apply a new method in assigning professors to courses next semester. As a criterion for judging who should teach each course, Professor Davis reviews the past two years’ teaching evaluations (which were filled out by students). Since each of the four professors taught each of the four courses at one time or another during the two- year period, Davis is able to record a course rating for each instructor. These ratings are shown in the table. Find the best assignment of professors to courses to maximize the overall teaching rating.
9-42 The hospital administrator at St. Charles General must appoint head nurses to four newly established departments: urology, cardiology, orthopedics, and obstetrics. In anticipation of this staffing problem, she had hired four nurses: Hawkins, Condriac,
VIEWING HOURS A B C INDEPENDENT
1–2 P.M. 27.1 18.1 11.3 9.5
2–3 P.M. 18.9 15.5 17.1 10.6
3–4 P.M. 19.2 18.5 9.9 7.7
4–5 P.M. 11.5 21.4 16.8 12.8
WORKER 1 2 3
Adams $11 $14 $6
Brown 8 10 11
Cooper 9 12 7
Davis 10 13 8
9-44 The Fix-It Shop (see Section 9.8) has added a fourth repairman, Davis. Solve the accompanying cost table for the new optimal assignment of workers to projects. Why did this solution occur?
DISCUSSION QUESTIONS AND PROBLEMS 389
9-45 The Patricia Garcia Company is producing seven new medical products. Each of Garcia’s eight plants can add one more product to its current line of med- ical devices. The unit manufacturing costs for pro- ducing the different parts at the eight plants are shown in the table above. How should Garcia assign the new products to the plants to minimize manufac- turing costs?
9-46 Haifa Instruments, an Israeli producer of portable kidney dialysis units and other medical products, develops an 8-month aggregate plan. Demand and capacity (in units) are forecast as shown in the table below.
The cost of producing each dialysis unit is $1,000 on regular time, $1,300 on overtime, and $1,500 on a subcontract. Inventory carrying cost is $100 per unit per month. There is no beginning or ending in- ventory in stock. (a) Set up a production plan, using the transporta-
tion model, that minimizes cost. What is this plan’s cost?
(b) Through better planning, regular time produc- tion can be set at exactly the same value, 275 per month. Does this alter the solution?
(c) If overtime costs rise from $1,300 to $1,400, does this change your answer to part (a)? What if they fall to $1,200?
9-47 NASA’s astronaut crew currently includes 10 mis- sion specialists who hold a doctoral degree in either astrophysics or astromedicine. One of these special- ists will be assigned to each of the 10 flights sched- uled for the upcoming nine months. Mission specialists are responsible for carrying out scientific and medical experiments in space or for launching, retrieving, or repairing satellites. The chief of astro- naut personnel, himself a former crew member with three missions under his belt, must decide who should be assigned and trained for each of the very different missions. Clearly, astronauts with medical educations are more suited to missions involving bi- ological or medical experiments, whereas those with engineering- or physics-oriented degrees are best suited to other types of missions. The chief assigns each astronaut a rating on a scale of 1 to 10 for each possible mission, with a 10 being a perfect match for the task at hand and a 1 being a mismatch. Only one specialist is assigned to each flight, and none is reas- signed until all others have flown at least once. (a) Who should be assigned to which flight? (b) NASA has just been notified that Anderson is
getting married in February and has been granted a highly sought publicity tour in Europe that month. (He intends to take his wife and let
CAPACITY SOURCE JAN. FEB. MAR. APR. MAY JUNE JULY AUG.
Regular time 235 255 290 300 300 290 300 290
Overtime 20 24 26 24 30 28 30 30
Subcontract 12 15 15 17 17 19 19 20
Demand 255 294 321 301 330 320 345 340
Data for Problem 9-46
COMPONENT 1 2 3 4 5 6 7 8
C53 $0.10 $0.12 $0.13 $0.11 $0.10 $0.06 $0.16 $0.12
C81 0.05 0.06 0.04 0.08 0.04 0.09 0.06 0.06
D5 0.32 0.40 0.31 0.30 0.42 0.35 0.36 0.49
D44 0.17 0.14 0.19 0.15 0.10 0.16 0.19 0.12
E2 0.06 0.07 0.10 0.05 0.08 0.10 0.11 0.05
E35 0.08 0.10 0.12 0.08 0.09 0.10 0.09 0.06
G99 0.55 0.62 0.61 0.70 0.62 0.63 0.65 0.59
Data for Problem 9-45
390 CHAPTER 9 • TRANSPORTATION AND ASSIGNMENT MODELS
the trip double as a honeymoon.) How does this change the final schedule?
(c) Certo has complained that he was misrated on his January missions. Both ratings should be 10s, he claims to the chief, who agrees and re- computes the schedule. Do any changes occur over the schedule set in part (b)?
(d) What are the strengths and weaknesses of this approach to scheduling?
9-48 The XYZ Corporation is expanding its market to include Texas. Each salesperson is assigned to
potential distributors in one of five different areas. It is anticipated that the salesperson will spend about three to four weeks in each area. A statewide mar- keting campaign will begin once the product has been delivered to the distributors. The five sales peo- ple who will be assigned to these areas (one person for each area) have rated the areas on the desirability of the assignment as shown in the following table. The scale is 1 (least desirable) to 5 (most desirable). Which assignments should be made if the total of the ratings is to be maximized?
JAN. JAN. FEB. FEB. MAR. APR. MAY JUN. AUG. SEP. ASTRONAUT 12 27 5 26 26 12 1 9 20 19
Vincze 9 7 2 1 10 9 8 9 2 6
Veit 8 8 3 4 7 9 7 7 4 4
Anderson 2 1 10 10 1 4 7 6 6 7
Herbert 4 4 10 9 9 9 1 2 3 4
Schatz 10 10 9 9 8 9 1 1 1 1
Plane 1 3 5 7 9 7 10 10 9 2
Certo 9 9 8 8 9 1 1 2 2 9
Moses 3 2 7 6 4 3 9 7 7 9
Brandon 5 4 5 9 10 10 5 4 9 8
Drtina 10 10 9 7 6 7 5 4 8 8
Data for Problem 9-47
EL PASO/WEST TEXAS
CORPUS CHRISTI/RIO GRANDE VALLEY
Erica 5 3 2 3 4
Louis 3 4 4 2 2
Maria 4 5 4 3 3
Paul 2 4 3 4 3
Orlando 4 5 3 5 4
See our Internet home page, at www.pearsonhighered.com/render, for additional problems, Problems 9-49 through 9-55.
Internet Homework Problems
CASE STUDY 391
Andrew–Carter, Inc. (A–C), is a major Canadian producer and distributor of outdoor lighting fixtures. Its fixture is distributed throughout North America and has been in high demand for several years. The company operates three plants that manufac- ture the fixture and distribute it to five distribution centers (warehouses).
During the present recession, A–C has seen a major drop in demand for its fixture as the housing market has declined. Based on the forecast of interest rates, the head of operations feels that demand for housing and thus for its product will re- main depressed for the foreseeable future. A–C is considering closing one of its plants, as it is now operating with a forecasted excess capacity of 34,000 units per week. The forecasted weekly demands for the coming year are
Warehouse 1 9,000 units
Warehouse 2 13,000 units
Warehouse 3 11,000 units
Warehouse 4 15,000 units
Warehouse 5 8,000 units
The plant capacities in units per week are
Plant 1, regular time 27,000 units
Plant 1, on overtime 7,000 units
Plant 2, regular time 20,000 units
Plant 2, on overtime 5,000 units
Plant 3, regular time 25,000 units
Plant 3, on overtime 6,000 units
If A–C shuts down any plants, its weekly costs will change, as fixed costs are lower for a nonoperating plant. Table 9.34 shows production costs at each plant, both variable at regular time and overtime, and fixed when operating and shut down. Table 9.35 shows distribution costs from each plant to each warehouse (distribution center).
Discussion Questions 1. Evaluate the various configurations of operating and
closed plants that will meet weekly demand. Determine which configuration minimizes total costs.
2. Discuss the implications of closing a plant.
Source: Professor Michael Ballot, University of the Pacific.
FIXED COST PER WEEK
PLANT VARIABLE COST OPERATING NOT OPERATING
No. 1, regular time $2.80/unit $14,000 $6,000
No. 1, overtime 3.52
No. 2, regular time 2.78 12,000 5,000
No. 2, overtime 3.48
No. 3, regular time 2.72 15,000 7,500
No. 3, overtime 3.42
TABLE 9.34 Andrew–Carter, Inc., Variable Costs and Fixed Production Costs per Week
TO DISTRIBUTION CENTER
FROM PLANT W1 W2 W3 W4 W5
No. 1 $0.50 $0.44 $0.49 $0.46 $0.56
No. 2 0.40 0.52 0.50 0.56 0.57
No. 3 0.56 0.53 0.51 0.54 0.35
TABLE 9.35 Andrew–Carter, Inc., Distribution Costs per Unit
392 CHAPTER 9 • TRANSPORTATION AND ASSIGNMENT MODELS
Old Oregon Wood Store
In 1992, George Brown started the Old Oregon Wood Store to manufacture Old Oregon tables. Each table is carefully con- structed by hand using the highest-quality oak. Old Oregon tables can support more than 500 pounds, and since the start of the Old Oregon Wood Store, not one table has been returned because of faulty workmanship or structural problems. In addi- tion to being rugged, each table is beautifully finished using a urethane varnish that George developed over 20 years of work- ing with wood-finishing materials.
The manufacturing process consists of four steps: prepara- tion, assembly, finishing, and packaging. Each step is per- formed by one person. In addition to overseeing the entire operation, George does all of the finishing. Tom Surowski per- forms the preparation step, which involves cutting and forming the basic components of the tables. Leon Davis is in charge of the assembly, and Cathy Stark performs the packaging.
Although each person is responsible for only one step in the manufacturing process, everyone can perform any one of the steps. It is George’s policy that occasionally everyone should complete several tables on his or her own without any help or assistance. A small competition is used to see who can complete an entire table in the least amount of time. George maintains average total and intermediate completion times. The data are shown in Figure 9.5.
It takes Cathy longer than the other employees to construct an Old Oregon table. In addition to being slower than the other employees, Cathy is also unhappy about her current responsi- bility of packaging, which leaves her idle most of the day. Her first preference is finishing, and her second preference is preparation.
In addition to quality, George is concerned with costs and efficiency. When one of the employees misses a day, it causes major scheduling problems. In some cases, George assigns an- other employee overtime to complete the necessary work. At other times, George simply waits until the employee returns to work to complete his or her step in the manufacturing process. Both solutions cause problems. Overtime is expensive, and waiting causes delays and sometimes stops the entire manufac- turing process.
To overcome some of these problems, Randy Lane was hired. Randy’s major duties are to perform miscellaneous jobs and to help out if one of the employees is absent. George has given Randy training in all phases of the manufacturing process, and he is pleased with the speed at which Randy has been able to learn how to completely assemble Old Oregon tables. Total and intermediate completion times are given in Figure 9.6.
Preparation Assembly Finishing Packaging
100 160 250 275
Preparation Assembly Finishing Packaging
80 160 220 230
Preparation Assembly Finishing Packaging
110 200 290
Preparation Assembly Finishing Packaging
120 190 290 315
FIGURE 9.5 Manufacturing Time in Minutes
Preparation Assembly Finishing Packaging
110 190 290 300 FIGURE 9.6 Randy’s Completion Times in Minutes
APPENDIX 9.1: USING QM FOR WINDOWS 393
Discussion Questions 1. What is the fastest way to manufacture Old Oregon tables
using the original crew? How many could be made per day? 2. Would production rates and quantities change signifi-
cantly if George would allow Randy to perform one of the four functions and make one of the original crew the backup person?
3. What is the fastest time to manufacture a table with the original crew if Cathy is moved to either preparation or finishing?
4. Whoever performs the packaging function is severely un- derutilized. Can you find a better way of utilizing the four- or five-person crew than either giving each a single job or allowing each to manufacture an entire table? How many tables could be manufactured per day with this scheme?
See our Internet home page, at www.pearsonhighered.com/render, for these additional case studies: (1) Northwest General Hospital: This case involves improving the food distribution system in
a hospital to reduce the chances of food getting cold before it is delivered to the patients.
(2) Custom Vans, Inc: This case involves finding the best location for a plant that will manu- facture showers used in customized vans.
Internet Case Studies
Adlakha, V., and K. Kowalski. “Simple Algorithm for the Source-Induced Fixed-Charge Transportation Problem,” Journal of the Operational Research Society 55, 12 (2004): 1275–1280.
Awad, Rania M., and John W. Chinneck. “Proctor Assignment at Carleton University,” Interfaces 28, 2 (March–April 1998): 58–71.
Bowman, E. “Production Scheduling by the Transportation Method of Linear Programming,” Operations Research 4 (1956).
Dawid, Herbert, Johannes Konig, and Christine Strauss. “An Enhanced Ros- tering Model for Airline Crews,” Computers and Operations Research 28, 7 (June 2001): 671–688.
Domich, P. D., K. L. Hoffman, R. H. F. Jackson, and M. A. McClain. “Locat- ing Tax Facilities: A Graphics-Based Microcomputer Optimization Model,” Management Science 37 (August 1991): 960–979.
Hezarkhani, Behzad, and Wieslaw Kubiak. “A Coordinating Contract for Transshipment In a Two-Company Supply Chain,” European Journal of Operational Research 207, 1 (2010): 232–237.
Koksalan, Murat, and Haldun Sural. “Efes Beverage Group Makes Location and Distribution Decisions for Its Malt Plants,” Interfaces 29, 2 (March–April, 1999): 89–103.
Liu, Shiang-Tai. “The Total Cost Bounds of the Transportation Problem with Varying Demand and Supply,” Omega 31, 4 (2003): 247–251.
Martello, Silvano. “Jeno Egervary: From the Origins of the Hungarian Algo- rithm to Satellite Communication,” Central European Journal of Opera- tions Research 18, 1 (2010): 47–58.
McKeown, P., and B. Workman. “A Study in Using Linear Programming to Assign Students to Schools,” Interfaces 6, 4 (August 1976).
Pooley, J. “Integrated Production and Distribution Facility Planning at Ault Foods,” Interfaces 24, 4 (July–August 1994): 113–121.
Render, B., and R. M. Stair. Introduction to Management Science. Boston: Allyn & Bacon, Inc., 1992.
Appendix 9.1: Using QM for Windows
QM for Windows has both a transportation module and an as- signment module in its menu. Both are easy to use in terms of data entry and easy to interpret in terms of output. Program 9.6A shows the input screen for the Executive Furniture transportation example. The starting solution technique may be specified. The
results are shown in Figure 9.6B. By clicking Window, you have the option of seeing the iterations that are performed to reach the final solution. Program 9.7A provides the input screen for the Fix-It Shop assignment example. Simply enter the costs and then click Solve. Program 9.7B gives the solution to this.