103 TEACHING SUGGESTIONS Teaching Suggestion 8.1: Importance of Formulating Large LP Problems. Since computers are used to solve virtually all business LP prob- lems, the most important thing a student can do is to get experi- ence in formulating a wide variety of problems. This chapter pro- vides such a variety. Teaching Suggestion 8.2: Note on Production Scheduling Problems. The Greenberg Motor example in this chapter is the largest prob- lem in the book in terms of constraints, so it provides a good prac- tice environment. An interesting feature to point out is that LP constraints are capable of tying one production period to the next. Teaching Suggestion 8.3: Solving Assignment Problems by LP. The example of the law firm of Ivan and Ivan in this chapter can clearly be solved more quickly using QM for Windows’ assign- ment program than by the LP program. Students should be asked why anyone would choose to use the LP approach. There are two answers: (1) many commercial LP programs do not contain as- signment algorithms (which are more popular in academic soft- ware such as QM for Windows); and (2) the LP program can pro- vide more sensitivity analysis and economic interpretation than is available in the assignment module. The assignment problem is treated in Chapter 10. Teaching Suggestion 8.4: Labor Planning Problem—Arlington Bank. This example is a good practice tool and lead-in for the Chase Manhattan Bank case at the end of the chapter. Without this exam- ple, the case would probably overpower most students. Teaching Suggestion 8.5: Ingredient Blending Applications. Three points can be made about the two blending examples in this chapter. First, both the diet and fuel blending problems presented here are tiny compared to huge real-world blending problems. But they do provide some sense of the issues to be faced. Second, diet problems that are missing the constraints that force variety into the diet can be terribly embarrassing. It has been said that a hospital in New Orleans ended up with an LP solution to feed each patient only castor oil for dinner because analysts ne- glected to add constraints forcing a well-rounded diet. ALTERNATIVE EXAMPLES Alternative Example 8.1: Natural Furniture Company manu- factures three outdoor products, chairs, benches, and tables. Each product must pass through the following departments before it is shipped: sawing, sanding, assembly, and painting. The time re- quirements (in hours) are summarized in the tables below. The production time available in each department each week and the minimum weekly production requirement to fulfill con- tracts are as follows: The production manager has the responsibility of specifying pro- duction levels for each product for the coming week. Let X 1 Number of chairs produced X 2 Number of benches produced X 3 Number of tables produced The objective function is Maximize profit 15X 1 10X 2 20X 3 Constraints 1.5X 1 1.5X 2 2.0X 3 450 hours of sawing available 1.0X 1 1.5X 2 2.0X 3 400 hours of sanding available 2.0X 1 2.0X 2 2.5X 3 625 hours of assembly available 1.5X 1 2.0X 2 2.0X 3 550 hours of painting available X 1 2.0X 2 2.0X 3 100 chairs X 2 2.0X 3 50 benches X 3 50 tables X 1 , X 2 , X 3 0 Alternative Example 8.2: A phosphate manufacturer produces three grades, A, B, and C, which cost the firm $40, $50, and $60 per kilogram, respectively. The products require the labor and ma- terials per batch that are shown on the following page. 8 C H A P T E R Linear Programming Modeling Applications: With Computer Analyses in Excel and QM for Windows Minimum Capacity Production Department (In Hours) Product Level Sawing 450 Chairs 100 Sanding 400 Benches 50 Assembly 625 Tables 50 Painting 550 Unit Product Sawing Sanding Assembly Painting Profit Chairs 1.5 1.0 2.0 1.5 $15 Benches 1.5 1.5 2.0 2.0 $10 Tables 2.0 2.0 2.5 2.0 $20 Hours Required M08_REND6289_10_IM_C08.QXD 5/7/08 2:26 PM Page 103 REVISED
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103
TEACHING SUGGESTIONS
Teaching Suggestion 8.1: Importance of Formulating Large LP Problems.Since computers are used to solve virtually all business LP prob-lems, the most important thing a student can do is to get experi-ence in formulating a wide variety of problems. This chapter pro-vides such a variety.
Teaching Suggestion 8.2: Note on Production Scheduling Problems.The Greenberg Motor example in this chapter is the largest prob-lem in the book in terms of constraints, so it provides a good prac-tice environment. An interesting feature to point out is that LPconstraints are capable of tying one production period to the next.
Teaching Suggestion 8.3: Solving Assignment Problems by LP.The example of the law firm of Ivan and Ivan in this chapter canclearly be solved more quickly using QM for Windows’ assign-ment program than by the LP program. Students should be askedwhy anyone would choose to use the LP approach. There are twoanswers: (1) many commercial LP programs do not contain as-signment algorithms (which are more popular in academic soft-ware such as QM for Windows); and (2) the LP program can pro-vide more sensitivity analysis and economic interpretation than isavailable in the assignment module. The assignment problem istreated in Chapter 10.
Teaching Suggestion 8.4: Labor Planning Problem—ArlingtonBank.This example is a good practice tool and lead-in for the ChaseManhattan Bank case at the end of the chapter. Without this exam-ple, the case would probably overpower most students.
Teaching Suggestion 8.5: Ingredient Blending Applications.Three points can be made about the two blending examples in thischapter. First, both the diet and fuel blending problems presentedhere are tiny compared to huge real-world blending problems. Butthey do provide some sense of the issues to be faced.
Second, diet problems that are missing the constraints thatforce variety into the diet can be terribly embarrassing. It has beensaid that a hospital in New Orleans ended up with an LP solutionto feed each patient only castor oil for dinner because analysts ne-glected to add constraints forcing a well-rounded diet.
ALTERNATIVE EXAMPLES
Alternative Example 8.1: Natural Furniture Company manu-factures three outdoor products, chairs, benches, and tables. Eachproduct must pass through the following departments before it is
shipped: sawing, sanding, assembly, and painting. The time re-quirements (in hours) are summarized in the tables below.
The production time available in each department each weekand the minimum weekly production requirement to fulfill con-tracts are as follows:
The production manager has the responsibility of specifying pro-duction levels for each product for the coming week. Let
X1 � Number of chairs produced
X2 � Number of benches produced
X3 � Number of tables produced
The objective function is
Maximize profit � 15X1 � 10X2 � 20X3
Constraints
1.5X1 � 1.5X2 � 2.0X3 � 450 hours of sawing available
1.0X1 � 1.5X2 � 2.0X3 � 400 hours of sanding available
2.0X1 � 2.0X2 � 2.5X3 � 625 hours of assembly available
1.5X1 � 2.0X2 � 2.0X3 � 550 hours of painting available
X1 � 2.0X2 � 2.0X3 � 100 chairs
X2 � 2.0X3 � 50 benches
X3 � 50 tables
X1, X2, X3 � 0
Alternative Example 8.2: A phosphate manufacturer producesthree grades, A, B, and C, which cost the firm $40, $50, and $60per kilogram, respectively. The products require the labor and ma-terials per batch that are shown on the following page.
8C H A P T E R
Linear Programming Modeling Applications: WithComputer Analyses in Excel and QM for Windows
CHAPTER 8 LINEAR PROGRAMMING MODEL ING APPL ICAT IONS 105
8-5. Let Xij � 1 if pitcher i is scheduled to go against opponent j,0 otherwise
where i � 1, 2, 3, 4 stands for Jones, Baker, Parker, andWilson, respectively, and
j � 1, 2, 3, 4 stands for Des Moines, Davenport,Omaha, and Peoria, respectively.
Objective: maximize sum of ratings �
0.6X11 � 0.8X12 � 0.5X13 � 0.4X14
� 0.7X21 � 0.4X22 � 0.8X23 � 0.3X24
� 0.9X31 � 0.8X32 � 0.7X33 � 0.8X34
� 0.5X41 � 0.3X42 � 0.4X43 � 0.2X44
subject to
X11 � X12 � X13 � X14 � 1 (“Dead-Arm” Jones)
X21 � X22 � X23 � X24 � 1 (“Spitball” Baker)
X31 � X32 � X33 � X34 � 1 (“Ace” Parker)
X41 � X42 � X43 � X44 � 1 (“Gutter” Wilson)
X11 � X21 � X31 � X41 � 1 (Des Moines)
X12 � X22 � X32 � X42 � 1 (Davenport)
X13 � X23 � X33 � X43 � 1 (Omaha)
X14 � X24 � X34 � X44 � 1 (Peoria)
Solution: X12 � 1, X23 � 1, X34 � 1, X41 � 1, Total P � 2.9
8-6. Let
T � number of TV ads
R � number of radio ads
B � number of billboard ads
N � number of newspaper ads
Maximize total audience � 30,000T � 22,000R � 24,000B �8,000N
Subject to
800T � 400R � 500B � 100N � 15,000
� 10
R 10
10
� 10
� � R � 6
500B � 100N 800T
�, R, , � � 0
Solution: T � 6.875; R � 10; B � 9; N � 10; Audience reached� 722,250.If integer solutions are necessary, integer programming (see Chapter 11) could be used.
Note that the problem is not limited to unduplicated exposure(e.g., one person seeing the Sunday newspaper three weeks in arow counts for three exposures).
Problem 8-7 solved by computer:
Buy 20 Sunday newspaper ads (X1)
Buy 0 TV ads (X2)
This has a cost of $18,500. Perhaps the paint store should considera blend of TV and newspaper, not just the latter.
8-8. Let Xij � number of new leases in month i for j-months, i � 1, . . . , 6; j � 3, 4, 5
8-11. Maximize number of rolls of Supertrex sold �20X1 � 6.8X2 � 12X3 � 65,000X4
where X1 � dollars spent on advertising
X2 � dollars spent on store displays
X3 � dollars in inventory
X4 � percent markup
subject to
X1 � X2 � X3 � $17,000 (budgeted)
X1 � $3,000 (advertising constraint)
X2 � 0.05X3 (or X2 � 0.05X3 � 0)(ratio of displays to inventory)
(markup ranges)
X1, X2, X3, X4 � 0Problem 8-11 solved by computer:
Spend $17,000 on advertising (X1).
Spend nothing on in-store displays or on-hand inventory(X2 and X3).
Take a 20% markup.
The store will sell 327,000 rolls of Supertrex.
X
X4
4
0 20
0 45
�
.
.
⎫⎬⎪
⎭⎪
This solution implies that no on-hand inventory or displaysare needed to sell the product, probably due to an oversight on Mr. Kruger’s part. Perhaps a constraint indicating that X3 �$3,000 of inventory should be held might be needed.
The meal is fairly well-balanced (two meats, a green veg-etable, and a potato). The weight of each item is realistic. Thisproblem is very sensitive to changing food prices.
Sensitivity analysis when prices change:
Milk increases 10 cents/lb: no change in price or dietMilk decreases 10 cents/lb: no change in price or dietMilk decreases 30 cents/lb (to 30 cents): potatoes drop out and
milk enters, price � $1.42/mealGround meat increases from $2.35 to $2.75: price � $1.93 and
spinach leaves the optimal solutionGround meat increases to $5.25/lb: price � $2.07 and meat
leaves; milk, chicken, and potatoes in solutionFish decreases from $2.25 to $2.00/lb: no changeChicken increases to $3.00/lb: price � $1.91 and meat, fish,
spinach, and potatoes in solution
If meat and fish are omitted from the problem, the solution is
CHAPTER 8 LINEAR PROGRAMMING MODEL ING APPL ICAT IONS 107
8-13. a. Let X1 � no. of units of internal modems produced perweek
X2 � no. of units of external modems producedper week
X3 � no. of units of circuit boards produced perweek
X4 � no. of units of floppy disk drives producedper week
X5 � no. of units of hard drives produced perweek
X6 � no. of units of memory boards produced perweek
Objective function analysis: First find the time used on each testdevice:
hours on test device 1
hours on test device 2
hours on test device 3
Thus, the objective function is
maximize profit � revenue � material cost � test cost
� 200X1 � 120X2 � 180X3 � 130X4 � 430X5
� 260X6
� 35X1 � 25X2 � 40X3 � 45X4 � 170X5 � 60X6
This can be rewritten as
maximize profit � $161.35X1 � 92.95X2 � 135.50X3
� 82.50X4 � 249.80X5 � 191.75X6
subject to
All variables � 0
b. The solution is
X1 � 496.55 internal modems
X2 � 1,241.38 external modems
X3 through X6 � 0
profit � $195,504.80
5 1 3 2 9 2
601001 2 3 4 5 6X X X X X X� � � � �
hours
2 5 3 2 15 17
601201 2 3 4 5 6X X X X X X� � � � �
hours
7 3 12 6 18 17
601201 2 3 4 5 6X X X X X X� � � � �
hours
�� � � � �
185 1 3 2 9 2
601 2 3 4 5 6X X X X X X
�� � � � �
122 5 3 2 15 17
601 2 3 4 5 6X X X X X X
�� � � � �
157 3 12 6 18 17
601 2 3 4 5 6X X X X X X
�� � � � �5 1 3 2 9 2
601 2 3 4 5 6X X X X X X
�� � � � �2 5 3 2 15 17
601 2 3 4 5 6X X X X X X
=7 3 12 6 18 17
601 2 3 4 5 6X X X X X X� � � � �
c. The shadow prices, as explained in Chapters 7 and 9,for additional time on the three test devices are $21.41,$5.75, and $0, respectively, per minute.
8-14. a. Let Xi � no. of trained technicians available at start ofmonth i
Yi � no. of trainees beginning in month i
Minimize total salaries paid � $2,000X1
� 2,000X2 � 2,000X3 � 2,000X4 � 2,000X5
� 900Y1 � 900Y2 � 900Y3 � 900Y4 � 900Y5
subject to
130X1 � 90Y1 � 40,000 (Aug. need, hours)
130X2 � 90Y2 � 45,000 (Sept. need)
130X3 � 90Y3 � 35,000 (Oct. need)
130X4 � 90Y4 � 50,000 (Nov. need)
130X5 � 90Y5 � 45,000 (Dec. need)
X1 � 350 (starting staff on Aug. 1)
X2 � X1 � Y1 � 0.05X1 (staff on Sept. 1)
X3 � X2 � Y2 � 0.05X2 (staff on Oct. 1)
X4 � X3 � Y3 � 0.05X3 (staff on Nov. 1)
X5 � X4 � Y4 � 0.05X4 (staff on Dec. 1)
All Xi, Yi � 0
b. The computer-generated results are:
Total salaries paid over the five-month period � $3,627,279.
8-15. a. Let Xij � acres of crop i planted on parcel j
where i � 1 for wheat, 2 for alfalfa, 3 for barley
Profit will be $337,862.10. Multiple optimal solutions exist.
c. Yes, need only 500 more water-feet.
8-16. Amalgamated’s blending problem will have eight variablesand 11 constraints. The eight variables correspond to the eight materi-als available (three alloys, two irons, three carbides) that can be se-lected for the blend. Six of the constraints deal with maximum andminimum quality limits, one deals with the 2,000 pound total weightrestriction, and four deal with the weight availability limits for alloy 2(300 lb), carbide 1 (50 lb), carbide 2 (200 lb), and carbide 3 (100 lb).
Let X1 through X8 represent pounds of alloy 1 through poundsof carbide 3 to be used in the blend.
8-17. This problem refers to Problem 8-16’s infeasibility. Someinvestigative work is needed to track down the issues. From a finalsimplex tableau, we find that constraints 5 and 11 still have artifi-cial variables in the final solution. The two issues are:
1. Requiring at least 5.05% carbon is not possible.2. Producing 1 ton from the materials is not possible.
If constraints 5 and 11 are relaxed (or removed), one solutionis X2 � $83.6 (alloy 2), X6 � 50 lb (carbide 1), X7 � $83.6 (car-bide 2), and X8 � 100 lb (carbide 3). Cost � $34.91.
Each student may take a different approach and other recom-mendations may result.
8-18. X1 � number of medical patients
X2 � number of surgical patients
Maximize revenue � $2,280X1 � $1,515X2
subject to
8X1 � 2.5X2 � 32,850 (patient-days available � 365 days 90 new beds)
3.1X1 � 2.6X2 � 15,000 (lab tests)
1X1 � 2.2X2 � 7,000 (x-rays)
X2 � 2,800 (operations/surgeries)
X1, X2 � 0
Problem 8-18 solved by computer:
X1 � 2,791 medical patients
X2 � 2,105 surgical patients
revenue� $9,551,659 per year
To convert X1 and X2 to number of medical versus surgical beds,find the total number of hospital days for each type of patient:
medical � (2,791 patients)(8 days/patient)
� 22,328 days
surgical � (2,105 patients)(5 days/patient)
� 10,525 days
total � 32,853 days
This represents 68% medical days and 32% surgical days, whichyields 61 medical beds and 29 surgical beds. (Note that an alterna-tive approach would be to formulate with X1, X2 as number of beds.)
See the printout on the next page for the solution and sensi-tivity analysis.
8-19. This problem, suggested by Professor C. Vertullo, is anexcellent exercise in report writing. Here is a chance for studentsto present management science results in a management format.Basically, the following issues need to be addressed in any report:
(a) As seen in Problem 8-18, there should be 61 medicaland 29 surgical beds, yielding $9,551,659 per year.
110 CHAPTER 8 LINEAR PROGRAMMING MODEL ING APPL ICAT IONS
(b) Referring to the QM for Windows printout, there areno empty beds.(c) There are 876 lab tests of unused capacity.(d) The x-ray is used to its maximum and has a $65.45shadow price.(e) The operating room still has 695 operations available.
8-20. 8-20. Let
Si � 1 if Smith is assigned to Job i for i � 1, 2, 3, 4
� 0 otherwise
Ji � 1 if Jones is assigned to Job i for i � 1, 2, 3, 4
� 0 otherwise
Di � 1 if Davis is assigned to Job i for i � 1, 2, 3, 4
� 0 otherwise
Ni � 1 if Nguyen is assigned to Job i for i � 1, 2, 3, 4
All variables 0, 1There are multiple optimal solutions. All of these require a total of31 days. One solution is to assign Smith to Job 2, Jones to Job 4,Davis to Job 1, and Nguyen to Job 3.
8-21. a. Let
A1 � tons of ore from mine A to plant 1
A2 � tons of ore from mine A to plant 2
B1 � tons of ore from mine B to plant 1
B2 � tons of ore from mine B to plant 2
X1 � tons shipped to Builder’s Home from plant 1
X2 � tons shipped to Builder’s Home from plant 2
Y1 � tons shipped to Homeowners’ Headquarters fromplant 1
Y2 � tons shipped to Homeowners’ Headquarters fromplant 2
A1 � B1 � X1 � Y1 � Z1 units shipped into plant 1 mustequal units shipped out of plant 1
A2 � B2 � X2 � Y2 � Z2 units shipped into plant 2 mustequal units shipped out of plant 2
All variables � 0
b. Solving this on the computer, we find the following solution:
A1 � 50 tons of ore from mine A to plant 1
A2 � 270 tons of ore from mine A to plant 2
B1 � 450 tons of ore from mine B to plant 1
X1 � 200 tons shipped to Builder’s Home from plant 1
Y1 � 240 tons shipped to Homeowners’ Headquartersfrom plant 1
Z1 � 60 tons shipped to Hardware City from plant 1
Z2 � 270 tons shipped to Hardware City from plant 2
All other variables equal 0.
Minimum total cost � $19,160
8-22. a. The formulation is the same as the formulation in prob-lem 8-21 except for a change in the objective function. We add theprocessing cost in the objective function, and the new objectivefunction is:
Minimize cost � 28A1 � 30A2 � 25B1 � 28B2 � 13X1
� 19X2 � 17Y1 � 22Y2 � 20Z1 � 21Z2
All the constraints are the same as in the previous problem.
b. The solution is the same as problem 8-21 except thevalue of the objective function is $34,300.
A � FA � 36 maximum amount of fuel board whenleaving Atlanta
L � FL � 15 minimum amount of fuel board whenleaving Los Angeles
L � FL � 23 maximum amount of fuel board whenleaving Los Angeles
H � FH � 9 minimum amount of fuel board whenleaving Houston
H � FH � 17 maximum amount of fuel board whenleaving Houston
N � FN � 11 minimum amount of fuel board whenleaving New Orleans
N � FN � 20 maximum amount of fuel board whenleaving New Orleans
FL � A � FA � (12 � 0.05(A � FA � 24))
This says that the fuel on board when the plane lands in Los Angeleswill equal the amount on board at take-off minus the fuel consumedon that flight. The fuel consumed is 12 (thousand gallons) plus 5% ofthe excess above 24 (thousand gallons). This simplifies to:
Solving this with QM for Windows, we have S2 � 1, J4 � 1, D1 �1, and N3 � 1. So, Smith does Job 2, Jones does Job 4, Davis doesJob 1, and Nguyen does Job 3. The total time is 31 days.
8-30. Let Xi � number of BR54 produced in month i, for i � 1,2, 3.
Yi � number of BR49 produced in month i, for i � 1, 2, 3.
IXi � number of BR54 units in inventory at end of month i,for i � 0, 1, 2, 3.
IYi � number of BR49 units in inventory at end of month i,for i � 0, 1, 2, 3.
CHAPTER 8 LINEAR PROGRAMMING MODEL ING APPL ICAT IONS 113
SOLUTION TO RED BRAND CANNERS CASE
1. The main issue in this case is how to allocate 3 million poundsof tomatoes. The overall objective is to maximize total sales lessvariable costs. These costs include production and selling ex-penses. Twenty percent of the crop was grade A and the rest wasgrade B. In setting up the constraints, the amount of grade A toma-toes cannot exceed 20% of 3 million pounds. Thus not more than600,000 pounds of grade A tomatoes can be used. Similarly, notmore than 2,400,000 pounds of grade B tomatoes can be used.Furthermore, the demand for 50,000 cases of tomato juice and80,000 cases of tomato paste should be met. The demand forwhole tomatoes is not a constraint in this problem. Finally, mini-mum quality requirements should be met. This includes an aver-age of 8 points per pound for whole tomatoes and 6 points perpound for tomato juice. There is no constraint for tomato paste.
Another issue is whether or not to buy 80,000 additionalpounds of grade A tomatoes. This would increase the amount ofavailable grade A tomatoes from 600,000 pounds to 680,000pounds. To answer this question, a new formulation can be madeusing the new 680,000-pound constraint and a price of 8.5 centsper pound for the 80,000 additional pounds of grade A tomatoes inthe objective function. A faster way to resolve this issue is to usepostoptimality analysis, or shadow prices. Using this approach,you compare the value of the 80,000 additional tomatoes with thecost, which is 8.5 cents per pound.
2. The problem can be formulated using LP as follows:
The first constraint refers to the 14 million pounds of wholetomatoes—800,000 cases at 18 pounds per case—that constitutesmaximum demand. Similarly, the maximum demand for tomatojuice is 50,000 cases at 20 pounds per case or 1 million pounds,and the maximum demand for tomato paste is 80,000 cases at 25pounds per case or 2 million pounds, and these are constraints 2and 3. Constraints 4 and 5 reflect the availability of grade A andgrade B tomatoes, respectively, and the last two constraints are thequality constraints. The requirements that canned tomatoes mustaverage at least 8 points means that at least three-fourths of thetomatoes must be grade A:
X1 � 0.75(X1 � X2) �� X1 � 3X2 � 0
Similarly, the requirements that tomato juice must average at least6 points means that at least one-fourth of the tomato juice must begrade A, and that is the last constraint.
The coefficients in the objective function are the unit profits. Acase of whole tomatoes (grade A and grade B) sells for $4. The vari-able cost (less the tomatoes) is $2.52. Since the tomatoes are alreadyon hand (and no salvage appears to be possible), they represent asunk cost and are not part of the decision process. Since there are 18pounds per case, the unit profit is (4.00 � 2.52)/18 � 0.0822. Simi-lar analyses hold for the other terms in the objective function.
The maximum profit is $225,340.All of the grade A tomatoes are used. The shadow price for
the slack variable in constraint 4 is 0.0903. Each additional poundof grade A tomatoes costing 8.5 cents will increase profits by0.093 � 0.0850 � 0.0053. A sensitivity analysis indicates that upto an additional 600,000 pounds of grade A tomatoes could bepurchased without affecting the solution basis.
SOLUTION TO CHASE MANHATTAN BANK CASE
This very advanced and challenging scheduling problem can besolved most expeditiously using linear programming, preferablyinteger programming. Let F denote the number of full-time em-ployees. Some number, F1, of them will work 1 hour of overtimebetween 5 P.M. and 6 P.M. each day and some number, F2, of thefull-time employees will work overtime between 6 P.M. and 7 P.M.There will be seven sets of part-time employees; Pj will be thenumber of part-time employees who begin their workday at hour j,j � 1, 2, . . . , 7, with P1 being the number of workers beginning at9 A.M., P2 at 10 A.M., . . . , P7 at 3 P.M. Note that because part-timeemployees must work a minimum of 4 hours, none can start after3 P.M. since the entire operation ends at 7 P.M. Similarly, somenumber of part-time employees, Qj, leave at the end of hour j, j �4, 5, . . . , 9.
The workforce requirements for the first two hours, 9 A.M.and 10 A.M., are:
F � P1 � 14
F � P1 � P2 � 25
At 11 A.M. half of the full-time employees go to lunch; the remain-ing half go at noon. For those hours:
0.5F � P1 � P2 � P3 � 26
0.5F � P1 � P2 � P3 � P4 � 38
Starting at 1 P.M., some of the part-time employees begin to leave.For the remainder of the straight-time day:
If the left-hand sides of these 10 constraints are added, one findsthat 7F hours of full-time labor are used in straight time (although 8Fare paid for), F1 � F2 full-time labor hours are used and paid for atovertime rates, and the total number of part-time hours is
Total overtime for a full-time employee is restricted to 5 hours orless, an average of 1 hour or less per day per employee. Thus thenumber of overtime hours worked per day cannot exceed the num-ber of full-time employees:
F1 � F2 � F
Since part-time employees must work at least 4 hours per day,
Q4 � P1
for those leaving at the end of the fourth hour. At the end of thefifth hour, those leaving must be drawn from the P1 � Q4 remain-ing plus the P2 that arrived at the start of the second hour:
The resulting problem has 16 integer variables and 22 con-straints. If integer programming software is not available, the linearprogramming problem can be solved and the solution rounded,making certain that none of the constraints have been violated. Notethat the integer programming solution might also need to be ad-justed—if F is an odd integer, 0.5F will not be an integer and the re-quirement that “half” of the full-time employees go to lunch at 11A.M. and the other half at noon will have to be altered by assigningthe extra employee to the appropriate hour.
1. The least-cost solution requires 29 full-time employees, 9 ofwhom work two hours of overtime per day. In actuality, 18 of thefull-time employees would work overtime on two different daysand 9 would work overtime on one day. Fourteen of the full-timeworkers would take lunch at 11 A.M. and the other 15 would take itat noon. Eleven part-timers would begin at 11 A.M., with 9 of themleaving at 3 P.M. and the other 2 at 4 P.M. Fifteen part-time em-ployees would work from noon until 4 P.M., and 5 would workfrom 2 P.M. until 6 P.M. The resulting cost of 232 hours of straighttime, 18 hours of overtime, and 126 hours of part-time work is$3,476.28 per day.
This solution is not unique—other work assignments can befound that result in this same cost.
2. The same staffing would be used every day. In fact, onewould expect different patterns to present themselves on differentdays; for example, Fridays are usually much busier bank days thanthe others. In addition, the person-hours required for each hour ofthe day are assumed to be deterministic. In a real situation, widefluctuations will be experienced in a stochastic manner.
The optimal solution results in a considerable amount of idletime, partly caused by the restriction that employees can start atthe beginning of an hour and leave at the end. Eliminating this re-striction might yield better results at the risk of increasing theproblem size.