linear programming models in services. learning objectives describe the features of constrained...
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Linear Programming Models in Services
Learning Objectives
• Describe the features of constrained optimization models.
• Formulate LP models for computer solution.• Solve two-variable models using graphics.• Explain the nature of sensitivity analysis.• Solve LP models with Excel Add-in Solver and
interpret the results.• Formulate a goal programming model.
Stereo Warehouse
Let x = number of receivers to stock
y = number of speakers to stock
Maximize 50x + 20y gross profit
Subject to 2x + 4y 400 floor space
100x + 50y 8000 budget
x 60 sales limit
x, y 0
Diet ProblemLakeview Hospital
Let E = units of egg custard base in the shake
C = units of ice cream in the shake
S = units of butterscotch syrup in the shake
Minimize
Subject to cholesterol
fat
protein
calories
015 0 25 010. . .E C S
50 150 90 175E C S
100 50 150C S 70 10E C 200
30 80 200 100E C S E C S, , 0
Shift-Scheduling ProblemGotham City Police Patrol
Let xi = number of officers reporting at period i for i =1, 2, 3, 4, 5, 6
Minimize
x1 + x6 6 period 1
x1 + x2 4 period 2
x2 + x3 14 period 3
x3 + x4 8 period 4
x4 + x5 12 period 5
x5 + x6 16 period 6
x x x x x x1 2 3 4 5 6
Workforce-Planning ProblemLast National Drive-in Bank
Let Tt = number of trainees hired at the beginning of period t
for t = 1,2,3,4,5,6
At = number of tellers available at the beginning of period t
for t = 1,2,3,4,5,6
Minimize
subject to
A1 = 12
for t = 2,3,4,5,6
At , Tt 0 and integer for t = 1,2,3,4,5,6
( )600 3001
6
A Ttt
t
160 80 1500
160 80 1800
160 80 1600
160 80 2000
160 80 1800
160 80 2200
1 1
2 2
3 3
4 4
5 5
6 6
A T
A T
A T
A T
A T
A T
0 9 01 1. A T At t t
Transportation ProblemLease-a-Lemon Car Rental
Let xij = number of cars sent from city i to city j for i = 1,2,3 and j = 1,2,3,4
Minimize 439x11 + 396 x12 + . . . +479x33 + 0x34
subject to x11 + x12 + x13 + x14 = 26
x21 + x22 + x23 + x24 = 43
x31 + x32 + x33 + x34 = 31
x11 + x21 + x31 = 32
x12 + x22 + x32 = 28
x13 + x23 + x33 = 26
x14 + x24 + x34 = 14
xij 0 for all i , j
Graphical SolutionStereo Warehouse
0
50
100
150
200
0 50 100 150 200
Z=2000
Z=3000
Z=3600
Z=3800
A B
C
D
EOptimal solution( x = 60, y = 40)
Model in Standard Form
Let s1 = square feet of floor space not used
s2 = dollars of budget not allocated
s3 = number of receivers that could have been sold
Maximize Z = 50x + 20y
subject to 2x + 4y + s1 = 400 (constraint 1)
100x + 50y + s2 = 8000 (constraint 2)
x + s3 = 60 ( constraint 3)
x, y, s1, s2, s3 0
Stereo WarehouseExtreme-Point Solutions
Extreme Nonbasic Basic Variable Objective-function
point variables variables value value Z
A x, y s1 400 0
s2 8000
s3 60
B s3, y s1 280 3000
s2 2000
x 60
C s3, s2 s1 120 3800
y 40
x 60
D s1, s2 s3 20 3600
y 80
x 40
E s1, x s3 60 2000
y 100
s2 3000
Sensitivity AnalysisObjective-Function Coefficients
0
50
100
150
200
0 50 100 150 200
z = 50x + 20y
x 60 (constraint 3 )
2 4 400x y (constraint 1)
100 50 8000x y (constraint 2)
A B
C
D
Sensitivity AnalysisRight-Hand-Side Ranging
0
50
100
150
200
0 50 100 150 200
x 60 (constraint 3 )
100 50 8000x y (constraint 2)
A B I
C
DH
Goal ProgrammingStereo Warehouse Example
Let x = number of receivers to stock
y = number of speakers to stock
dd
d
d
d
d
d
d
Pk
1
1
2
2
3
3
4
4
= amount by which profit falls short of $99,999= amount by which profit exceeds $99,999= amount by which floor space used falls short of 400 square feet= amount by which floor space used exceeds 400 square feet= amount by which budget falls short of $8000= amount by which budget exceeds $8000= amount by which sales of receivers fall short of 60= amount by which sales of receivers exceed 60= priority level with rank k
Minimizesubject to
Z Pd P d d P d d 1 4 2 1 3 3 1 22 2 2( ) ( )
50 20 99 999
2 4 400
100 50 8000
1 1
2 2
3 3
x y d d
x y d d
x y d d
,
x d d 4 4 60
x y d d d d d d d d, , , , , , , , ,1 1 2 2 3 3 4 4 0
profit goal
floor-space goal
budget goal
sales-limit goal
Topics for Discussion
• How can the validity of LP models be evaluated?
• Interpret the meaning of the opportunity cost for a nonbasic decision variable that did not appear in the LP solution.
• Explain graphically what has happened when a degenerate solution occurs in an LP problem.
• Is LP a special case of goal programming? Explain.
• What are some limitations to the use of LP?