1. Linear Programming Applications in Services

2. 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.

3. Stereo Warehouse Let x = number of receivers to stock
y = number of speakers to stock
Maximize 50x + 20y gross profit
Subject to 2x + 4y floor space
100x + 50y budget
x sales limit
x, y

4. Diet Problem Lakeview 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

5. Shift-Scheduling Problem Gotham City Police Patrol
Let xi = number of officers reporting at period i for i =1, 2, 3, 4, 5, 6
Minimize
x x period 1
x1 + x period 2
x2 + x period 3
x3 + x period 4
x4 + x period 5
x5 + x period 6

6. Workforce-Planning Problem Last 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
Minimize
subject to
A1 = 12
for t = 2,3,4,5,6
At , Tt and integer for t = 1,2,3,4,5,6

7. Transportation Problem Lease-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 x x x33 + 0x34
subject to x11 + x12 + x13 + x = 26
x21 + x22 + x23 + x = 43
x31 + x32 + x33 + x34 = 31
x x x = 32
x x x = 28
x x x = 26
x x x34 = 14
xij for all i , j

8. Graphical Solution Stereo Warehouse
Z=3800
Z=3600
Z=3000
Z=2000 E
Optimal solution
( x = 60, y = 40) D C A B

9. 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 x y + s = 400 (constraint 1)
100x + 50y s = 8000 (constraint 2)
x s3 = ( constraint 3)
x, y, s1, s2, s

10. Stereo Warehouse Extreme-Point Solutions
Extreme Nonbasic Basic Variable Objective-function
point variables variables value value Z
A x, y s s s
B s3, y s s x
C s3, s s y
D s1, s s y x
E s1, x s y s

11. Sensitivity Analysis Objective-Function Coefficients
z = 50x + 20y
(constraint 3 ) D
(constraint 1)
(constraint 2) C A B

12. Sensitivity Analysis Right-Hand-Side Ranging
(constraint 3 ) D H
(constraint 2) C A B I

13. Goal Programming Stereo Warehouse Example
Let x = number of receivers to stock
y = number of speakers to stock
= 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
Minimize
subject to
profit goal
floor-space goal
budget goal
sales-limit goal

14. 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?