1. OPSM301 Spring 2012 Class11: LP Model Examples
Evrim Didem Güneş

2. Announcements
Grades are announced in KUAIS- you can check your quizzes in TA office hours
Midterm on March 30, Friday, 18:30
Today:More examples for LP formulations
Product mix problem
Employee Scheduling Problem
Investment Problem
Transportation Problem
Next time (Thursday): Class in SOS180

3. Remember from last time
Product-mix problems
Several examples discussed:Hockey sticks vs chess sets, soldiers vs. Trains, sweet vs dill pickles.
Common Problem definition (product-mix problem ):
Given a set of products, how many of each should we produce to maximize profit subject to capacity availability constraints and other constraints.

4. Example 1:Product Mix ProblemQ
Sales price: 100 $/unit
Max Demand:50 units/week
Sales price:90 $/unit
Max demand:100 units/week D
10 min/un D
5 min/un
Products P and Q are produced using the given process routing.
4 machines are used:A,B,C,D. (available for 2400 min/week)
The price and raw material costs are given.
Problem:
Formulate an LP to find the product mix that maximizes weekly profit.
i.e. How many of each product should we produce given the capacity and demand constraints?
What is the bottleneck of this process?
Purchase
Part 5$/un C
10 min/un C
5 min/un B
15 min/un A
15 min/un. B
15 min/un. A
10 min/un.
RM1
20$/un
RM2
20$/un
RM3
20$/un
Source: Paul Jensen

5. LP Formulation Decision variables:
P:Amount of product P to produce per week
Q:Amount of product Q to produce per week
Objective Function: Maximize Profit
Max 45P+60 Q
Constraints:
Machine hours used should be less than or equal to 2400 minutes:
A: 15 P + 10 Q <= 2400
B: 15 P + 30 Q <= 2400
C: 15 P Q <= 2400
D: 10 P + 5 Q <= 2400
Production should not exceed demand:
P<=100
Q<=50
Non-negativity
P>=0,
Q>=0

6. Solver Solution Maximum possible value Machine B is used atP Q
changing cells
100 30
objective Coefficients 45 60
Profit
6300
Constraint Coefficients:
L.H.S. Value
R.H.S.
Machine A 15 10
1800
<=
2400
Machine B
Machine C 5
1650
Machine D
1150
Demand P 1
Demand Q 50
Realized amount
Maximum possible value
Machine B is used at
Full capacity
This constraint is “binding”
Another binding constraint

7. Some Definitions
Solution: Any chosen values for the decision variables
Feasible Set: Set of solutions that satisfiy all the constraints
Optimal Solution: The solution(s) that gives the best (maximum or minimum) value for the objective function

8. Binding Constraint and Shadow Price
Binding Constraint: A constraint that is satisfied as equality at a given solution
The shadow price (for a constraint): the amount that the objective function value changes per unit change in the constraint’s right hand side.
Example: shadow price of Machine B is the change in the optimal profit by increasing the availability of Machine B.
If a constraint’s shadow price is positive than the constraint has to be binding.
If the constraint is not binding, the shadow price has to be zero. (If we did not use all the availabilty capacity, there’s no value in increasing the resource capacity)

9. Example 2: An Employee Scheduling Problem: Air-Express
Each worker should work for 5 consecutive days. There are 7 shifts possible,
defined according to the first day of work.
The objective is to find the minimum costly staffing plan.
Day of Week Workers Needed
Sunday 18
Monday 27
Tuesday 22
Wednesday 26
Thursday 25
Friday 21
Saturday 19
Shift Days Off Wage
1 Sun & Mon $680
2 Mon & Tue $705
3 Tue & Wed $705
4 Wed & Thr $705
5 Thr & Fri $705
6 Fri & Sat $680
7 Sat & Sun $655

10. LP formulation Minimize the total wage expense. Decision Variables
X1 = the number of workers assigned to shift 1
X2 = the number of workers assigned to shift 2
X3 = the number of workers assigned to shift 3
X4 = the number of workers assigned to shift 4
X5 = the number of workers assigned to shift 5
X6 = the number of workers assigned to shift 6
X7 = the number of workers assigned to shift 7
Objective Function:
Minimize the total wage expense.
MIN: 680X1 +705X2 +705X3 +705X4 +705X5 +680X6 +655X7

11. Defining the Constraints
Shift Days Off Wage
1 Sun & Mon $680
2 Mon & Tue $705
3 Tue & Wed $705
4 Wed & Thr $705
5 Thr & Fri $705
6 Fri & Sat $680
7 Sat & Sun $655
On Sunday, shifts 1 and 7 are off
On Tuesday shifts 2 and 3 are off
On Wed shifts 3 and 4 are off
...
Enough workers should be assigned for each day
0X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 + 0X7 >= 18 } Sunday
0X1 + 0X2 + 1X3 + 1X4 + 1X5 + 1X6 + 1X7 >= 27 } Monday
1X1 + 0X2 + 0X3 + 1X4 + 1X5 + 1X6 + 1X7 >= 22 }Tuesday
1X1 + 1X2 + 0X3 + 0X4 + 1X5 + 1X6 + 1X7 >= 26 } Weds.
1X1 + 1X2 + 1X3 + 0X4 + 0X5 + 1X6 + 1X7 >= 25 } Thurs.
1X1 + 1X2 + 1X3 + 1X4 + 0X5 + 0X6 + 1X7 >= 21 } Friday
1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 0X6 + 0X7 >= 19 } Saturday
Nonnegativity constraints
Xi >= 0 for all i

12. Solver Solution X1 X2 X3 X4 X5 X6 X7 chaning cells 5 0.333333 6.333333X1 X2 X3 X4 X5 X6 X7
chaning cells 5 4
coefficients
680
705
655
Cost
Constraints
Sunday 1 18
>=
Monday 27
Tueday 22
Wednesday 26
Thursday 25
Friday 21
Saturday 19

13. Example 3: An Investment Problem: Retirement Planning Services, Inc.
A client wishes to invest $750,000 in the following bonds.
Years to
Company Return Maturity Rating
1-Acme Chemical 8.65% 11 1-Excellent
2-DynaStar 9.50% 10 3-Good
3-Eagle Vision 10.00% 6 4-Fair
4-Micro Modeling 8.75% 10 1-Excellent
5-OptiPro 9.25% 7 3-Good
6-Sabre Systems 9.00% 13 2-Very Good

14. Investment Restrictions
No more than 25% can be invested in any single company.
At least 50% should be invested in long-term bonds (maturing in 10+ years).
No more than 35% can be invested in DynaStar, Eagle Vision, and OptiPro.
We would like to find the optimal investment portfolio to maximize return

15. LP Formulation Decision Variables
X1 = amount of money to invest in Acme Chemical
X2 = amount of money to invest in DynaStar
X3 = amount of money to invest in Eagle Vision
X4 = amount of money to invest in MicroModeling
X5 = amount of money to invest in OptiPro
X6 = amount of money to invest in Sabre Systems
Objective Function:
Maximize the total annual investment return.
MAX: X X X X X X6

16. Defining the Constraints
Total amount is invested
X1 + X2 + X3 + X4 + X5 + X6 = 750,000
No more than 25% in any one investment
Xi <= 187,500, for all i
50% long term investment restriction.
X1 + X2 + X4 + X6 >= 375,000
35% Restriction on DynaStar, Eagle Vision, and OptiPro.
X2 + X3 + X5 <= 262,500
Nonnegativity conditions
Xi >= 0 for all i

17. Solver Solution X1 X2 X3 X4 X5 X6 changing cells 112500 75000 187500X1 X2 X3 X4 X5 X6
changing cells
112500
75000
187500
coefficients
0,0865
0,095
0,1
0,0875
0,0925
0,09
Return
68887,5
constraints 1
750000 =
562500
>=
375000
262500
<=

18. Example 4: Transportation Problem U.S. Pharmaceuticals Example
From/To
Columbus
St. Louis
Denver
Los Angeles
Factory Supply
Indianapolis 25 35 36 60 15
Phoenix 55 30 6
New York 40 50 80 90 14
Atlanta 66 75 11
Requirements 10 12 9 Si Dj
Define Xij: # units transported from location i to location j
Cij: Unit cost of transportation between i and j
Objective:
See the Excel file in the
Course share folder
For the solution
Constraints: Supply constraints:
Demand Constraints

19. Next time Class on Thursday: We will meet in SOS180 (Computer Lab)
Will discuss Excel Solver solutions for the example problems
Check the coursepack for more examples