1. OPSM 301 Operations Management
Koç University
OPSM 301 Operations Management
Class 25:
Applied LP continued
Zeynep Aksin

2. A Transportation Problem: Tropicsun
Ocala 4
Orlando 5
Leesburg 6
Processing
Plants
Mt. Dora 1
Eustis 2
Clermont 3
Groves
Distances (in miles)
Supply
Capacity 21
225,000
600,000
200,000
275,000
400,000
300,000 50 40 35 30 22 55 20 25

3. Defining the Decision Variables
Xij = # of bushels shipped from node i to node j
Specifically, the nine decision variables are:
X14 = # of bushels shipped from Mt. Dora (node 1) to Ocala (node 4)
X15 = # of bushels shipped from Mt. Dora (node 1) to Orlando (node 5)
X16 = # of bushels shipped from Mt. Dora (node 1) to Leesburg (node 6) X24 = # of bushels shipped from Eustis (node 2) to Ocala (node 4)
X25 = # of bushels shipped from Eustis (node 2) to Orlando (node 5)
X26 = # of bushels shipped from Eustis (node 2) to Leesburg (node 6)
X34 = # of bushels shipped from Clermont (node 3) to Ocala (node 4)
X35 = # of bushels shipped from Clermont (node 3) to Orlando (node 5)
X36 = # of bushels shipped from Clermont (node 3) to Leesburg (node 6)

4. Defining the Objective Function
Minimize the total number of bushel-miles.
MIN: 21X X X16 +
35X X X26 +
55X X X36

5. Defining the Constraints
Capacity constraints
X14 + X24 + X34 <= 200,000 } Ocala
X15 + X25 + X35 <= 600,000 } Orlando
X16 + X26 + X36 <= 225,000 } Leesburg
Supply constraints
X14 + X15 + X16 = 275,000 } Mt. Dora
X24 + X25 + X26 = 400,000 } Eustis
X34 + X35 + X36 = 300,000 } Clermont
Nonnegativity conditions
Xij >= 0 for all i and j

6. Implementing the Model

7. An Employee Scheduling Problem: Air-Express
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

8. Defining the 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

9. Defining the Objective Function
Minimize the total wage expense.
MIN: 680X1 +705X2 +705X3 +705X4 +705X5 +680X6 +655X7

10. Defining the Constraints
Workers required 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 conditions
Xi >= 0 for all i

11. Implementing the Model

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

13. 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.

14. Defining the 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

15. Defining the 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. Implementing the Model

18. Product Mix Decisions: Kristen Cookies offers 2 products
Sale Price of Chocolate Chip Cookies: $5.00/dozen
Cost of Materials: $2.50/dozen
Sale Price of Oatmeal Raisin Cookies: $5.50/dozen
Cost of Materials: $2.40/dozen
Maximum weekly demand of
Chocolate Chip Cookies: 100 dozen
Oatmeal Raisin Cookies: 50 dozen
Total weekly operating expense $270

19. Product Mix Decisions
Total time available in week: 20 hrs

20. Margin per dozen Chocolate Chip cookies = $2.50
Product Mix Decisions
Margin per dozen Chocolate Chip cookies = $2.50
Margin per dozen Oatmeal Raisin cookies = $3.10
Margin per oven minute from Chocolate Chip cookies = $2.50 / 10 = $ 0.250
Margin per oven minute from Oatmeal Raisin cookies = $3.10 / 15 = $ 0.207

21. LP for Optimal Product Mix Selection
xcc: Dozens of chocolate chip cookies sold.
xor: Dozens of oatmeal raisin cookies sold.
Max 2.5 xcc xor
subject to
8 xcc xor < 1200
10 xcc xor < 1200
4 xcc xor < 1200
xcc < 100
xor <
Technology Constraints
Market Constraints

22. over the next four months:
Anadolu Truck A. Ş. operates a small facility that assembles street cleaning trucks.
The company has the following firm orders for its product and production capacity
over the next four months:
February
March
April
May
Demand
200
350
400
Capacity
300
450
Many of the parts used in the assembly of the trucks are imported, and the company estimates that
product costs will change (due to the fluctuations in the exchange rates) over the next four months
as follows:
February
March
April
May
Cost per Truck (TL)
15 Billion
14 Billion
16 Billion
17 Billion
The cost of holding one completed truck in the inventory for one month is estimated to be 1 Billion TL.
Write a Linear Program to minimize the total cost of the company.

23. Decision Variables:
XFF: Quantity of trucks produced and sold in February
XFM: Produced in Feb. sold in March
XFA: Produced in Feb. sold in April XFY: Produced in Feb. sold in May
XMM: March/March
XMA: March/April XMY: March/May XAA: April/April XAY: April/May
XYY: May/May
Objective Function:
Minimize Total Cost =
($1*XFM+$2*XFA+$3*XFY+$15(XFF+XFM+XXF+XFY)+
$1*XMA+$2*XMY+$14(XMM+ XMA+XMY) +
$1*XAY+$16 (XAA+XAY)+$17*XYY )

24. All variables should be ≥ 0
Constraints:
XFF+XFM+XFA+XFY ≤ 300
XFF ≥ 200
XMM+XMA+XMY ≤ 400
XFM+XMM ≥ 350
XAA+XAY ≤ 450
XFA+XMA+XAA ≥ 400
XYY ≤ 450
XFY+XMY+XAY+XYY ≥ 350
All variables should be ≥ 0

25. Better Products, Inc., manufactures three products on two machines.
In a typical week, 40 hours are available on each machine.
The profit contribution and production time in hours per unit are as follows:
Category
Product 1
Product 2
Product 3
Profit/unit
$30
$50
$20
Machine 1 time/unit
0.5
2.0
0.75
Machine 2 time/unit
1.0
Two operators are required for machine 1; thus, 2 hours of labor
must be scheduled for each hour of machine 1 time. Only one operator
is required for machine 2 time. A maximum of 100 labor-hours is
available for assignment to the machines during the coming week.
Other production requirements are that product 1 cannot account for
more than 50% of the units produced and that product 3 must account
for at least 20% of the units produced. Formulate a linear programming
model that can be used to determine the number of units of each
product to produce to maximize the total profit contribution

26. Unit Production Requirements:
X1: Number of Product1 to be produced.
X2: Number of Product2 to be produced.
X3: Number of Product3 to be produced.
Objective; maximize profits: (30. X1+50. X2+20. X3)
Simply unit profits x number of units to be produced
Subject to: Machine working hours constraint (machines have limited capacity)
0,5X1 + 2X2 + 0,75X3 ≤ for machine 1
X1 + X2 + 0,5X3 ≤ for machine 2
Labor Hours Constraint: (we have limited labor hours=100 to be allocated over machines)
[0,5X1 + 2X2 + 0,75X3] [X1 + X2 + 0,5X3] ≤ 100
labor time req. for machine1 + 2x labor time req. for machine2 ≤ Total labor hrs available
Unit Production Requirements:
And,
X1, X2, X3 are non-negative