OPSM 301 Operations Management

Class 25:Applied LP continued

Koç University

Zeynep Aksinzaksin@ku.edu.tr

A Transportation Problem:Tropicsun

Distances (in miles)CapacitySupply

275,000

400,000

300,000 225,000

600,000

200,000Mt. Dora

1

Eustis

2

Clermont

3

Groves

Ocala

4

Orlando

5

Leesburg

6

Processing Plants

21

50

40

3530

22

55

25

20

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)

Defining the Objective Function

Minimize the total number of bushel-miles.

MIN: 21X14 + 50X15 + 40X16 +

35X24 + 30X25 + 22X26 +

55X34 + 20X35 + 25X36

Defining the Constraints

Capacity constraintsX14 + X24 + X34 <= 200,000 } Ocala

X15 + X25 + X35 <= 600,000 } Orlando

X16 + X26 + X36 <= 225,000 } Leesburg

Supply constraintsX14 + X15 + X16 = 275,000 } Mt. Dora

X24 + X25 + X26 = 400,000 } Eustis

X34 + X35 + X36 = 300,000 } Clermont

Nonnegativity conditionsXij >= 0 for all i and j

Implementing the Model

An Employee Scheduling Problem:Air-Express

Day of Week Workers NeededSunday 18

Monday 27

Tuesday 22

Wednesday 26

Thursday 25

Friday 21

Saturday 19

Shift Days Off Wage1 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

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

Defining the Objective Function

Minimize the total wage expense.

MIN: 680X1 +705X2 +705X3 +705X4 +705X5 +680X6 +655X7

Defining the Constraints

Workers required each day0X1 + 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 conditionsXi >= 0 for all i

Implementing the Model

An Investment Problem:Retirement Planning Services, Inc.

A client wishes to invest $750,000 in the following bonds.

Years toCompany 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

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.

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

Defining the Objective Function

Maximize the total annual investment return.

MAX: .0865X1 + .095X2 + .10X3 + .0875X4 + .0925X5 + .09X6

Defining the Constraints

Total amount is investedX1 + X2 + X3 + X4 + X5 + X6 = 750,000

No more than 25% in any one investmentXi <= 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 conditionsXi >= 0 for all i

Implementing the Model

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

Maximum weekly demand of

Oatmeal Raisin Cookies: 50 dozen

Total weekly operating expense $270

Product Mix Decisions

ProcessingTimes

Mix &Dish

Load Oven& Set

Bake Cool Pack Receive Total

ChocolateChip

8 mins.(6+2/tray)

1 9 5 2/tray 1 26

OatmealRaisin

5 mins.(3+2/tray)

1 14 2 2/tray 1 25

Resource You RM+Oven Oven RM RM

Total time available in week: 20 hrs

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

LP for Optimal Product Mix Selection

xcc: Dozens of chocolate chip cookies sold.

xor: Dozens of oatmeal raisin cookies sold.

Max 2.5 xcc + 3.1 xor

subject to

8 xcc+ 5 xor < 1200

10 xcc + 15 xor < 1200

4 xcc+ 4 xor < 1200

xcc < 100

xor < 50

Technology Constraints

Market Constraints

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 200 350 400 350

Capacity 300 400 450 450

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.

Decision Variables:XFF: Quantity of trucks produced and sold in February XFM: Produced in Feb. sold in MarchXFA: Produced in Feb. sold in April XFY: Produced in Feb. sold in May XMM: March/MarchXMA: 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 )

Constraints:XFF+XFM+XFA+XFY ≤ 300

XFF ≥ 200 XMM+XMA+XMY ≤ 400

XFM+XMM ≥ 350XAA+XAY ≤ 450

XFA+XMA+XAA ≥ 400XYY ≤ 450

XFY+XMY+XAY+XYY ≥ 350 All variables should be ≥ 0

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:

CategoryProduct 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 1.0 0.5

 

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

0,5 321

1 XXX

X

0,2 321

3 XXX

X

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 producedSubject to: Machine working hours constraint (machines have limited capacity)

0,5X1 + 2X2 + 0,75X3 ≤ 40 for machine 1

X1 + X2 + 0,5X3 ≤ 40 for machine 2

Labor Hours Constraint: (we have limited labor hours=100 to be allocated over machines)

[0,5X1 + 2X2 + 0,75X3] + 2 . [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