1

1

Course Objectives

Engineers and managers are constantly attempting to

optimize, particularly in the design, analysis, and operation

of complex systems. The course seeks to:

to present a range of applications of linear programming and

network optimization problem in many scientific domains and

industrial setting;

provide an in-depth understanding of the underlying theory of

linear programming and network flows;

to present a range of algorithms available to solve such

problems;

to give exposure to the diversity of applications of these

problems in engineering and management;

to help each student develop his or her intuition about algorithm

design, development and analysis.

2

Course Topics

Linear Programming Formulating linear programs Applications of linear programming Linear algebra Simplex algorithm Duality theory Sensitivity analysis Integer programming: Applications and algorithms

Network Optimization

Shortest path problem Minimum spanning tree problem Maximum flow problem Minimum cost flow problem

Text Books: M.S. Bazaraa, J. J. Jarvis, and H.D. Sherali, “Linear Programming

and Network Flows : Second Edition," ISBN: 0-471-63681-9 Winston, Wayne L., “Operations Research : Applications and

Algoritms”, Fourth Edition, Thomson, 2004

2

3

Grading and Regrading

Grading Tugas: 20%

Quiz: 30%

Midterm Exam: 25%

Final Exam: 25%

Regrading

If I have made a mistake in grading something, I will be happy

to correct it.

In order to receive a re-grade, you must contact/email me

within 48 hours of my handing back the test.

If a test is submitted for regrading, I have the right to regrade

the entire test. So it is possible to lose additional points.

Therefore, it is strongly recommended that you do not ask for

regrading unless you have substantial reason to believe that I

made a mistake when originally grading the test.

4

OR Definition

Scientific approach to solve decision

making problem for finding the best system

design and operation, with limited resources

(can also be defined as Management

Science)

3

5

Characteristics of OR

Decision making is the main focus

Economy si the effective criteria.

Depends on the formal mathematical model

Depends on computer

6

The steps of OR study

1. Model formulation:

Objective function (Max atau Min)

Decision variables (controllable)

Parameters (uncontrollable)

Constraints

2. Build mathematical model

3. Do analytical

4. Validity test of the model and solution

5. Conduct sensitivitis analysis

6. Implementation

4

TI091306/OR1/sew/2012/#1

The steps of OR study

SYSTEM

Management

Problems Mathematic

Model Solu-

tion ?

C

O

N

T

R

O

L

IMPLEMENTATION

Observation

Data Collection

Y

N

7

8

INTRODUCTION TO LINEAR PROGRAMMING

CONTENTS

Introduction to Linear Programming

Applications of Linear Programming

Reference: Chapter 1 in BJS book.

5

9

A Typical Linear Programming Problem

Linear Programming Formulation: Minimize c1x1 + c2x2 + c3x3 + …. + cnxn

subject to a11x1 + a12x2 + a13x3 + …. + a1nxn b1

a21x1 + a22x2 + a23x3 + …. + a2nxn b2 : : am1x1 + am2x2 + am3x3 + …. + amnxn bm x1, x2, x3 , …., xn 0 or,

Minimize j=1, n cjxj

subject to

j=1, n aijxj +xn+i = bi for all i = 1, …, m

xj 0 for all j =1, …, n+m Note:- xn+i is the slack variable corresponding to ith equation.

10

Matrix Notation

Minimize cx

subject to

Ax = b

x 0

where

a11 a12 ….. a1n 1

a21 a22 ….. a2n 1

A = ::

::

::

::

am1 am2 amn 1

b1

b2

b = ::

bm

x1

x2

::

x = xn

Xn+1

::

Xn+m

c1

c2

::

c = cn

0

::

0

6

11

Features of a Linear Programming (LP) Problem

Decision Variables

We minimize (or maximize) a linear function of decision

variables, called objective function.

The decision variables must satisfy a set of constraints.

Decision variables have sign restrictions.

12

An Example of a LP

Giapetto’s woodcarving manufactures two types of wooden toys: soldiers and trains

Constraints:

100 finishing hour per week available 80 carpentry hours per week available produce no more than 40 soldiers per week

Objective: maximize profit

Soldier Train

Selling Price $27 $21

Raw Material

required

$10 $9

Variable Cost $14 $10

Finishing Labor

required

2 hrs 1 hr

Carpenting labor

required

1 hr 1 hr

7

13

An Example of a LP (cont.)

Linear Programming formulation:

Maximize z = 3x1 + 2x2 (Obj. Func.)

subject to

2x1 + x2 100 (Finishing constraint)

x1 + x2 80 (Carpentry constraint)

x1 40 (Bound on soldiers)

x1 0 (Sign restriction)

x2 0 (Sign restriction)

14

Assumptions of Linear Programming

Proportionality Assumption

Contribution of a variable is proportional to its value.

Additivity Assumptions

Contributions of variables are independent.

Divisibility Assumption

Decision variables can take fractional values.

Certainty Assumption

Each parameter is known with certainty.

8

15

Linear Programming Modeling and Examples

Stages of an application:

Problem formulation

Mathematical model

Deriving a solution

Model testing and analysis

Implementation

16

Capital Budgeting Problem

Five different investment opportunities are available for

investment.

Fraction of investments can be bought and cannot more

than one

Money available for investment:

Time 0: $40 million

Time 1: $20 million

Maximize the NPV of all investments.

Inv.1 Inv. 2 Inv. 3 Inv. 4 Inv. 5

Time 0 cash Outflow

$11 $5 $5 $5 $29

Time 1 cash Outflow

$3 $6 $5 $1 $3

NPV $17 $16 $16 $14 $39

9

17

Solution: Capital Budgeting Problem

Decision Variables:

xi: fraction of investment i purchased

Formulation:

Maximize z = 13x1 + 16x2 + 16x3 + 14x4 + 39x5

subject to

11x1 + 53x2 + 5x3 + 5x4 + 29x5 40

3x1 + 6x2 + 5x3 + x4 + 34x5 20

x1 1

x2 1

x3 1

x4 1

x5 1

x1, x2, x3, x4, x5 0

18

Transportation Problem

The Brazilian coffee company processes coffee beans into

coffee at m plants. The production capacity at plant i is ai.

The coffee is shipped every week to n warehouses in major

cities for retail, distribution, and exporting. The demand at

warehouse j is bj.

The unit shipping cost from plant i to warehouse j is cij.

It is desired to find the production-shipping pattern xij from

plant i to warehouse j, i = 1, .. , m, j = 1, …, n, that minimizes

the overall shipping cost.

10

19

Solution: Transportation Problem

Decision Variables: xij: amount shipped from plant i to warehouse j Formulation: Minimize z =

subject to

= ai, i = 1, … , m bj, j = 1, … , n xij 0, i = 1, … , m, j = 1, … , n

m n

ij iji=1j=1

c x

n

ijj=1

x

m

iji=1

x

20

Static Workforce Scheduling

Number of full time employees on different days of the week

are given below.

Each employee must work five consecutive days and then

receive two days off.

The schedule must meet the requirements by minimizing

the total number of full time employees.

Day 1 = Monday 17

Day 2 = Tues. 13

Day 3 = Wedn. 15

Day 4 = Thurs. 19

Day 5 = Friday 14

Day 6 = Satur. 16

Day 7 = Sunday 11

11

21

Solution: Static Workforce Scheduling

LP Formulation:

Min. z = x1+ x2 + x3 + x4 + x5 + x6 + x7

subject to

x1 + x4 + x5 + x6 + x7 17

x1+ x2 + x5 + x6 + x7 13

x1+ x2 + x3 + x6 + x7 15

x1+ x2 + x3 + x4 + x7 19

x1+ x2 + x3 + x4 + x5 14

x2 + x3 + x4 + x5 + x6 16

x3 + x4 + x5 + x6 + x7 11

x1, x2, x3, x4, x5, x6, x7 0

22

Multi-Period Financial Models

Determine investment strategy for the next three years

Money available for investment at time 0 = $100,000

Investments available : A, B, C, D & E

No more than $75,000 in one invest

Uninvested cash earns 8% interest

Cash flow of these investments:

0 1 2 3A -1 + 0.5 + 1 0

B 0 -1 + 0.5 + 1

C -1 + 1.2 0 0

D -1 0 0 + 1.9

E 0 0 -1 + 1.5

12

23

Multi-Period Workforce Scheduling

Requirement of skilled repair time (in hours) is given below.

At the beginning of the period, 50 skilled technicians are

available.

Each technician is paid $2,000 and works up to 160 hrs per

month.

Each month 5% of the technicians leave.

A new technician needs one month of training, is paid

$1,000 per month, and requires 50 hours of supervision of a

trained technician.

Meet the service requirement at minimum cost.

Month 1 Month 2 Month 3 Month 4 Month 5

6,000 7,000 8,000 9,500 11,000

24

Solution: Multiperiod Financial Model

Decision Variables:

A, B, C, D, E : Dollars invested in the investments A, B, C, D, and E

St: Dollars invested in money market fund at time t (t = 0, 1, 2)

Formulation:

Maximize z = B + 1.9D + 1.5E + 1.08S2

subject to

A + C + D + S0 = 100,000

0.5A + 1.2C + 1.08S0 = B + S1

A + 0.5B + 1.08S1 = E + S2

A 75,000

B 75,000

C 75,000

D 75,000

E 75,000

A, B, C, D, E, S0, S1, S2 0

13

25

Solution: Multiperiod Workforce Scheduling

Decision Variables:

xt: number of technicians trained in period t

yt: number of experienced technicians in period t

Formulation:

Minimize z = 1000(x1 + x2 + x3 + x4 + x5) + 2000(y1 + y2 + y3 + y4 + y5)

subject to

160y1 - 50 x1 6000 y1 = 50

160y2 - 50 x2 7000 0.95y1 + x1 = y2

160y3 - 50 x3 8000 0.95y2 + x2 = y3

160y4 - 50 x4 9500 0.95y3 + x3 = y4

160y5 - 50 x5 11000 0.95y4 + x4 = y5

xt, yt 0, t = 1, 2, 3, … , 5

26

Cutting Stock Problem (contd.)

Formulation:

Minimize i=1,n xi

subject to

i=1,n aij xi bi i = 1, …, m

xi 0 j = 1, …, n

xi integer j = 1, …, n

14

27

Feasible Region

Feasible Region: Set

of all points satisfying

all the constraints

and all the sign

restrictions

Example:

Max. z = 3x1 + 2x2

subject to

2x1 + x2 100

x1 + x2 80

x1 40

x1 0

x2 0

28

Example 1

Maximize z = 50x1 + 100x2

subject to

7x1 + 2x2 28

2x1 + 12x2 24

x1, x2 0

Feasible region in this example is unbounded.

15

29

Example 2

Maximize z = 3x1+ 2x2

subject to

1/40x1 + 1/60x2 1

1/50x1 + 1/50x2 1

x1 30

x2 20

x1, x2 0

This linear program does not have any feasible solutions.

30

Example 3

Max. z = 3x1 + 2x2

subject to 1/40 x1 + 1/60x2 1 1/50 x1 + 1/50x2 1 x1, x2 0

This linear program has multiple or alternative optimal

solutions.

16

31

Cutting Stock Problem

A manufacturer of metal sheets produces rolls of standard

fixed width w and of standard length l.

A large order is placed by a customer who needs sheets of

width w and varying lengths. He needs bi sheets of length

li, i = 1, …, m.

The manufacturer would like to cut standard rolls in such a

way as to satisfy the order and to minimize the waste.

Since scrap pieces are useless to the manufacturer, the

objective is to minimize the number of rolls needed to

satisfy the order.

32

Cutting Stock Problem (contd.)

Given a standard sheet of length l, there are many ways of

cutting it. Each such way is called a cutting pattern.

The jth cutting pattern is characterized by the column

vector aj, where the ith component, namely, aij, is a

nonnegative integer denoting the number of sheets of

length li in the jth pattern.

Note that the vector aj represents a cutting pattern if and

only if i=1,n aijli l and each aij is a nonnegative number.