Tran Van Hoai 1

Applications of LP/ILP Models

Tran Van HoaiFaculty of Computer Science & Engineering

HCMC University of Technology

2010-2011

Tran Van Hoai 2

Evolution of LP/ILP• Millions USD saved from applying LP/ILP to

business/government• Motivated by– Simplex– Digital computers

• Applied to

2010-2011

Aircraft fleet assignment

Health care Fire protection Diary production

Telecom network expansion

Bank portfolio selection

Defense/aerospace contracting

Military deployment

Air pollution control

Agriculture Land use planning

Tran Van Hoai 3

Building good LP/ILP models

• Familiarity– Limited resources– Overall (tradeoff) objective– Different perspectives

• Simplification– Models always simplify real-life, but which is

simplified is important• Clarity– Model must be clear

2010-2011

What constitutes the proper simplification is subject to individual judgment and experience(George Dantzig)

Tran Van Hoai 4

Summation variables/constraints

• Introduce new variables to be easier to understand/debug– Summation of variables/constraints

2010-2011

Production of 3 TV models• resource: 7000 pounds plastic

2 pounds/TV1, 3 pounds/TV2, 4 pounds/TV3• profit:

$23/TV1, $34/TV2, $45/TV3• management constraint: not any TV model exceed 40% total production

Tran Van Hoai 5

First modelMAX 23X1 + 34X2 + 45X3

S.T. 2X1 + 3X2 + 4X3 ≤ 7000

X1 ≤ .4(X1+X2+X3)

X2 ≤ .4(X1+X2+X3)

X3 ≤ .4(X1+X2+X3)

X1,X2,X3 ≥ 0

2010-2011

MAX 23X1 + 34X2 + 45X3

S.T. 2X1 + 3X2 + 4X3 ≤ 7000

.6X1 - .4X2 - .4X3 ≤ 0

-.4X1 + .6X2 - .4X3 ≤ 0

-.4X1 - .4X2 + .6X3 ≤ 0

X1,X2,X3 ≥ 0

No meaning as natural input(especially on spreadsheet)

Not management

constraint anymore

Tran Van Hoai 6

• Define summation variableX4 = total production of TVs

• Add summation constraintX1 + X2 + X3 - X4 = 0

MAX 23X1 + 34X2 + 45X3

S.T. 2X1 + 3X2 + 4X3 ≤ 7000X1 + X2 + X3 - X4 = 0X1 - .4X4 ≤ 0

X2 - .4X4 ≤ 0X3 - .4X4 ≤ 0

X1,X2,X3 ≥ 0

Revised model

2010-2011

Summation variable

Summation constraint

Clarity(although more variables/constraints)

Tran Van Hoai 7

Applications of LP/ILP

• More realistic example• Reduced version in different practical

applications– Portfolio model

2010-2011

Tran Van Hoai 8

Financial portfolio model

• Consider return projections of investment– Measure of risk,

volatility, liquidity, short/long term

• Highly nonlinear in nature, but we consider a linear case

2010-2011

Tran Van Hoai 9

Jones investment service(advise clients on investment)

2010-2011

Potential investment Expected return

John’s rating

Liquidity analysis

Risk factor

Savings account 4.0% A Immediate 0

Certificate of deposit 5.2% A 5-year 0

Atlantic Lighting 7.1% B+ Immediate 25

Arkansas REIT 10.0% B Immediate 30

Bedrock insurance annuity 8.2% A 1-year 20

Nocal mining bond 6.5% B+ 1-year 15

Minicomp systems 20.0% A Immediate 65

Antony hotels 12.5% C Immediate 40

Problem summary• Determine amount to be placed in each investment• Minimize total risk• Invest all $100,000• Meet the goals developed with client

annual return at least 7% at least 50% in A-rated investments at least 40% in immediately liquid investments no more $30,000 in savings and deposit

Tran Van Hoai 10

LP modelMIN 25X1 + 30X4 + 20X5 + 15X6 + 65X7 + 40X8

S.T. X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 = 100000

.04X1 + .052X2 + .71X3 + .1X4 + .082X5 + .065X6 + .2X7 + .125X8 ≥ 7500

X1 + X2 + X5 + X7 ≥ 50000

X1 + X3 + X4 + X7 + X8 ≥ 40000

X1 + X2 ≤ 30000

All Xi ≥ 0

2010-2011

Tran Van Hoai 11

Analysis (1)

• Binding constraints

• What exceed minimum requirements

• Investment

2010-2011

Expected annual return $7500Liquid investment $40000Savings & certificate deposit $30000

A-rated investment $50000 $77333

Savings $17333 Arkansas $22666Certificate deposit

$12666 Bedrock insurance

$47333

Tran Van Hoai 12

Analysis (2)• Reduced costs: in order to be included, risk

factor must be lowered

• Optimality range

• Shadow price: risk increased by

2010-2011

Atlantic 4.67 Minicomp 1.67Nocal 0.67 Antony 1.67

Bedrock 19.5 (20-0.5) -> 20.43 (20+0.43)Savings 0 -> 1.17

$1 extra to $100000 -7.33$1 extra liquid investments 4