Ip Extended Intro

8/12/2019 Ip Extended Intro

1/18

Integer Programming

Integer programming is a solution method for manydiscrete optimization problems

Programming = Planning in this context

Origins go back to military logistics in WWII (1940s).

In a survey of Fortune 500 firms, 85% of thoseresponding said that they had used linear or integerprogramming.

Why is it so popular? Many different real-life situations can be modeled as integer

programs (IPs).

There are efficient algorithms to solve IPs.


2/18

Standard form of integer program (IP)

maximize c1x1+c2x2++cnxn (objective function)

subject to

a11x1+a12x2++a1nxnb1 (functional constraints)

a21x1+a22x2++a2nxnb2.

am1x1+am2x2++amnxnbm

x1, x2, , xn

Z+ (set constraints)

Note: Can also have equalityor constraint

in non-standard form.


3/18

Standard form of integer program (IP)

In vector form:

maximize cx (objective function)subject to Axb (functional constraints)

x (set constraints)

Inputfor IP: 1n vector c, mn matrice A , m1 vector b.

Outputof IP: n1 integer vector x .

Note: More often, we will considermixed integer programs (MIP),

that is, some variables are integer,

the others are continuous.

n

Z

E


4/18

Examp e o Integer Program(Production Planning-Furniture Manufacturer)

Technological data:

Production of 1 table requires 5ft pine, 2ft oak, 3hrs labor1 chair requires 1ft pine, 3ft oak, 2hrs labor

1 desk requires 9ft pine, 4ft oak, 5hrs labor

Capacities for 1 week: 1500ft pine, 1000ft oak,

20employees (each works 40hrs).

Market data:

Goal:Find a production schedule for 1 week

that will maximize the profit.

profit demand

table $12/unit 40

chair $5/unit 130

desk $15/unit 30

P P F


5/18

Pro uction P anning-Furniture Manu acturer:modeling the problem as integer program

The goal can be achievedby making appropriate decisions.

First define decision variables:

Let xtbe the number of tables to be produced;

xcbe the number of chairs to be produced;

xdbe the number of desks to be produced.

(Always define decision variables properly!)

P P F M


6/18

Pro uction P anning-Furniture Manu acturer:modeling the problem as integer program

Objective is to maximize profit:

max 12xt + 5xc+ 15xd Functional Constraints

capacity constraints:

pine: 5xt + 1xc+ 9xd 1500oak: 2xt + 3xc+ 4xd 1000

labor: 3xt + 2xc+ 5xd 800

market demand constraints:

tables: xt 40chairs: xc 130

desks: xd 30

Set Constraints

xt , xc, xdZ+


7/18

Solutions to integer programs

A solutionis an assignment of values to variables.

A feasible solutionis an assignment of values tovariables such that all the constraints are satisfied.

The objective function valueof a solution is obtained

by evaluating the objective function at the given point. An optimal solution(assuming maximization) is one

whose objective function value is greater than or equal

to that of all other feasible solutions. There are efficient algorithms for finding the optimal

solutions of an integer program.


8/18

Next: IP modeling techniques

Modeling techniques:

Using binary variables

Restrictions on number of options

Contingent decisions

Variables with k possible values

Applications:

Facility Location Problem

Knapsack Problem


9/18

Example of IP: Facility Location

A company is thinking about building new facilities inLA and SF.

Relevant data:

Total capital available for investment: $10M

Question:Which facilities should be built

to maximize the total profit?

capital needed expected profit

1. factory in LA $6M $9M

2. factory in SF $3M $5M

3. warehouse in LA $5M $6M

4. warehouse in SF $2M $4M


10/18

Example of IP: Facility Location

Define decision variables (i= 1, 2, 3, 4):

Then the total expected benefit: 9x1+5x2+6x3+4x4the total capital needed: 6x1+3x2+5x3+2x4

Summarizing, the IP model is:

max9x1+5x

2+6x

3+4x

4s.t.6x1+3x2+5x3+2x4 10

x1, x2, x3, x4 binary ( i.e., xi{0,1} )

notif0

builtisfacilityif1x i

i


11/18

Knapsack problem

Any IP, which has only one constraint,

is referred to as a knapsack problem.

n items to be packed in a knapsack.

The knapsack can hold up to W lbof items.

Each item has weight wilband benefit bi.

Goal:Pack the knapsack such that

the total benefit is maximized.


12/18

IP model for Knapsack problem Define decision variables (i= 1, , n):

Then the total benefit:

the total weight:

Summarizing, the IP model is:

max

s.t.

xibinary (i= 1, , n)

notif0packedisitemif1x i

i

n

i

iixb1

n

i ii

xw1

n

iiixb

1

Wxw

n

i

ii 1


13/18

Connection between the problems

Note: The version of the facility locationproblem is a special case of the knapsackproblem.

Important modeling skill: Suppose you know how to model Problem A1,,Ap;

You need to solve Problem B;

Notice the similarities between Problems Aiand B; Build a model for Problem B, using the model for

Problem Aias a prototype.


14/18

The Facility Location Problem:adding new requirements

Extra requirement:

build at most oneof the two warehouses.

The corresponding constraint is:

x3 +x4 1

Extra requirement:

build at least oneof the two factories.

The corresponding constraint is:

x1 +x2 1


15/18

Modeling Technique:Restrictions on the number of options

Suppose in a certain problem, n differentoptions are considered. For i=1,,n

Restrictions:At leastpand at most qof

the options can be chosen. The corresponding constraints are:

notif0

chosenisoptionif1x i

i

pxn

i

i 1

qxn

i

i 1


16/18

Modeling Technique: Contingent Decisions

Back to the facility location problem.

Requirement:Cant build a warehouseunlessthere is a factory in the city.

The corresponding constraints are:

x3 x1 (LA) x4 x2 (SF) Requirement: Cant select option 3 unless

at least one of options 1 and 2 is selected.

The constraint: x3x1 + x2 Requirement: Cant select option 4 unless

at least two of options 1, 2 and 3 are selected.

The constraint: 2x4

x1 + x2+x3

M d li T h i


17/18

Modeling Technique:Variables with k possible values

Suppose variable y should take

one of the values d1, d2, , dk.

How to achieve that in the model?

Introduce new decision variables. For i=1,,k,

Then we need the following constraints.

otherwise0

dvaluey takesif1x

i

i

value)oneonlycan take(11

yxk

i

i

1)xifdvaluetakeshould( ii1

yxdyk

i

ii


18/18

More on Integer Programming and otherdiscrete optimization problems andtechniques:

Math 4620

Linear and Nonlinear Programming

Math 4630Discrete Modeling and Optimization

Documents

Ip Extended Intro