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IP Modeling Techniques: Facility Location and Knapsack Problem, Slides of Discrete Structures and Graph Theory

Aliah UniversityDiscrete Structures and Graph Theory

Information on ip modeling techniques using binary variables, restrictions on number of options, contingent decisions, and variables with k possible values. It includes examples of the facility location problem and knapsack problem, and explains how to model additional requirements and functions with k possible values.

Typology: Slides

2012/2013

Uploaded on 04/27/2013

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IP modeling techniques I
In this handout,
Modeling techniques:
Using binary variables
Restrictions on number of options
Contingent decisions
Variables (functions) with k possible values
Applications:
Facility Location Problem
Knapsack Problem
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Download IP Modeling Techniques: Facility Location and Knapsack Problem and more Slides Discrete Structures and Graph Theory in PDF only on Docsity!

IP modeling techniques I

In this handout,

Modeling techniques:

Using binary variables

Restrictions on number of options

Contingent decisions

Variables (functions) with k possible values

Applications:

Facility Location Problem

Knapsack Problem

Example of IP: Facility Location

  • A company is thinking about building new facilities in

LA 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

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 lb of items.
  • Each item has weight w (^) i lb and benefit bi.
  • Goal: Pack the knapsack such that

the total benefit is maximized.

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.

x (^) i binary ( i = 1, …, n)

0 if not

1 if item is packed x (^) i

i

∑

n

i

bi xi 1 ∑ =

n

i

wi xi 1

∑

n

i

bi xi 1

w x W

n

i

∑ i i ≤ = 1

The Facility Location Problem:

adding new requirements

  • Extra requirement:

build at most one of the two warehouses.

The corresponding constraint is:

x 3 +x 4 ≤ 1

  • Extra requirement:

build at least one of the two factories.

The corresponding constraint is:

x 1 +x 2 ≥ 1

Modeling Technique:

Restrictions on the number of options

  • Suppose in a certain problem, n different

options are considered. For i=1,…,n

  • Restrictions: At least p and at most q of

the options can be chosen.

  • The corresponding constraints are:

0 if not

1 if option is chosen x (^) i

i

x p

n

i

∑ i ≥ = 1

x q

n

i

∑ i ≤ = 1

Modeling Technique:

Variables with k possible values

  • Suppose variable y should take

one of the values d 1 , d 2 , …, d (^) k.

  • How to achieve that in the model?
  • Introduce new decision variables. For i=1,…,k,
  • Then we need the following constraints.

0 otherwise

1 if y takes value d x

i i

1 ( can take only one value) 1

x y

k

i

∑ i =

( should take value di if xi 1) 1

= (^) ∑ =

y d x y

k

i

i i

Modeling Technique:

Functions with k possible values

  • The technique of the previous slide can be extended

to functions.

  • Suppose the linear function f( y )=a 1 y 1 +…+a (^) n y (^) n should

take one of the values d 1 , d 2 , …, d (^) k.

  • Introduce new decision variables. For i=1,…,k,
  • Then we need the following constraints.

0 otherwise

1 if f(y) takes value d x

i i

1 (f(y) can take only one value ) 1

∑^ =

k

i

xi

( f(y) should take value di if xi 1) 1 1

∑ =^ ∑^ = = =

k

i

i i

n

j

aj yj d x Docsity.com