Download presentation

Presentation is loading. Please wait.

Published byKaylin Oakey Modified over 2 years ago

1
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

2
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 neededexpected 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

3
Example of IP: Facility Location Define decision variables ( i = 1, 2, 3, 4): Then the total expected benefit: 9x 1 +5x 2 +6x 3 +4x 4 the total capital needed: 6x 1 +3x 2 +5x 3 +2x 4 Summarizing, the IP model is: max 9x 1 +5x 2 +6x 3 +4x 4 s.t. 6x 1 +3x 2 +5x 3 +2x 4 10 x 1, x 2, x 3, x 4 binary ( i.e., x i {0,1} )

4
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 b i. Goal: Pack the knapsack such that the total benefit is maximized.

5
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)

6
Connection between the problems Note: The version of the facility location problem is a special case of the knapsack problem. Important modeling skill: –Suppose we know how to model Problems A 1,…,A p ; –We need to solve Problem B; –Notice the similarities between Problems A i and B; –Build a model for Problem B, using the model for Problem A i as a prototype.

7
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

8
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:

9
Modeling Technique: Contingent Decisions Back to the facility location problem. Requirement: Can’t build a warehouse unless there is a factory in the city. The corresponding constraints are: x 3 x 1 (LA) x 4 x 2 (SF) Requirement: Can’t select option 3 unless at least one of options 1 and 2 is selected. The constraint:x 3 x 1 + x 2 Requirement: Can’t select option 4 unless at least two of options 1, 2 and 3 are selected. The constraint:2x 4 x 1 + x 2 + x 3

10
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.

11
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.

Similar presentations

OK

BU.520.601 Decision Models Integer_LP1 Integer Optimization Summer 2013.

BU.520.601 Decision Models Integer_LP1 Integer Optimization Summer 2013.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on basic concepts of chemistry Ppt on diode as rectifier circuit Ppt on central nervous system Ppt on engineering marvels in india Ppt on bluetooth hacking downloads Ppt on hindu religion diet Ppt online compressors Ppt on different occupations in nursing Ppt on complex numbers class 11th Ppt on column chromatography uses