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15.053 Tuesday, May 7

• Integer Programming Formulations

Handouts: Lecture Notes

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Quick Summary of DP

• DP often works for decision making over time

• Objectives of the past two lectures – introduce DP recursions – students should learn ˙how to carry out a DP recursion, given the

optimal value function ˙to understand DP recursions when they se

e them• Note: there is a graduate level subject 6.231

dedicated to

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Integer Programming Again

• Representing piecewise linear functions

• A product design & marketing problem

• A configuration problem

• A vehicle routing problem

• Note: about 50% of the final exam will be a mixture of LP and IP formulations.

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The λ-method (same as in NLP lecture)

subject to adjacency conditions.

• What are the adjacency conditions and how can one express them using 0-1 variables?

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The adjacency conditions

• At most two of the λ variables are nonzero.(How do we express this?)

• If i and j are not adjacent, then it is not the case that > 0 and > 0.

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Illustration: x = 8 and y = 10

Constraints are satisfied

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λ method for representing NLP

• represents x as a convex combination of breakpoints

• represents y as a convex combination of function values

• adjacency conditions

– enforced using properties of integer variables

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Representing non-linear functions: the δ method.

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Representing non-linear functions: the δ method.

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More on the δ-method

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Rule: δj = 0 unless δj-1 is at is upper bound for j > 1.

• Let wj = 0 if δj-1 is not at its upper bound.

w2≤ δ1/4; w3 ≤ δ2/8; w2 , w3 ∈ {0,1}

• Let δj = 0 if wj = 0

0 ≤ δ1 ≤ 4; 0 ≤ δ2 ≤ 8 w2; 0 ≤ δ3 ≤ 4 w3

(this replaces the other upper bounds)

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Illustration: x = 8 and y = 10

• x = δ1 + δ2 + δ3

• 0 ≤ δ1 ≤ 4; 0 ≤ δ2 ≤ 8 w2; 0 ≤ δ3 ≤ 4 w3

So, δ1 = 4 ; δ2 = 4; δ3 = 0 .

and w2 = 1 ; w3 = 0

• y = 2.5 δ1 + 0 δ2 + 2.5 δ3

and so y = 10

Constraints are all satisfied.

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δ-method for representing IPs

represents x as a sum of intervals

x = δ1 + … + δk; 0 ≤ δj ≤ uj for each j

represents y using slopes from the intervals

y = a1δ1 + … + akδk

cannot use an interval unless the previous

interval is ‘used up’ (λ at its upper bound)

If δj < uj then δj+1 = 0

enforced using properties of integer variables

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Selecting products optimally for stocking

A computer manufacturer has 8 different attributes that can be varied when buying a computer. (RAM memory size, disk size, computer speed, peripherals, etc). Taking into account all possibilities, there are more than 10,000 variations.

Each customer can special order a computer, or can order from stock. Customers may prefer to order from stock because it is more convenient, and they get the computer quicker.

What computers should be stocked? (At most 25 computer configurations will be stocked.)

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Assumptions for this model

• The computer manufacturer has detailed knowledge on 1000 random customers

– For each customer i and for each possible computer configuration j, u(i,j) is the customer’s utility for selecting j from stock

– u*(i) = utility for person i special ordering – p(j) is the profit for a purchase of j if j is purchased

from stock (higher than if it is made to order since one can manufacture more at once

– v(i) is the profit if person i selects a made to order

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More assumptions

• There is a fixed charge f(j) for stocking item j

• Company’s goal: maximize profit from stocking computers: taking into account revenues not made in special orders and the fixed charges

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The integer programming model

• Let x(i,j) = 1 if customer i selects product j

• Let w(i) = 1 if customer i special orders

• Let y(j) = 1 if product j is stocked

• Work with your partners:

• What is the objective function.

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What are the constraints?

• Each person selects at most 1 product

• The manufacturer stocks at most 25 different configurations

• A person cannot select product j unless product j is stocked

• Variables are binary

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How do we enforce that each person selects the product with highest utility?

• If item j is stocked, and if u(i,j) > u(i,k), then person i does not select item k.

• If item j is stocked, and if u(i,j) is greater than u*(i), then person i does not do a made to order

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Please don’t look ahead

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The model

max Σi,j p(j) x(i,j) + Σi v(i) w(i) - Σj f(j) y(j)

s.t. w(i) + Σj x(i,j) = 1 Σj y(j) ≤ 25 x(i,j) ≤ y(j) for all i, j x(i,j) ≤ (1 – y(k)) if u(i,j) < u(i,k) w(i) ≤ (1 – y(k)) if u*(i) < u(i,k) w(i), x(i,j), y(j) ∈ {0, 1}

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Comments on the model

• Utilities can be estimated using “conjoint analysis”

• The problem is widespread

– Where may it come up?

• Any suggestions on variations to consider?

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More on Modeling of Integer Programs

• Integer programming can be used whenever the optimization problem looking for solutions in n variables and where the solution can be described in binary solutions.

• But, some times it takes skill in transforming problems into an IP

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Some other areas where IPs are used

• Project management

• Product configuration

• Vehicle routing or packet routing and

scheduling

• Manufacturing

• Finance

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A Product Configuration Problem

• Kaya wants to buy a Porsche. There are a number of possible attributes that can be varied:

– there are 14 options for color – one can have 2 door or 4-door – one can vary the trim – there are four different stereo systems – Kaya has specified the utility for each attribute – She has a budget of B for the car

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Notation for the configuration problem

• Attributes A1, …, AK

• Each attribute has options.

– We write j ∈ Ai if j is an option of Ai.• u(j) = utility of option j• Exactly one option of any attribute must beselected -- occasionally, one of the options is “null”• There is a set P of pairs. If (i,j) ∈ P, this means that j

must be selected if i is selected.• We have a set E of exclusions. If (i,j) ∈ E, this means

that one cannot select both i and j.• There is a budget B for the configuration.

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Towards a formulation for the configuration problem

let x(j) = 1 if option j is selected = 0 if option j is not selected

Maximize UtilitySubject to: select one option from each attribute obey precedence constraints obey exclusion constraints do not spend more than the budget B

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A Fleet Routing Problem

• Vehicles 1, 2, … , m• qk = capacity of vehicle k• Locations 1, 2, … , n• d(i,j) = distance from location i to location j• Depot D• Each vehicle starts at the depot D and makes

a tour of length at most T• Each location must be visited by some vehicl

e• Minimize the total distance

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A vehicle Routing Problem

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A vehicle Routing Problem

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Towards a model

• let k be the index for the vehicle

• let = 1 if vehicle k travels from i to j

= 0 otherwise

• How can we formulate the vehicle routing problem?

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Summary

• IP is quite widespread• can model piecewise linear costs• can model most problems in which the

solution is integer value• applications – marketing – product configuration – vehicle routing