6 Integer Graphical

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Integer Programming ModelsTypes of Models

Total Integer Model: All decision variables required to have integer solution values.

0–1 Integer Model: All decision variables required to have integer values of zero or one.

Also referred to as Boolean,True/False, Binary

Mixed Integer Model: Some of the decision variables (but not all) required to have integer values.

The owner of a machine shop is planning to expand by purchasing some new machines presses and lathes. The owner has estimated that each press purchased will increase profit by $100 per day and each lathe will increase profit by $150 daily. The number of machines the owner can purchase is limited by the cost of the machines and the available floor space in the shop. The machine purchase prices and space requirements are as follows:

A Total Integer Model Example

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Machine

Required Floor Space (sq. ft.)

Purchase Price

Press Lathe

15

30

$8,000

4,000

A Total Integer Model (1 of 2)

Machine shop obtaining new presses and lathes.

Marginal profitability: each press $100/day; each lathe $150/day.

Resource constraints: $40,000, 200 sq. ft. floor space.

Machine purchase prices and space requirements:

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A Total Integer Model (2 of 2)

Integer Programming Model:

Maximize Z = $100x1 + $150x2

subject to:

8,000x1 + 4,000x2 $40,000

15x1 + 30x2 200 ft2

x1, x2 0 and integer

x1 = number of presses x2 = number of lathes

A 0-1 Integer Model Example

A community council must decide which recreation facilities to construct in its community. Four new recreation facilities have been proposed a swimming pool, a tennis center, an athletic field, and a gymnasium. The council wants to construct facilities that will maximize the expected daily usage by the residents of the community, subject to land and cost limitations. The expected daily usage and cost and land requirements for each facility follow:

Recreation Facility

Expected Usage

(people/day) Cost

Land Requirements

(acres)

Swimming pool

300 $35,000 4

Tennis center 90 10,000 2

Athletic field 400 25,000 7

Gymnasium 150 90,000 3

The community has a $120,000 construction budget and 12 acres of land. Because the swimming pool and tennis center must be built on the same part of the land parcel, however, only one of these two facilities can be constructed. The council wants to know which of the recreation facilities to construct to maximize the expected daily usage.

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Recreation facilities selection to maximize daily usage by residents.

Resource constraints: $120,000 budget; 12 acres of land.

Selection constraint: either swimming pool or tennis center (not both).

Data:

Recreation Facility

Expected Usage (people/day)

Cost ($) Land

Requirement (acres)

Swimming pool Tennis Center Athletic field Gymnasium

300 90 400 150

35,000 10,000 25,000 90,000

4 2 7 3

A 0–1 Integer Model (1 of 2)

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Integer Programming Model:

Maximize Z = 300x1 + 90x2 + 400x3 + 150x4

subject to:

$35,000x1 + 10,000x2 + 25,000x3 + 90,000x4 $120,000

4x1 + 2x2 + 7x3 + 3x4 12 acres

x1 + x2 1 facility

x1, x2, x3, x4 = 0 or 1 (either do or don’t)

x1 = construction of a swimming pool x2 = construction of a tennis center x3 = construction of an athletic field x4 = construction of a gymnasium

A 0–1 Integer Model (2 of 2)

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A Mixed Integer Model (1 of 2)

$250,000 available for investments providing greatest return after one year.

Data:

Condominium cost $50,000/unit, $9,000 profit if sold after one year.

Land cost $12,000/ acre, $1,500 profit if sold after one year.

Municipal bond cost $8,000/bond, $1,000 profit if sold after one year.

Only 4 condominiums, 15 acres of land, and 20 municipal bonds available.

A Mixed Integer Model Example

Nancy Smith has $250,000 to invest in three alternative investments: condominiums, land, and municipal bonds. She wants to invest in the alternatives that will result in the greatest return on investment after 1 year.

Each condominium costs $50,000 and will return a profit of $9,000 if sold at the end of 1 year; each acre of land costs $12,000 and will return a profit of $1,500 at the end of 1 year; and each municipal bond costs $8,000 and will result in a return of $1,000 if sold at the end of 1 year. In addition, there are only 4 condominiums, 15 acres of land, and 20 municipal bonds available for purchase.

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Integer Programming Model:

Maximize Z = $9,000x1 + 1,500x2 + 1,000x3

subject to:

50,000x1 + 12,000x2 + 8,000x3 $250,000 x1 4 condominiums x2 15 acres x3 20 bonds x2 0 x1, x3 0 and integer

x1 = condominiums purchased x2 = acres of land purchased x3 = bonds purchased

A Mixed Integer Model (2 of 2)

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Rounding non-integer solution values up (down) to the nearest integer value can result in an infeasible solution

A feasible solution may be found by rounding down (up)non-integer solution values, but may result in a less than optimal (sub-optimal) solution.

Whether a variable is to be rounded up or down depends on where the corresponding point is with respect to the constraint boundaries.

Integer Programming Graphical Solution

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Integer Programming ExampleGraphical Solution of Maximization Model

Maximize Z = $100x1 + $150x2

subject to: 8,000x1 + 4,000x2 $40,000 15x1 + 30x2 200 ft2

x1, x2 0 and integer

Optimal Solution:Z = $1,055.56x1 = 2.22 pressesx2 = 5.55 lathes

Feasible Solution Space with Integer Solution PointsBoth of these may be found by judiciousrounding, but the value of the objective function needs to be checked for each.

Harrison Electric Company Example of Integer Programming by Enumeration

The Company produces two products popular with home renovators, old-fashioned chandeliers and ceiling fans

Both the chandeliers and fans require a two-step production process involving wiring and assembly

It takes about 2 hours to wire each chandelier and 3 hours to wire a ceiling fan

Final assembly of the chandeliers and fans requires 6 and 5 hours respectively

The production capability is such that only 12 hours of wiring time and 30 hours of assembly time are available

Harrison Electric Company Example of Integer Programming

Each chandelier produced nets the firm $7 and each fan $6

Harrison’s production mix decision can be formulated using LP as follows

Maximize profit = $7X1 + $6X2

subject to 2X1 + 3X2 ≤ 12 (wiring hours)

6X1 + 5X2 ≤ 30 (assembly hours)

X1, X2 ≥ 0 (nonnegative)

where

X1 = number of chandeliers produced

X2 = number of ceiling fans produced

Harrison Electric Company Example of Integer Programming

The Harrison Electric Problem

6 –

5 –

4 –

3 –

2 –

1 –

0 –

| | | | | | |

1 2 3 4 5 6 X1

X2

+

++

++++

+

6X1 + 5X2 ≤ 30

2X1 + 3X2 ≤ 12

+ = Possible Integer Solution

Optimal LP Solution

(X1 =3.75, X2 = 1.5, Profit = $35.25)

Harrison Electric Company Example of Integer Programming

The production planner Wes recognizes this is an integer problem

His first attempt at solving it is to round the values to X1 = 4 and X2 = 2

However, this is not feasible

Rounding X2 down to 1 gives a feasible solution, but it may not be optimaloptimal

This could be solved using the enumerationenumeration method

Enumeration is generally not possible for large problems

Harrison Electric Company Example of Integer Programming

Integer solutions

CHANDELIERS (X1) CEILING FANS (X2) PROFIT ($7X1 + $6X2)

0 0 $0

1 0 7

2 0 14

3 0 21

4 0 28

5 0 35

0 1 6

1 1 13

2 1 20

3 1 27

4 1 34

0 2 12

1 2 19

2 2 26

3 2 33

0 3 18

1 3 25

0 4 24

Optimal solution to integer programming

problem

Solution if rounding is used

Table 11.1

Harrison Electric Company Example of Integer Programming

The rounding solution of X1 = 4, X2 = 1 gives a profit of $34

The optimal solution of X1 = 5, X2 = 0 gives a profit of $35

The optimal integer solution is less than the optimal LP solution

An integer solution can nevernever be better than the LP solution and is usuallyusually a lesser solution