1 introduction to linear programming

8/8/2019 1 Introduction to Linear Programming

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1 Introduction to Linear Programming


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Prototype Example

The Windor Glass Co. produces high-quality

glass products, including windows and glass

doors. It has three plants. Alumunium frames

and hardware are made in Plant 1, wood

frames are made in Plant 2, and Plant 3

produces the glass and assembles the

products.


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Prototype Example

Top management has decided to launch two

new products having large sales potential:

Product 1: An 8-foot glass door with alumuniumframing

Product 2: A 4 x 6 foot double-hung wood

framed window


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Prototype Example

Product 1 requires some of the production

capacity in Plants 1 and 3, but none in Plant 2.

Product 2 needs only Plants 2 and 3.


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Prototype Example

Plant

Production Time per Batch, Hours

Production Time

per Week, Hours

Product

1 2

1 1 0 4

2 0 2 12

3 3 2 18

Profit per batch $3,000 $5,000


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Formulation as a Linear Programming

Problem

To formulate the mathematical (linear

programming) model for this problem, let

x1 = number of batches of product 1 producedper week

x2 = number of batches of product 2 produced

per week

Z = total profit per week (in thousands of

dollars) from producing these two products


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Problem

Thus, x1 and x2 are the decision variables for the

model. Finally we have to maximize Z.

To summarize, the problem is to choose values

of x1 and x2 so as to

Maximize Z = 3x1 + 5x2


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Problem

subject to restrictions

x1e 4

2x2 e 123x1 + 2x2 e 8

and

x1 u 0, x2 u 0


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The Linear Programming Model

Certain symbols are commonly used to denote

the various components of a linear

programming model. These symbols are listed

below, along with their interpretation for the

general problem of allocating resources to

activities.

Z : value of overall measure of performance.xj : level of activity j (j = 1,2, , n).


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cj : increase in Z that would result from each unit

increase in level of activity j.

bi: amount of resource i that is available for

allocation to activities (i = 1,2, , m).

aij : amount of resource i consumed by each unit

of activity j.


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The model poses the problem in terms of

making decisions about the levels of the

activities, so x1, x2, , xn are called the

decision variables. The values of cj, bi, and aij(for i = 1,2, , m and j = 1,2, , n) are the

input constants for the model. The cj, bi, and

aij are also referred to as the parameters ofthe model.


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A Standard Form of the Model

Maximize Z = c1x1 + c2x2 + + cnxn

subject to restrictions

a11 x1 + a12 x2 + + a1n xne b1

a21 x1 + a22 x2 + + a2n xne b2

am1 x1 + am2 x2 + + amn xne bm

and x1 u 0, x2 u 0 , , xn u 0


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Common terminology for the linear

programming model can now be summarized.

The function being maximized, c1x

1+ c

2x

2+

+ cnxn, is called the objective function. The

restrictions normally are referred to as

constraints.


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The first m constraints are sometimes called

functional constraints (structural constraints).

Similarly, the xju 0 restrictions are called

nonnegativity constraints (nonnegativity

conditions).


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Other Forms

The other legitimate forms are the following:

1 Minimizing rather than maximizing the

objective function:Minimize Z = c1x1 + c2x2 + + cnxn.

2 Some functional constraints with a greater-

than-or-equal inequality:ai1 x1 + ai2 x2 + + ain xnu bi for some values of i.


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Terminology for Solutions of the

Model

Any specification of values for the decision

variables (x1, x2, , xn) is called a solution.

A feasible solution is a solution for which all theconstraints are satisfied.

An infeasible solution is a solution for which at

least one constraint is violated.


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Model

In the example, the points (2, 5) and (4, 3) are

feasible solutions, while the points (-2, 4)

and(5,6) are infeasible solutions.

The feasible region is the collection of all

feasible solutions.


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Model

It is possible for a problem to have no feasible

solutions. This would have happened in the

example if the new products had been

required to return a net profit of at least

$50,000 per week. The corresponding

constraint, 3x1 + 2x2 u 50, would eliminate the

entire feasible region.


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Model

An optimal solution is a feasible solution that

has the most favorable value of the objective

function.

The most favorable value is the largest value if

the objective function is to be maximized,

whereas it is the smallest value if the objective

function is to be minimized.


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Model

It is possible for a problem to have multiple

optimal solutions. This would occur in the

example if the profit per batch produced of

product 2 were changed to $2,000. This

changes the objective function to Z = 3x1 +

2x2, so that all the points on line segment

connecting (2, 6) and (4, 3) would be optimal.


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Model

Another possibility is that a problem has no

optimal solutions. This occurs only if (1) it has

no feasible solutions or (2) the constraints do

not prevent improving the value of the

objective function. The latter case is referred

to as having an unbounded Z (unbounded

objective).


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Model

This would occur in the example if the only

functional constraint were x1e 4, because x2then could be increased indefinitely in the

feasible region without ever reaching the

maximum value of 3x1 + 5x2.


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Model

A corner-point feasible (CFP) solution is a

solution that lies at a corner of the feasible

region.

1 introduction to linear programming

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