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Linear Programming Problem

Optimization Problem

Problems which seek to maximize or minimize an

objective function of a finite number of variables

subject to certain constraints are called optimization

problems

Example

1 1 2 2 n n Maximize z = c x +c x +.............+c x

Subject to;

11 1 12 2 1n n 1

21 1 22 2 2n n 2

m1 1 m2 2 mn n m

a x +a x +.............+a x b

a x +a x +.............+a x b

.

.

.

a x +a x +.............+a x b

jx > 0 j (j = 1, 2,............,n)

Feasible Solution

Any solution of a linear programming problem that

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satisfies all the constraints of the model is called a

feasible solution.

1 1 2 2 n n Maximize z = c x +c x +.............+c x

Subject to;

11 1 12 2 1n n 1

21 1 22 2 2n n 2

m1 1 m2 2 mn n m

a x +a x +.............+a x b

a x +a x +.............+a x b

.

.

.

a x +a x +.............+a x b

jx > 0 j (j = 1, 2,............,n)

The solution, 1 1 2 2 n n x = s , x = s ,..........,x = s will be a

feasible solution of the given problem if it does not

violate any of the constraints of the given problem.

Programming Problem

Programming problems always deal with determining

optimal allocations of limited resources to meet given

objectives. The constraints or limited resources are

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given by linear or non-linear inequalities or equations.

The given objective may be to maximize or minimize

certain function of a finite number of variables.

Linear Programming and Linear Programming

Problem

Suppose we have given m linear inequalities or

equations in n unknown variables 1 2 nx , x ,............and,x

and we wish to find non-negative values of these

variables which will satisfy the constraints and

maximize or minimize some linear functions of these

variables (objective functions), then this procedure is

known as linear programming and the problem which is

described is known as linear programming problem.

Mathematically it can be described as, suppose we have

m linear inequalities or equations in n unknown

variables 1 2 nx , x ,............and,x of the form

n

ij j i

j=1

a x { ,=, }b (i= 1, 2,....,m) where for each constraint one

and only one of the signs ,=, holds. Now we wish to

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find the non-negative values of jx , j = 1, 2,,n.

which will satisfy the constraints and maximize or

minimize a linear function n

j j

j=1

z = c x . Here ija , ib and

jc are known constant.

At the short-cut method mathematically the linear

programming problem can be written as

Optimize (maximize or minimize) n

j j

j=1

z = c x

Subject to

n

ij j i

j=1

a x { ,=, }b (i= 1, 2,....,m)

jx > 0 j (j = 1, 2,............,n)

Application of LPM

(i) Linear programming problem is widely applicable in

(ii) It is also applicable in government, military and

industrial operations

(iii) It is also extensively used in development and

distribution of planning.

• 5

Objective Function

In a linear programming problem, a linear function of

the type n

j j

j=1

z = c x of the variables jx , j = 1, 2,,n.

which is to be optimized is called objective function. In

an objective function no constant term will be appeared.

i. e. we cannot write the objective function of the type

n

j j

j=1

z = c x +k

Example of Linear Programming Problem:

Suppose m types of machines 1 2 m A , A ,.........,A are producing

n products namely 1 2 nP , P ,.........P . Let (i) ij a is the hours

required of the ith machine (i = 1, 2,,m) to produce

per unit of the jth product (j=1, 2,..,n) (ii) ib (i= 1,

2,.,m) is the total available hours per week for

machine i, and (iii) jc is the per unit profit on sale of

each of the jth product.

Machines 1P 2P nP Total Available Time

1A

2A

.

.

11a 12 a 1na

21a 22a 2na

.

1b

2b

.

.

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.

m A m1a m2a mna

.

m b

Unit Profits 1c 2c nc

Construct the LPP

Suppose jx (j = 1, 2, 3,..,n) is the no. of units of

the jth product produced per week. The objective

function is given by;

1 1 2 2 n n z = c x +c x +.............+c x

The constraints are given by;

11 1 12 2 1n n 1

21 1 22 2 2n n 2

m1 1 m2 2 mn n m

a x +a x +.............+a x b

a x +a x +.............+a x b

.

.

.

a x +a x +.............+a x b

Since the amount of production cannot be negative so,

jx 0 (j = 1, 2, 3, 4) .

The weekly profit is given by 1 1 2 2 n n z = c x +c x +.............+c x .

Now we wish to determine the values of the variables

jx 's for which all the constraints are satisfied and the

objective function will be at maximum. That is

1 1 2 2 n n Maximize z = c x +c x +.............+c x

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Subject to

11 1 12 2 1n n 1

21 1 22 2 2n n 2

m1 1 m2 2 mn n m

a x +a x +.............+a x b

a x +a x +.............+a x b

.

.

.

a x +a x +.............+a x b

and

jx 0 (j = 1, 2, 3, 4)

Formulation of Linear Programming Problem

(i) Transportation Problem

Suppose given amount of uniform product are available

at each of a no. of origins say warehouse. We wish to

send specified amount of the products to each of a no.

of different destinations say retail stores. We are

interested in determining the minimum cost-routing

from warehouse to the retail stores.

Let use define,

m = no. of warehouses

n = no. of retail stores

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ijx the amount of product shipped from the ith

warehouse to the jth retail store.

Since negative amounts cannot be shipped so we have

ijx 0 i, j

ia = total no. of units of the products available for

shipment at the ith (i= 1, 2,,m)warehouse.

jb = the no. of units of the product required at the jth

retail store.

Since we cannot supply more than the available amount

of the product from ith warehouse to the different retail

stores, therefore we have

11 11c :x

1a 11x 1b

21x 12 x

2a 22x 2b

.

m Origins m2 x n Destinations

.

m1x 2n x 1nx

ma mn mnc :x nb

1 1

2 2

m n

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i1 i2 in ix +x +............+x a i= 1, 2,..,m

We must supply at each retail store with the no. of units

desired, therefore. The total amount received at any

retail store is the sum over the amounts received from

each warehouse. That is

1j 2j mj jx +x +.............+x =b ; j = 1, 2,.,n

The needs of the retail stores can be satisfied

m n

i

i=1 j=1

a b j

Let us define ij c is the per unit cost of shifting from ith

warehouse to the jth retail store, then the total cost of

shifting is given by;

m n

ij ij

i=1 j=1

z= c x

We wish to determine ijx s which minimize the cost m n

ij ij

i=1 j=1

z= c x

subject to the constraints

i1 i2 in ix +x +............+x a ;i=1, 2,....,m

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1j 2j mj jx +x +.............+x =b ; j= 1, 2,........,n

It is a linear programming problem in mn variables with

(m+n) constraints.

(2) The Diet Problem

Suppose we have given the nutrient content of a no. of

different foods. We have also given the minimum daily

requirement for each nutrient and quantities of nutrient

contained in one of each food being considered. Since

we know the cost per ounce of food, the problem is to

determine the diet that satisfy the minimum daily

requirement of nutrient and also the minimum cost diet.

Let us define

m = the no. of nutrients

n = the no. of foods

ija = the quantity (mg) of ith nutrient per (oz) of the jth

food

ib = the minimum quantity of ith nutrient

jc = the cost per (oz) of the jth food

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jx = the quantity of jth food to be purchased

The total amount of ith nutrient contained in all the

purchased foods cannot be less than the minimum daily

requirements

Therefore we have

n

i1 1 i2 2 in n ij j i

j=1

a x +a x +............+a x = a x b

The total cost for all purchased foods is given by;

n

j j

j=1

z = c x

Now our problem is to minimize cost n

j j

j=1

z = c x subject

to the constraints

n

i1 1 i2 2 in n ij j i

j=1

a x +a x +............+a x = a x b and

jx 0

This is called the linear programming problem.

Feasible Solution

Any set of values of the variables jx w ##### Fletcher Building Limited Analysts Presentation › assets › incoming › ... · PDF file Flooring Dardanup LPM Ballarat Customers LPM Brisbane DARDANUP PBD TLG DISTRIBUTION Processed
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