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Operation Management

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  • 1

    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

    business and economic activities

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

    industrial operations

    (iii) It is also extensively used in development and

    distribution of planning.

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    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

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