Linear Programming

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Chapter 2 Linear Programming
Dr. Alaa Sagheer 2010-2011
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Chapter Outline
Part I

• Introduction
• The Linear Programming Model
• Examples of Linear Programming Problems
• Developing Linear Programming Models
• Graphical Solution to LP Problems

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Introduction

• Mathematical programming is used to find the best
  or optimal solution to a problem that requires a
  decision or set of decisions about how best to
  use a set of limited resources to achieve a state
  goal of objectives.
• Steps involved in mathematical programming
• Conversion of stated problem into a mathematical
  model that abstracts all the essential elements
  of the problem.
• Exploration of different solutions of the
  problem.
• Finding out the most suitable or optimum
  solution.
• Linear programming requires that all the
  mathematical functions in the model be linear
  functions.

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Introduction
Example The Burroughs garment company
manufactures men's shirts and womens blouses for
Walmark Discount stores. Walmark will accept all
the production supplied by Burroughs. The
production process includes cutting, sewing and
packaging. Burroughs employs 25 workers in the
cutting department, 35 in the sewing department
and 5 in the packaging department. The factory
works one 8-hour shift, 5 days a week. The
following table gives the time requirements and
the profits per unit for the two garments
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Introduction
Minutes per unit
Garment Cutting Sewing Packaging Unit profit()
Shirts 20 70 12 8.00
Blouses 60 60 4 12.00
Determine the optimal weekly production schedule
for Burroughs.
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Introduction
Solution Assume that Burroughs produces x1
shirts and x2 blouses per week
8 x1 12 x2
Profit got
Time spent on cutting
20 x1 60 x2 mts
Time spent on sewing
70 x1 60 x2 mts
Time spent on packaging
12 x1 4 x2 mts
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Introduction
The objective is to find x1, x2 so as to maximize
the profit z 8 x1 12 x2 satisfying the
constraints
20 x1 60 x2 25 ? 40 ? 60 70 x1 60 x2
35 ? 40 ? 60 12 x1 4 x2 5 ? 40 ? 60
x1, x2 0, integers
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Introduction
This is a typical optimization problem
Any values of x1, x2 that satisfy all the
constraints of the model is called a feasible
solution. We are interested in finding the
optimum feasible solution that gives the maximum
profit while satisfying all the constraints.
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The Linear Programming Model

• Let x1, x2, x3, , xn are decision
  variables and
• Z Objective function or linear
  function
• Requirement Maximization of the linear function
  Z
• Z c1x1 c2x2 c3x3 cnxn
• subject to the following constraints

where aij, bi, and cj are given constants.
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The Linear Programming Model

• The linear programming model can be written in
  more efficient notation as

The decision variables, xI, x2, ..., xn,
represent levels of n competing activities
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The Linear Programming Model
An LPP satisfies the three basic properties
1. Proportionality It means, the contributions
of each decision variable in the objective
function and its requirements in the constraints
are directly proportional to the value of the
variable
2. Additivity This property requires the
contribution of all the variables in the
objective function and its constraints to be the
direct sum of individual contributions of each
variables.
3. Certainty All the objective and constraint
coefficient of the LP model are deterministic.
This mean that they are known constants- a rare
occurrence in real life.
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Examples of LP Problems
1. A Product Mix Problem

• A manufacturer has fixed amounts of different
  resources such as raw material, labor, and
  equipment.
• These resources can be combined to produce any
  one of several different products.
• The quantity of the ith resource required to
  produce one unit of the jth product is known.
• The decision maker wishes to produce the
  combination of products that will maximize total
  income.

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Examples of LP Problems
2. A Blending Problem

• Blending problems refer to situations in which a
  number of components (or commodities) are mixed
  together to yield one or more products.
• Typically, different commodities are to be
  purchased. Each commodity has known
  characteristics and costs.
• The problem is to determine how much of each
  commodity should be purchased and blended with
  the rest so that the characteristics of the
  mixture lie within specified bounds and the total
  cost is minimized.

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Examples of LP Problems
3. A Production Scheduling Problem

• A manufacturer knows that he must supply a given
  number of items of a certain product each month
  for the next n months.
• They can be produced either in regular time,
  subject to a maximum each month, or in overtime.
  The cost of producing an item during overtime is
  greater than during regular time. A storage cost
  is associated with each item not sold at the end
  of the month.
• The problem is to determine the production
  schedule that minimizes the sum of production and
  storage costs.

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Examples of LP Problems
4. A Transportation Problem

• A product is to be shipped in the amounts al, a2,
  ..., am from m shipping origins and received in
  amounts bl, b2, ..., bn at each of n shipping
  destinations.
• The cost of shipping a unit from the ith origin
  to the jth destination is known for all
  combinations of origins and destinations.
• The problem is to determine the amount to be
  shipped from each origin to each destination such
  that the total cost of transportation is a
  minimum.

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Examples of LP Problems
5. A Flow Capacity Problem

• One or more commodities (e.g., traffic, water,
  information, cash, etc.) are flowing from one
  point to another through a network whose branches
  have various constraints and flow capacities.
• The direction of flow in each branch and the
  capacity of each branch are known.
• The problem is to determine the maximum flow, or
  capacity of the network.

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Developing LP Model

• The variety of situations to which linear
  programming has been applied ranges from
  agriculture to zinc melting.
• Steps Involved
• Determine the objective of the problem and
  describe it by a criterion function in terms of
  the decision variables.
• Find out the constraints.
• Do the analysis which should lead to the
  selection of values for the decision variables
  that optimize the criterion function while
  satisfying all the constraints imposed on the
  problem.

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Developing LP Model
Graphical Solution
Example Product Mix Problem
The N. Dustrious Company produces two products I
and II. The raw material requirements, space
needed for storage, production rates, and selling
prices for these products are given in Table I.
The total amount of raw material available per
day for both products is 15751b. The total
storage space for all products is 1500 ft2, and a
maximum of 7 hours per day can be used for
production.
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Developing LP Model
Example Product Mix Problem
All products manufactured are shipped out of the
storage area at the end of the day. Therefore,
the two products must share the total raw
material, storage space, and production time. The
company wants to determine how many units of each
product to produce per day to maximize its total
income
Solution

• The company has decided that it wants to maximize
  its sale income, which depends on the number of
  units of product I and II that it produces.
• Therefore, the decision variables, x1 and x2 can
  be the number of units of products I and II,
  respectively, produced per day.

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Developing LP Model

• The object is to maximize the equation
• Z 13x1 11x2
• subject to the constraints on storage space, raw
  materials, and production time.
• Each unit of product I requires 4 ft2 of storage
  space and each unit of product II requires 5 ft2.
  Thus a total of 4x1 5x2 ft2 of storage space is
  needed each day. This space must be less than or
  equal to the available storage space, which is
  1500 ft2. Therefore,
• 4X1 5X2 ? 1500
• Similarly, each unit of product I and II produced
  requires 5 and 3 1bs, respectively, of raw
  material. Hence a total of 5xl 3x2 Ib of raw
  material is used.

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Developing LP Model

• This must be less than or equal to the total
  amount of raw material available, which is 1575
  Ib. Therefore,
• 5x1 3x2 ? 1575
• Prouct I can be produced at the rate of 60 units
  per hour. Therefore, it must take I minute or
  1/60 of an hour to produce I unit. Similarly, it
  requires 1/30 of an hour to produce 1 unit of
  product II. Hence a total of x1/60 x2/30 hours
  is required for the daily production. This
  quantity must be less than or equal to the total
  production time available each day. Therefore,
• x1 / 60 x2 / 30 ? 7
• or x1 2x2 ? 420
• Finally, the company cannot produce a negative
  quantity of any product, therefore x1 and x2 must
  each be greater than or equal to zero.

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Developing LP Model
Graphical Solution

• The linear programming model for this example can
  be summarized as

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Developing LP Model
Graphical Solution
Example (2) Reddy Mikks produce both interior
and exterior paints from two raw materials, M1
and M2. The following table provides the basic
data of the problem
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Developing LP Model
Example Problem
A market survey indicates that the daily demand
for the interior paint cannot exceed that for
extirior paint by more than one ton. Also, the
maximam daily demand for the interior paint is 2
tons. Reddy Mikks wants to determine the
optimum(best) product mix of the interior and
exterior paints that maximizes the total daily
profit.
Solution

• Reddy Mikks has decided that it wants to maximize
  its total daily profit, which depends on the
  product mix of the interior and exterior paints.
• Therefore, the decision variables, x1 and x2 can
  be the ton of exterior and interior paints,
  respectively, produced per day.

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Developing LP Model

• The object is to maximize the equation
• Z 5x1 4x2
• The daily raw material M1 is 6tons per ton of
  exterior paint and 4 tons per ton of interior
  paint must be equal the daily avaliability of raw
  material M1 (24 ton) , Therefore,
• 6x1 4x2 ? 24
• Similary, The daily usage of raw material M2 is
  1ton per ton of exterior paint and 2 tons per ton
  of interior paint must be equal the daily
  aviliability or raw material M2 (6 tons),
  Therefore,
• x1 2x2 ? 6

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Developing LP Model

• The first demand restriction stipulates that the
  excess the daily production of interior over
  exterior paint should not exceed 1 ton, Therefore
  ,
• x2 - x1 ? 1
• The second demand restrection stipulates that the
  maximum daily demand of interior paint is limited
  to 2 tons. Therefore,
• x2 ? 2

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Developing LP Model
Graphical Solution

• The linear programming model for this example can
  be summarized as

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Developing LP Model
Example (3)
Wild West produces two types of cowboy hats. Type
I hat requires twice as much labor as a Type II.
If all the available labor time is dedicated to
Type II alone, the company can produce a total of
400 Type II hats a day. The respective market
limits for the two types of hats are 150 and 200
hats per day. The profit is 8 per Type I hat and
5 per Type II hat. Formulate the problem as an
LPP so as to maximize the profit.
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Developing LP Model
Solution Assume that Wild West produces x1
Type I hats and x2 Type II hats per day.
8 x1 5 x2
Per day Profit got
Assume the time spent in producing one type II
hat is c minutes.
Labour Time spent is
(2 x1 x2) c minutes
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Developing LP Model
The objective is to find x1, x2 so as to
maximize the profit z 8 x1 5 x2
satisfying the constraints
(2 x1 x2 ) c 400 c x1
150 x2 200 x1, x2
0, integers
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Developing LP Model
Example (4)
BITS wants to host a Seminar for five days. For
the delegates there is an arrangement of dinner
every day. The requirement of napkins during the
5 days is as follows
Day 1 2 3 4 5
Napkins Needed 80 50 100 80 150
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Developing LP Model
Institute does not have any napkins in the
beginning. After 5 days, the Institute has no
more use of napkins. A new napkin costs Rs.
2.00. The washing charges for a used one are Rs.
0.50. A napkin given for washing after dinner is
returned the third day before dinner. The
Institute decides to accumulate the used napkins
and send them for washing just in time to be used
when they return. How shall the Institute meet
the requirements so that the total cost is
minimized ? Formulate as a LPP.
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Developing LP Model
Solution Let xj be the number of napkins
purchased on day j, j1,2,..,5 Let yj be the
number of napkins given for washing after dinner
on day j, j1,2,3 Thus we must have
x1 80, x2 50, x3 y1 100, x4 y2 80 x5
y3 150
Also we have y1 80, y2 (80 y1) 50
y3 (80 y1) (50 y2) 100
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Developing LP Model
Thus we have to Minimize z 2(x1x2x3x4x5)0.
5(y1y2y3) Subject to x1 80, x2 50, x3
y1 100, x4 y2 80, x5
y3 150, y1 80,
y1y2 130, y1y2y3 230,
all variables 0, integers
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Developing LP Model
Example (5)
A Post Office requires different number of
full-time employees on different days of the
week. The number of employees required on each
day is given in the table below. Union rules say
that each full-time employee must receive two
days off after working for five consecutive days.
The Post Office wants to meet its requirements
using only full-time employees. Formulate the
above problem as a LPP so as to minimize the
number of full-time employees hired.
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Developing LP Model
Requirements of full-time employees day-wise
Day No. of full-time employees required
1 - Monday 10
2 - Tuesday 6
3 - Wednesday 8
4 - Thursday 12
5 - Friday 7
6 - Saturday 9
7 - Sunday 4
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Developing LP Model
Solution Let xi be the number of full-time
employees employed at the beginning of day i (i
1, 2, , 7). Thus our problem is to find xi so
as to
Minimize
Subject to
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The Graphical Solution

1. The determination of the solution space that
   defines the feasible solutions that satisfy all
   the constrains of the model.
2. The determination of the optimum solution from
   among all the points in the feasible solution
   space.

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Graphical LP Solution Model
Example
Electra produces two types of electric motors,
each on a separate assembly line. The respective
daily capacities of the two lines are 150 and 200
motors. Type I motor uses 2 units of a certain
electronic component, and type II motor uses
only 1 unit. The supplier of the component can
provide 400 pieces a day.The profits per motor of
types I and II are 8 and 5 respectively.
Formulate the problem as a LPP and find the
optimal daily production.
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Graphical LP Solution Model
Let the company produce x1 type I motors and x2
type II motors per day.
The objective is to find x1 and x2 so as to
Maximize the profit
Subject to the constraints
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Graphical LP Solution Model
Step 1 Determination of the Feasible solution
space
The non-negativity restrictions tell that the
solution space is in the first quadrant. Then we
replace each inequality constraint by an equality
and then graph the resulting line (noting that
two points will determine a line uniquely).
Step 2 Determination of the optimal solution
The determination of the optimal solution
requires the direction in which the objective
function will increase (decrease) in the case of
a maximization (minimization) problem. We find
this by assigning two increasing (decreasing)
values for z and then drawing the graphs of the
objective function for these two values. The
optimum solution occurs at a point beyond which
any further increase (decrease) of z will make us
leave the feasible space.
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Graphical solution
x2
Maximize z8x15x2
Subject to the constraints
Optimum 1800 at
2x1x2 ? 400 x1 ? 150 x2 ? 200
x1,x2? 0
(100,200)
(0,200)
z1200
(150,100)
z1800
z1000
x1
z1700
z400
(150,0)
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Graphical LP Solution Model
Realted to Reddy Mikks problem
Problem
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Graphical Solution to LP Problems
Related to Product Mix Problem
Problem
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Graphical LP Solution Model
Feed Mix Problem Minimize z 2x1 3x2 Subject
to x1 3 x2 ? 15 2 x1 2 x2
? 20 3 x1 2 x2 ? 24
x1, x2 ? 0
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Graphical Solution of Feed Mix Problem
(0,12)
z 26
z 30
z 36
z 39
(4,6)
z 42
z 22.5
(7.5,2.5)
(15,0)
Minimum at
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Graphical LP Solution Model (5)
Maximize z 2x1 x2 Subject to x1 x2
40 4 x1 x2 100
x1, x2 0
Optimum 60 at (20, 20)
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The Graphical Solution
Graphical Solution
Ozark farms uses at least 800 Ib of special feed
is a mixture of corn and soybean meal with the
following compositions
The dietary requirement of the special feed are
at least 30 protein and at most 5 fiber. Ozark
Farms wishes to determine the daily minimm- cost
feed mix.
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The Graphical Solution
Graphical Solution
Solution
We can consider that,
And the objective function seeks to minimize the
total daily cost (in dollars) of the feed mix and
is thus expressed as
Subject to constrains
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The Graphical Solution

• Unlike previous examples, the second and the
  third constraints pass through the origin. To
  plot associated straight lines, we need one
  additional point, which can be obtained by
  assigning value to one of the variables and then
  solving for other variables. For example, in the
  second constraint, x1200, then x2140

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The Graphical Solution
Problem
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