Operations Research- LPP formulation with examples

Module-1
Formulation of Linear
Programming Problem
1

Introduction
• Many management decisions involve trying to make the most effective use of
an organization’s resources.
• Resources typically include machinery, labor, money, time, warehouse space,
or raw materials.
• Resources may be used to produce products (such as machinery, furniture,
food, or clothing) or services (such as schedules for shipping and production,
advertising policies, or investment decisions).
• Linear programming (LP) is a widely used mathematical technique designed to
help managers in planning and decision making relative to resource allocation.
• Despite the name, linear programming, and the more general category of
techniques called “mathematical programming”, have very little to do with
computer programming.
• In the world of Operations Research, programming refers to modeling and
solving a problem mathematically.
• Computer programming has, however, played an important role in the
advancement and use of LP to solve real-life LP problems
2

 Linear Programming Problem (LPP) is concerned with finding the
optimal value (maximum or minimum value) of a linear function (called
objective function) of several variables (say x and y), subject to the
conditions that the variables are non-negative and satisfy a set of linear
inequalities (called linear constraints)
 The objective function may be profit, cost, production capacity or any
other measure of effectiveness, which is to be obtained in the best
possible or optimal manner.
 The constraints may be imposed by different resources such as raw
material availability, market demand, production process and
equipment, storage capacity, etc.
3
LinearProgrammingProblem

 By linearity is meant a mathematical expression in which the expressions
among the variables are linear e.g., the expression a1x1 + а2x2 + a3x3 + ... +
an xn is linear. The variables obey the properties of proportionality (e.g., if a
product requires 3 hours of machining time, 5 units of it will require 15
hours) and additivity (e.g., amount of a resource required for a certain
number of products is equal to the sum of the resource required for each).
 A Linear Programming model seeks to maximize or minimize a linear
function, subject to a set of linear constraints. The linear model consists of
the following components:
 A set of decision variables
 An objective function
 A set of constraints
4

Importance of Linear Programming
 Many real world problems lend themselves to linear programming
modeling.
 There are well-known successful applications in:
Manufacturing - Product mix problems, Blending problems,
Production scheduling problems, Trim loss problems, Assembly-line
balancing.
Management - Media selection problems, Portfolio selection
problems, Profit planning problems, Transportation problems,
Assignment problems, Man-power scheduling problems
Finance (investment)
Advertising
Agriculture
5
LinearProgrammingProblem

6
LPP - Formulation
Max/Min Z = c1x1 + c2x2 + ... + cnxn
subject to:
a11x1 + a12x2 + ... + a1nxn (≤, =, ≥) b1
a21x1 + a22x2 + ... + a2nxn (≤, =, ≥) b2
:
am1x1 + am2x2 + ... + amnxn (≤, =, ≥) bm
x1, x2, ….., xn ≥ 0
xj = decision variables
bi = constraint levels
cj = objective function coefficients
aij = constraint coefficients

Q1
Suppose you consider academics and extra-curricular activities are the
two most important aspects that have direct impact on your prospects for
placements. You estimate the ratio of the impact of devoting 1 hour to
academics to the impact of devoting 1 hour to extra-curricular activities
on your placement prospects to be 3:5. You decide not to spend more
than 8 hours daily for these two activities. Moreover, you estimate that 1
hour of academics and 1 hour of extra-curricular activities burn, on an
average, 100 and 250 calories, respectively, and you cannot afford to burn
more than 1250 calories for these two activities based on your average
daily calorie intake. How many hours should you devote to academics and
extra-curricular activities daily?
7
LPP - Formulation

Q1
two most important aspects that have direct impact on your prospects for
placements. You estimate the ratio of the impact of devoting 1 hour to
daily calorie intake. How many hours should you devote to academics
and extra-curricular activities daily?
8
LPP - Formulation

Q1
two most important aspects that have direct bearings on your prospects
for placements. You estimate the ratio of the impact of devoting 1 hour to
daily calorie intake. How many hours should you devote to academics and
extra-curricular activities daily?
9
LPP - Formulation

Q1
two most important aspects that have direct bearings on your prospects
for placements. You estimate the ratio of the impact of devoting 1 hour to
than 8 hours daily for these two activities. Moreover, you estimate that
1 hour of academics and 1 hour of extra-curricular activities burn, on
an average, 100 and 250 calories, respectively, and you cannot afford
to burn more than 1250 calories for these two activities based on your
average daily calorie intake. How many hours should you devote to
academics and extra-curricular activities daily?
10
LPP - Formulation

Q1
Solution-
Decision variables:
Objective function:
Constraints:
11
LPP - Formulation

Q1
Solution-
Decision variables:
x1: No. of hours devoted to academics daily
x2: No. of hours devoted to extra-curricular activities daily
Objective function:
Maximize the impact on placement prospects
Max, Z = 3x1 + 5x2
Constraints:
x1 + x2 <= 8 (hours)
100x1 + 250x2 <= 1250 (calories)
x1, x2 >= 0
12
LPP - Formulation

13
Example:GiapettowoodcarvingInc.,
• Giapetto Woodcarving, Inc., manufactures two types of wooden toys:
soldiers and trains. A soldier sells for $27 and uses $10 worth of raw
materials. Each soldier that is manufactured increases Giapetto’s
variable labor and overhead cost by $14. A train sells for $21 and uses
$9 worth of raw materials. Each train built increases Giapetto’s
variable labor and overhead cost by $10. The manufacture of wooden
soldiers and trains requires two types of skilled labor: carpentry and
finishing. A soldier requires 2 hours of finishing labor and 1 hour of
carpentry labor. A train requires 1 hour of finishing and 1 hour of
carpentry labor. Each week, Giapetto can obtain all the needed raw
material but only 100 finishing hours and 80 carpentry hours.
Demand for trains is unlimited, but at most 40 soldiers are bought
each week. Giapetto wants to maximize weekly profit. Formulate a
linear programming model of Giapetto’s situation that can be used to
maximize Giapetto’s weekly profit

14
Solution:GiapettowoodcarvingInc.,
• Step 1: Model formulation
1. Decision variables: we begin by finding the decision
variables. In any LP, the decision variables should
completely describe the decisions to be made. Clearly,
Giapetto must decide how many soldiers and trains
should be manufactured each week. With this in mind,
we define:
X1 = number of soldiers produced each week
X2 = number of trains produced each week

15
2. Objective function: in any LP, the decision maker wants to
maximize (usually revenue or profit) or minimize (usually
costs) some function of the decision variables. The
function to be maximized or minimized is called the
objective function. For the Giapetto problem, we will
maximize the net profit (weekly revenues – raw materials
cost – labor and overhead costs).
Weekly revenues and costs can be expressed in terms of the
decision variables, X1 and X2 as following:

16
• Weekly revenues = weekly revenues from soldiers +
weekly revenues from trains
= 27 X1 + 21 X2
Also,
Weekly raw materials costs = 10 X1 + 9 X2
Other weekly variable costs = 14 X1 + 10 X2
Therefore, the Giapetto wants to maximize:
(27 X1 + 21 X2) – (10 X1 + 9 X2) – (14 X1 + 10 X2) = 3 X1 + 2
X2
Hence, the objective function is:
Maximize Z = 3 X1 + 2 X2

17
3. Constraints: as X1 and X2 increase, Giapetto’s objective
function grows larger. This means that if Giapetto were
free to choose any values of X1 and X2, the company
could make an arbitrarily large profit by choosing X1 and
X2 to be very large. Unfortunately, the values of X1 and X2
are limited by the following three restrictions (often
called constraints):
Constraint 1: each week, no more than 100 hours of
finishing time may be used.
Constraint 2: each week, no more than 80 hours of
carpentry time may be used.
Constraint 3: because of limited demand, at most 40
soldiers should be produced.

18
• The three constraints can be expressed in terms of
the decision variables X1 and X2 as follows:
Constraint 1: 2 X1 + X2  100
Constraint 2: X1 + X2  80
Constraint 3: X1  40
Note:
The coefficients of the decision variables in the
constraints are called technological coefficients. This
is because its often reflect the technology used to
produce different products. The number on the
right-hand side of each constraint is called Right-
Hand Side (RHS). The RHS often represents the
quantity of a resource that is available.

19
• Sign restrictions: to complete the formulation of the
LP problem, the following question must be
answered for each decision variable: can the
decision variable only assume nonnegative values,
or it is allowed to assume both negative and positive
values?
If a decision variable Xi can only assume a nonnegative
values, we add the sign restriction (called
nonnegativity constraints)
Xi  0.
If a variable Xi can assume both positive and negative
values (or zero), we say that Xi is unrestricted in
sign (urs).
In our example the two variables are restricted in sign,
i.e., X1  0 and X2  0

20
• Combining the nonnegativity constraints with the
objective function and the structural constraints yield
the following optimization model (usually called LP
model):
Max Z = 3 X1 + 2 X2 (objective function)
subject to (st)
2 X1 + X2  100 (finishing constraint)
X1 + X2  80 (carpentry constraint)
X1  40 (soldier demand constraint)
X1  0 and X2  0 (nonnegativity constraint)
The optimal solution to this problem is :
X1 = 20, and X2 = 60, Z = 180

TYPES OF FORMULATION PROBLEMS
 A Product Mix Example
 A Diet Example
 An Investment Example
 A Marketing Example
 A Transportation Example
 A Blend Example
 A Multi-Period Scheduling Example
 A Data Envelopment Analysis Example
21
LPP - Formulation

Q2
Suppose that a farmer has a piece of farm land, say A square kilometers
large, to be planted with either wheat or barley or some combination of
the two. The farmer has a limited permissible amount F of fertilizer and P
of insecticide which can be used, each of which is required in different
amounts per unit area for wheat (F1, P1) and barley (F2, P2). Let S1 be
the selling price of wheat, and S2 the price of barley. If we denote the area
planted with wheat and barley with x1 and x2 respectively, then the
optimal number of square kilometers to plant with wheat vs. barley can
be expressed as a linear programming problem.
22
LPP - Formulation

Q2
Solution-
Maximize, Z = S1x1 + S2x2 (maximize the revenue )
subject to x1 +x2 < A (limit on total area)
F1x1 + F2x2 < F (limit on fertilizer)
P1x2 + P2x2 < P (limit on insecticide)
x1 , x2 > 0 (cannot plant a negative area)
23
LPP - Formulation

Q3
Cycle Trends is introducing two new lightweight bicycle frames, the
Deluxe and the Professional, to be made from aluminum and steel alloys.
The anticipated unit profits are Rs10 for the Deluxe and Rs15 for the
Professional. The number of kg of each alloy needed per frame is
summarized below. A supplier delivers 100 kg of the aluminum alloy and
80 kg of the steel alloy weekly. How many Deluxe and Professional frames
should Cycle Trends produce each week?
Kg of each alloy needed per frame
24
LPP - Formulation
Aluminum Alloy Steel Alloy
Deluxe 2 3
Professional 2 4

Q3
Solution-
Max Z = 10 x + 15 y
Subject To
2 x + 2 y <= 100 (aluminum constraint)
3 x + 4 y <= 80 (steel constraint)
x , y > 0 (non-negativity constraints)
25
LPP - Formulation

Q4
An animal feed company must produce exactly 200 kg of a mixture
consisting of ingredients G1 and G2. The ingredient G1 costs Rs.3 per kg
and G2 costs Rs.5 per kg. Not more than 80 kg of G1 can be used and
atleast 60 kg of G2 must be used. Find the minimum cost mixture.
Formulate this as a linear programming model.
26
LPP - Formulation

Q4
Solution-
Let x1 = No. of units (in Kg) of ingredient G1.
& x2 = No. of units (in Kg) of ingredient G2.
Objective function is, Minimize, Z = 3x1 + 5x2
Subject to constraints,
x1 + x2 = 200
x1 ≤ 80
x2 ≥ 60
x1, x2 ≥ 0 27
LPP - Formulation

Q5
A company supplying three types of parts to an automatic manufacturing company,
purchases castings of three parts from a nearby foundry and performs three types
of operators before selling these and cost per hour of these machines is given in the
table below:
The cost of the castings for A, Rs 120, for B Rs 200, for C Rs 400 and the selling
price of these parts is Rs 200, Rs 350 and Rs 500 respectively. All the parts that are
processed by the company can be sold, What quantity of various parts should the
company process for selling in order to maximize their profits? 28
LPP - Formulation
Machine Capacity/hour Capacity/hour Capacity/hour Cost/hour (Rs.)
A B C
Cutting 20 60 25 150
Drilling 40 20 40 100
Polishing 50 50 20 200

Q5
Solution-
Let x , y, z be the number of parts the company should process.
Profit from part A = 200 – 120 – cost of (Cutting+ Drilling+ Polishing)
= 200 – 120 – [(150/20) + (100/40) + (200/50)]
= 200 – 120 – 14 = Rs 66
Profit from part B = Rs 118.5
Profit from part C = Rs 81.5
29
LPP - Formulation

Q5
Solution-
Max,
Z = 66 x + 118.5 y + 81.5 z
Subject to:
Cutting machine constraint; (x/20) + (y/60) + (z/25) ≤ 1
Drilling machine constraint; (x/40) + (y/20) + (z/40) ≤ 1
Polishing machine constraint; (x/50) + (y/50) + (z/20) ≤ 1
x, y, z ≥ 0
30
LPP - Formulation

Q6
The Sky shop promotes its products from a large city to different parts in the state.
The Sky shop has budgeted up to Rs. 8,000 per week for local advertising. The
money is to be allocated among four promotional media: TV spots, newspaper ads,
and two types of radio advertisements. Sky shop’s goal is to reach the largest
possible high-potential audience through the various media. The following table
presents the number of potential customers reached by making use of an
advertisement in each of the four media. It also provides the cost per
advertisement placed and the maximum number of ads that can be purchased per
week.
31
LPP - Formulation

Q6
Sky shop’s contractual arrangements require that at least five radio spots be placed
each week. To ensure a broad-scoped promotional campaign, management also
insists that no more than Rs.1,800 be spent on radio advertising every week.
Formulate this problem as a LPP.
32
LPP - Formulation

Q6
Solution-
Let
X1 = number of 1-minute TV spots taken each week.
X2 = number of full-page daily newspaper ads taken each week
X3 = number of 30-second prime-time radio spots taken each week
X4 = number of 1-minute afternoon radio spots taken each week
Objective: Maximize audience coverage
33
LPP - Formulation

Q6
Solution-
Objective: Maximize audience coverage,
Z = 5,000X1 + 8,500X2 + 2,400X3 + 2,800X4
Subject to:
X1 <= 12
X2 <= 05
X3 <= 25
X4 <= 20
800X1 + 925X2 + 290X3 + 380X4 <= 8000
X3 + X4 >= 5
290X3 + 380X4 <= 1800
X1, X2, X3, X4 >= 0
34
LPP - Formulation

Operations Research- LPP formulation with examples

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