Thesis on Linear Programming1

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“Thesis on Linear Programming”
INTRODUCTION OF LINEAR PROGRAMMING:
LINEAR PROGRAMMING, a pacific class of mathematical problems, in which a linear function
is maximized (or minimized) subject to given linear constraints. This problem class is broad
enough to encompass many interesting and important applications, yet specific enough to be
tractable even if the number of variables is large. Linear programming theory falls within convex
optimization theory and is also considered to be an important part of operations research. Linear
programming is extensively used in business and economics, but may also be used to solve
certain engineering problems.
Linear programming, sometimes known as linear optimization, is the problem of maximizing or
minimizing a linear function over a convex polyhedron specified by linear and non-negativity
constraints. Simplistically, linear programming is the optimization of an outcome based on some
set of constraints using a linear mathematical mode
Linear Programming (LP) is a mathematical procedure for determining optimal allocation of
scarce resources. LP is a procedure that has found practical application in almost all facets of
business, from advertising to production planning. Transportation, distribution, and aggregate
production planning problems are the most typical objects of LP analysis.
Linear programming deals with a class of programming problems where both the objective
function to be optimized is linear and all relations among the variables corresponding to
resources are linear. Rarely has a new mathematical technique found such a wide range of
practical business, commerce, and industrial applications and simultaneously received so
thorough a theoretical development, in such a short period of time. Today, this theory is being
successfully applied to problems of capital budgeting, design of diets, conservation of resources,
games of strategy, economic growth prediction, and transportation systems. In very recent times,
linear programming theory has also helped resolve and unify many outstanding applications.
Function: A function is a thing that does something. For example, a coffee grinding machine is a
function that transforms the coffee beans into powder. The (objective) function maps and
translates the input domain (called the feasible region) into output range, with the two end-values
called the maximum and the minimum values.
When we formulate a decision-making problem as a linear program, we must check the
following conditions:

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The objective function must be linear. That is, check if all variables have power of 1 and they are
added or subtracted (not divided or multiplied)
The objective must be either maximization or minimization of a linear function. The objective
must represent the goal of the decision-maker
The constraints must also be linear. Moreover, the constraint must be of the following forms (,
, or =, that is, the LP-constraints are always closed).
Constraints: Element, factor or subsystem that works as a bottleneck. It restricts an entity,
project or system (such as a manufacturing or decision making process) from achieving its
potential (or higher level of output) with reference to its goal.
Linear Programming Assumptions:
In linear equations, each decision variable is to be multiplied by a constant coefficient with no
multiplying between those decision variables and no nonlinear functions such as logarithms.
 Proportionality - The change in a variable results in a proportionate changes in which that
variable's contribution to the value of the function.
 Additively - The function values that are the sum of the contributions of each term.
 Divisibility - The decision variables that can be divided into the non-integer values, taking on
the fractional values. Integer programming techniques that can be used if the divisibility
assumption that does not hold.
History of linear programming:
Linear programming was developed as a discipline in the 1940's, motivated initially by the need
to solve complex planning problems in wartime operations. Its development accelerated rapidly
in the postwar period as many industries found valuable uses for linear programming. The
founders of the subject are generally regarded as George B. Dantzig, is devised the simple
method in 1947, and John von Neumann, established the theory of duality that same year. The
Nobel Prize in economics was awarded in 1975 to the mathematician Leonid Kantorovich
(USSR) and the economist Tjalling Koopmans (USA) for their contributions to the theory of
optimal allocation of resources, in which linear programming played a key role. Many industries
use linear programming as a standard tool, e.g. to allocate a finite set of resources in an optimal
way. Examples of important application areas include airline crew scheduling, shipping or
telecommunication networks, oil refining and blending, and stock and bond portfolio selection

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Linear programming, sometimes known as linear optimization. George B. Dantzig is the founder
of the simplex method of linear programming, but it was kept secret and was not published until
1947 since it was being used as a war-time strategy. But once it was released, many industries
also found the method to be highly valuable. Another person who played a key role in the
development of linear programming is John von Neumann, who developed the theory of the
duality and Leonid Kantorovich, a Russian mathematician who used similar techniques in
economics before Dantzig and won the Nobel prize in 1975 in economics.
Figure 1: George.B.Dantzig
Dantzig's original example of finding the best assignment of 70 people to 70 jobs emphasizes the
practicality of linear programming. The computing power required to test all possible
combinations to select the best assignment is quite large. However, it takes only a moment to
find the optimum solution by modeling problem as a linear program and applying the simplex
algorithm. The theory behind linear programming is to drastically reduce the number of possible
optimal solutions that must be checked. Its development accelerated rapidly in the postwar
period as many industries found valuable uses for linear programming. The founders of the
subject are generally regarded as George B. Dantzig, who devised the simplex method in 1947,
and John von Neumann, who established the theory of duality that same year. The Nobel Prize in
economics was awarded in 1975 to the mathematician Leonid Kantorovich (USSR) and the
economist Tjalling Koopmans (USA) for their contributions to the theory of optimal allocation
of resources, in which linear programming played a key role. Many industries use linear
programming as a standard tool, e.g. to allocate a finite set of resources in an optimal way.
Examples of important application areas include airline crew scheduling, shipping or
telecommunication networks, oil refining and blending, and stock and bond portfolio selection.

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A short history of Linear Programming:
1. In 1762, Lagrange solved tractable optimization problems with simple
equality constraints.
2. In 1820, Gauss solved linear system of equations by what is now call
Causssian elimination. In 1866 Wilhelm Jordan refinmened the method to
finding least squared errors as ameasure of goodness-of-fit. Now it is
referred to as the Gauss-Jordan Method.
3. In 1945, Digital computer emerged.
4. In 1947, Dantzig invented the Simplex Methods.
5. In 1968, Fiacco and McCormick introduced the Interior Point Method.
6. In 1984, Karmarkar applied the Interior Method to solve Linear Programs
adding his innovative analysis.
Advantages of Linear Programming
 The linear programming technique helps to make the best possible use of
available productive resources (such as time, labour, machines etc.)
 In a production process, bottle necks may occur. For example, in a factory some
machines may be in great demand while others may lie idle for some time. A
significant advantage of linear programming is highlighting of such bottle necks.
 The linear programming technique helps to make the best possible use of
available productive resources (such as time, labour, machines etc.)
 In a production process, bottle necks may occur. For example, in a factory some
machines may be in great demand while others may lie idle for some time. A
significant advantage of linear programming is highlighting of such bottle necks.
 The quality of decision making is improved by this technique because the
decisions are made objectively and not subjectively.
 By using this technique, wastage of resources like time and money may be
avoided.
Disadvantages of Linear Programming
 Linear programming is applicable only to problems where the constraints and objective
function are linear i.e., where they can be expressed as equations which represent straight
lines. In real life situations, when constraints or objective functions are not linear, this
technique cannot be used.
 2. Factors such as uncertainty, weather conditions etc. are not taken into consideration.

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 Linear programming is applicable only to problems where the constraints and objective
function are linear i.e., where they can be expressed as equations which represent straight
lines. In real life situations, when constraints or objective functions are not linear, this
technique cannot be used.
 Factors such as uncertainty, weather conditions etc. are not taken into consideration.
 There may not be an integer as the solution, e.g., the number of men required may be a
fraction and the nearest integer may not be the optimal solution.
i.e., Linear programming technique may give practical valued answer which is not
desirable.
 Only one single objective is dealt with while in real life situations, problems come with
multi-objectives.
 Parameters are assumed to be constants but in reality they may not be so.
Linear Programming Applications:
It is necessary to know the applications of linear programming. Goods have to be transported
from sources (like factories) to destinations (like warehouses) on a regular basis. The
transportation problem deals with minimizing the costs in doing so. Linear programming
effectively deals with this problem.
Crew Scheduling:
An airline has to assign crews to its flights. Divisibility
• We make sure that each flight is covered.
• We Meet regulations, eg: each pilot can only fly a certain amount each day.
•Then Minimize costs, eg: accommodation for crews staying overnight out of town, crews
deadheading.
• Would like a robust schedule. The airlines run on small profit margins, so saving a few percent
through a good scheduling can make an enormous difference in terms of the profitability.
They also use linear programming for the yield management.

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Telecommunications:
Call routing:
Many telephone calls from the Kushtia to Chittagong, from Jessore to Gazipur etc. These calls
should be routed through the telephone network
Network design:
If we need to build extra capacity, we should concentrate on some Links and build a new
switching Station.
Internet traffic: For example, there was a great deal of construction
of new networks for carrying internet traffic a few years ago.
Military Applications
To provide the required protection at the minimum cost, linear programming is used. This
technique is useful to cause maximum damage to the enemy with minimum fuel/cost.
Operation of System of Dams
Linear programming is used to find the variations in water storage of dams which generate
power, thus maximizing the energy got from the entire system.
Personnel Assignment Problem
If we are given the number of persons, number of jobs and the expected productivity of a
particular person on a particular job, linear programming is used to maximize the average
productivity of a person.
A Diet Problem:
This type of problem usually involves the mixing of raw materials or other ingredients to obtain
an end product that has certain characteristics. For instance, food processors and dieticians
generally are concerned with meeting dietary needs in food products. There may be specific
recruitments pertaining to nutrients calories sodium content and so on. The general question to

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be answered by linear programming is- “What mix of inputs (e.g. different food types) will
achieve the desired result for the least”. For example:
Suppose the only foods available in your local store are potatoes and steak. The
decision about how much of each food to buy is to made entirely on dietary and
economic considerations. We have the nutritional and cost information in the
following table:
Raw Materials
Per unit
of potatoes
Per unit
of steak
Minimum
requirements
Units of carbohydrates 3 1 8
Units of vitamins 4 3 19
Units of proteins 1 3 7
Unit cost 25 50
The problem is to find a diet (a choice of the numbers of units of the two foods)
that meets all minimum nutritional requirements at minimal cost.
a. Formulate the problem in terms of linear inequalities and an objective
function.
b. Solve the problem geometrically.
c. Explain how the 2:1 cost ratio (steak to potatoes) dictates that the solution
must be where you said it is.
d. Find a cost ratio that would move the optimal solution to a different choice
of numbers of food units, but that would still require buying both steak
and potatoes.
e. Find a cost ratio that would dictate buying only one of the two foods in
order to minimize cost.
a) We begin by setting the constraints for the problem. The first constraint
represents the minimum requirement for carbohydrates, which is 8 units per some
unknown amount of time. 3 units can be consumed per unit of potatoes and 1 unit
can be consumed per unit of steak. The second constraint represents the minimum

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requirement for vitamins, which is 19 units. 4 units can be consumed per unit of
potatoes and 3 units can be consumed per unit of steak. The third constraint
represents the minimum requirement for proteins, which is 7 units. 1 unit can be
consumed per unit of potatoes and 3 units can be consumed per unit of steak. The
fourth and fifth constraints represent the fact that all feasible solutions must be
nonnegative because we can't buy negative quantities.
Constraints:
{3X1 + X2  8, 4X1+ 3X2  19, X1+ 3X2  7, X1 0, X2  0};
Next we plot the solution set of the inequalities to produce a feasible region of
possibilities.
c) The 2:1 cost ratio of steak to potatoes dictates that the solution must be here
since, as a whole, we can see that one unit of steak is slightly less nutritious than
one unit of potatoes. Plus, in the one category where steak beats potatoes in
healthiness (proteins), only 7 total units are necessary. Thus it is easier to fulfill
these units without buying a significant amout of steak. Since steak is more
expensive, buying more potatoes to fulfill these nutritional requirements is more
logical.
d) Now we choose a new cost ratio that will move the optimal solution to a
different choice of numbers of food units. Both steak and potatoes will still be
purchased, but a different solution will be found. Let's try a 5:2 cost ratio.
Thus, the optimal solution for this cost ratio is buying 8 steaks and no potatoes per
unit time to meet the minimum nutritional requirements.
A Blending Problem:
Blending problems are very similar to diet problems. In fact, Diet and blending could be lumped into the
same category. Strictly speaking , thought, blending problems have an additional requirement : To
achieve a mix that has a specific consistency. For example:
Bryant's Pizza, Inc. is a producer of frozen pizza products. The company makes a net income of
$1.00 for each regular pizza and $1.50 for each deluxe pizza produced. The firm currently has
150 pounds of dough mix and 50 pounds of topping mix. Each regular pizza uses 1 pound of
dough mix and 4 ounces (16 ounces= 1 pound) of topping mix. Each deluxe pizza uses 1 pound

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of dough mix and 8 ounces of topping mix. Based on the past demand per week, Bryant can sell
at least 50 regular pizzas and at least 25 deluxe pizzas. The problem is to determine the number of
regular and deluxe pizzas the company should make to maximize net income. Formulate this
problem as an LP problem.
Let X1 and X2 be the number of regular and deluxe pizza, then the LP formulation is:
Maximize X1 + 1.5 X2
Subject to:
X1 + X2  150
0.25 X1 + 0.5 X2  50
X1  50
X2  25
X1  0, X2  0
Other Common Applications of LP
Linear programming is a powerful tool for selecting alternatives in a decision problem
and, consequently, has been applied in a wide variety of problem settings. We will
indicate a few applications covering the major functional areas of a business organization.
Finance: The problem of the investor could be a portfolio-mix selection problem. In
general, the number of different portfolios can be much larger than the example indicates,
more and different kinds of constraints can be added. Another decision problem involves
determining the mix of funding for a number of products when more than one method of
financing is available. The objective may be to maximize total profits, where the profit
for a given product depends on the method of financing. For example, funding may be
done with internal funds, short-term debt, or intermediate financing (amortized loans).
There may be limits on the availability of each of the funding options as well as financial
constraints requiring certain relationships between the funding options so as to satisfy the
terms of bank loans or intermediate financing. There may also be limits on the production
capacity for the products. The decision variables would be the number of units of each
product to be financed by each funding option.
Production and Operations Management: Quite often in the process industries a given
raw material can be made into a wide variety of products. For example, in the oil
industry, crude oil is refined into gasoline, kerosene, home-heating oil, and various
grades of engine oil. Given the present profit margin on each product, the problem is to
determine the quantities of each product that should be produced. The decision is subject
to numerous restrictions such as limits on the capacities of various refining operations,
raw-material availability, demands for each product, and any government-imposed
policies on the output of certain products. Similar problems also exist in the chemical and
food-processing industries.

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Human Resources: Personnel planning problems can also be analyzed with linear
programming. For example, in the telephone industry, demands for the services of
installer-repair personnel are seasonal. The problem is to determine the number of
installer-repair personnel and line-repair personnel to have on the work force each month
where the total costs of hiring, layoff, overtime, and regular-time wages are minimized.
The constraints set includes restrictions on the service demands that must be satisfied,
overtime usage, union agreements, and the availability of skilled people for hire. This
example runs contrary to the assumption of divisibility; however, the work-force levels
for each month would normally be large enough that rounding to the closest integer in
each case would not be detrimental, provided the constraints are not violated.
Marketing: Linear programming can be used to determine the proper mix of media to
use in an advertising campaign. Suppose that the available media are radio, television,
and newspapers. The problem is to determine how many advertisements to place in each
medium. Of course, the cost of placing an advertisement depends on the medium chosen.
We wish to minimize the total cost of the advertising campaign, subject to a series of
constraints. Since each medium may provide a different degree of exposure of the target
population, there may be a lower bound on the total exposure from the campaign. Also,
each medium may have a different efficiency rating in producing desirable results; there
may thus be a lower bound on efficiency. In addition, there may be limits on the
availability of each medium for advertising.
Distribution: Another application of linear programming is in the area of distribution.
Consider a case in which there are m factories that must ship goods to n warehouses. A
given factory could make shipments to any number of warehouses. Given the cost to ship
one unit of product from each factory to each warehouse, the problem is to determine the
shipping pattern (number of units that each factory ships to each warehouse) that
minimizes total costs. This decision is subject to the restrictions that demand at each
factory cannot ship more products than it has the capacity to produce.
Different Types (Methods) of Linear Programming
Some of the methods of linear programming are:
 The Graphical Method
 The Analytical Method
 The Simplex Method
 The Synthetic Method
The Graphical Method:
In the graphical method, the constraints are actually drawn as straight lines and the optimal
solution is found. This method is explained in detail later.

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 So far we have learnt how to construct a mathematical model for a linear programming
problem. If we can find the values of the decision variables x1, x2, x3 ... xn, which can
optimize (maximize or minimize) the objective function Z, then we say that these values
of xi are the optimal solution of the LPP.
The graphical method is applicable to solve the LPP involving two decision variables x1, and x2,
we usually take these decision variables as x, y instead of x1, x2. To solve an LPP, the graphical
method includes two major steps.
a) The determination of the solution space that defines the feasible solution (Note that the set of
values of the variable x1, x2, x3,....xn which satisfy all the constraints and also the non-negative
conditions is called the feasible solution of the LPP)
b) The determination of the optimal solution from the feasible region.
a) To determine the feasible solution of an LPP, we have the following steps.
Step 1:
Since the two decision variable x and y are non-negative, consider only the first quadrant of xy-
plane
Draw the line ax + by = c ... (1)
For each constraint,
the line (1) divides the first quadrant in to two regions say R1 and R2, suppose (x1, 0) is a point in
R1. If this point satisfies the in equation ax + by £ c or (³ c), then shade the region R1. If (x1, 0)
does not satisfy the in equation, shade the region R2.
Step 3:
Corresponding to each constant, we obtain a shaded region. The intersection of all these shaded
regions is the feasible region or feasible solution of the LPP.
Let us find the feasible solution for the problem of a decorative item dealer whose LPP is to
maximise profit function.

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Z = 50x + 18y ... (1)
Subject to the constraints
Step 1:
Since x 0, y 0, we consider only the first quadrant of the xy - plane
Step 2:
We draw straight lines for the equation
2x+ y = 100 ... (2)
x + y = 80
To determine two points on the straight line 2x + y = 100
Put y = 0, 2x = 100
x = 50
(50, 0) is a point on the line (2)
Put x = 0 in (2), y =100
(0, 100) is the other point on the line (2)
Plotting these two points on the graph paper draw the line which represent the line 2x + y =100.

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This line divides the 1st
quadrant into two regions, say R1 and R2. Choose a point say (1, 0) in R1.
(1, 0) satisfy the in equation 2x + y 100. Therefore R1 is the required region for the constraint
2x + y 100.
Similarly draw the straight line x + y = 80 by joining the point (0, 80) and (80, 0). Find the
required region say R1', for the constraint x + y 80.
The intersection of both the region R1 and R1' is the feasible solution of the LPP. Therefore every
point in the shaded region OABC is a feasible solution of the LPP, since this point satisfies all
the constraints including the non-negative constraints.
b) There are two techniques to find the optimal solution of an LPP.
Corner Point Method
The optimal solution to a LPP, if it exists, occurs at the corners of the feasible region.
The method includes the following steps
Step 1:
Find the feasible region of the LLP.
Step 2:
Find the co-ordinates of each vertex of the feasible region.

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These co-ordinates can be obtained from the graph or by solving the equation of the lines.
Step 3:
At each vertex (corner point) compute the value of the objective function.
Step 4:
Identify the corner point at which the value of the objective function is maximum (or minimum
depending on the LPP)
The co-ordinates of this vertex is the optimal solution and the value of Z is the optimal value
Example:
Find the optimal solution in the above problem of decorative item dealer whose objective
function is Z = 50x + 18y.
In the graph, the corners of the feasible region are
O (0, 0), A (0, 80), B(20, 60), C(50, 0)
At (0, 0) Z = 0
At (0, 80) Z = 50 (0) + 18(80)
= Rs. 1440
At (20, 60), Z = 50 (20) +18 (60)
= 1000 + 1080 = Rs.2080
At (50, 0) Z = 50 (50 )+ 18 (0)
= Rs. 2500.
Since our object is to maximize Z and Z has maximum at (50, 0) the optimal solution is x = 50
and y = 0.
The optimal value is Rs. 2500.

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If an LPP has many constraints, then it may be long and tedious to find all the corners of the
feasible region. There is another alternate and more general method to find the optimal solution
of an LPP, known as 'ISO profit or ISO cost method'
ISO- PROFIT (OR ISO-COST)
Method of Solving Linear Programming Problems
Suppose the LPP is to
Optimize Z = ax + by subject to the constraints
This method of optimization involves the following method.
Step 1:
Draw the half planes of all the constraints
Step 2:
Shade the intersection of all the half planes which is the feasible region.
Step 3:
Since the objective function is Z = ax + by, draw a dotted line for the equation ax + by = k,
where k is any constant. Sometimes it is convenient to take k as the LCM of a and b.
Step 4:
To maximize Z draw a line parallel to ax + by = k and farthest from the origin. This line should
contain at least one point of the feasible region. Find the coordinates of this point by solving the
equations of the lines on which it lies.

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To minimize Z draw a line parallel to ax + by = k and nearest to the origin. This
line should contain at least one point of the feasible region. Find the co-ordinates
of this point by solving the equation of the line on which it lies.
Step 5:
If (x1, y1) is the point found in step 4, then
x = x1, y = y1, is the optimal solution of the LPP and
Z = ax1 + by1 is the optimal value.
The above method of solving an LPP is more clear with the following example.
Example:
Solve the following LPP graphically using ISO- profit method.
Maximize Z =100 + 100y.
Subject to the constraints
Suggested answer:
since x 0, y 0, consider only the first quadrant of the plane graph the following straight lines
on a graph paper
10x + 5y = 80 or 2x+y =16
6x + 6y = 66 or x + y =11

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4x+ 8y = 24 or x+ 2y = 6
5x + 6y = 90
Identify all the half planes of the constraints. The intersection of all these half planes is the
feasible region as shown in the figure.
Give a constant value 600 to Z in the objective function, then we have an equation of the line
120x + 100y = 600 ...(1)
or 6x + 5y = 30 (Dividing both sides by 20)
P1Q1 is the line corresponding to the equation 6x + 5y = 30. We give a constant 1200 to Z, then
the P2Q2 represents the line.
120x + 100y = 1200
6x + 5y = 60

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P2Q2 is a line parallel to P1Q1 and has one point 'M' which belongs to feasible region and farthest
from the origin. If we take any line P3Q3 parallel to P2Q2 away from the origin, it does not touch
any point of the feasible region.
The co-ordinates of the point M can be obtained by solving the equation 2x + y = 16
x + y =11 which give
x = 5 and y = 6
The optimal solution for the objective function is x = 5 and y = 6
The optimal value of Z
120 (5) + 100 (6) = 600 + 600
= 1200
Analytical Method:
The graphical method is not applicable to linear programming problems with more than 2
variables. Then, the analytical method may be used. Here, equations are solved by assuming
some variables to be zero. However, this method is tedious and time consuming.
The meaning of the word analysis is to “separate things that are together.” In this method
we start from what is to be found or proved. Thorndike says that,” Analysis is the highest
intellectual performance of the mind.” Analysis also means, “Breaking up of a given
problem, so that it connects with what is already known.” In analysis we proceed from,”
Unknown to Known.” Analysis is,” Unfolding of a problem to find its hidden aspect.”
_ This method is used under the given conditions:
_ when we have to prove any theorem.
_ Can be used for construction problems.
_ To find out solutions of new arithmetical problems.
Merits/Adv of this method are as follows:
_ Logical, leaves no doubt.
_ Facilitates understanding, as we discover facts.
_ Each step has reason and justification.

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_ Student gains confidence and understanding.
_ Method suits the learner and the subject.
Demerits of this method are:
_ Lengthy method and also time consuming.
_ Difficult to acquire efficiency and speed.
_ Not applicable to all topics.
_ Not suitable for students with weak conceptual knowledge.
The Simplex Method:
The simplex method overcomes these difficulties and gives successive solutions that improve
progressively to give the optimal solution.
Simplex method is the most general and powerful method among all available methods for
solving a linear programming problem. (LPP). The simplex computational procedure is as
follows:
Step 1: Formulate the given problem as standard maximization LPP
Step 2 : Select an initial basic feasible solution to initiate the iterations.
Step 3 : Check the objective function to see if there is some non-basic variable that would
improve the objective function, if brought into the basic solution. If such a variable exists, go to
the next step. Otherwise stop.
Step 4 : Test the given solution for optimality.
Step 5 : Continue the iterations until either an optimum solution is obtained or there is an
indication that an unbounded solution exists.
The Synthetic Method: The word synthesis simply means,” To place things together or to
join separate parts.” In this method we proceed from “known to unknown.”
It is the process of relating known bits of data to a point where the unknown
becomes true. It is the method of formulation, recording and presenting concisely
the solution without any trial and errors.
_ Merits/Adv of this method are as follows:
_ Short and precise method.
_ Saves time and labour.
_ Suits the needs of majority of the students.
_ Can be applied to a majority of topics in mathematics.
_ Omits trial and error as in analysis method.
_ Demerits of this method are:
_ Teacher–centered method, students are passive listeners.

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_ Students rely on rote memory.
_ No opportunity to develop the skills of thinking and reasoning, as
understanding is hampered.
_ Students lack confidence to do other type of sums.
From the above discussion we can see that both the methods of analysis and synthesis by
themselves have their advantages and disadvantages. In order to ensure the complete
understanding of mathematics in the learners that both the methods be used together to
teach mathematics. By using a combination of these two methods the teacher can ensure
that effecting teaching learning takes place.
Solving Process of Linear Programming Problem:
There are mainly four steps in the mathematical formulation of linear programming problem as a
mathematical model. We will discuss formulation of those problems which involve only two
variables.
 Identify the decision variables and assign symbols x and y to them. These decision variables
are those quantities whose values we wish to determine.
 Identify the set of constraints and express them as linear equations/in equations in terms of
the decision variables. These constraints are the given conditions.
 Identify the objective function and express it as a linear function of decision variables. It
might take the form of maximizing profit or production or minimizing cost.
 Add the non-negativity restrictions on the decision variables, as in the physical problems,
negative values of decision variables have no valid interpretation.
There are many real life situations where an LPP may be formulated. The following examples
will help to explain the mathematical formulation of an LPP.
A diet is to contain at least 4000 units of carbohydrates, 500 units of fat and 300 units of protein.
Two foods A and B are available. Food A costs 2 dollars per unit and food B costs 4 dollars per
unit. A unit of food A contains 10 units of carbohydrates, 20 units of fat and 15 units of protein.
A unit of food B contains 25 units of carbohydrates, 10 units of fat and 20 units of protein.
Formulate the problem as an LPP so as to find the minimum cost for a diet that consists of a
mixture of these two foods and also meets the minimum requirements.

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Suggested answer:
The above information can be represented as
Let the diet contain x units of A and y units of B.
Total cost = 2x + 4y
The LPP formulated for the given diet problem is
Minimize Z = 2x + 4y
subject to the constraints
In the production of 2 types of toys, a factory uses 3 machines A, B and C. The time required to
produce the first type of toy is 6 hours, 8 hours and 12 hours in machines A, B and C
respectively. The time required to make the second type of toy is 8 hours, 4 hours and 4 hours in
machines A, B and C respectively. The maximum available time (in hours) for the machines A,
B, C are 380, 300 and 404 respectively. The profit on the first type of toy is 5 dollars while that
21
Suggested answer:
21
Suggested answer:

22
on the second type of toy is 3 dollars. Find the number of toys of each type that should be
produced to get maximum profit.
Suggested answer:
Mathematical Formulation
The data given in the problem can be represented in a table as follows.
Let x = number of toys of type-I to be produced
y = number of toys of the type - II to be produced
Total profit = 5x + 3y
The LPP formulated for the given problem is:
Maximize Z = 5x + 3y subject to the constraints
22
Suggested answer:
22
Suggested answer:

23
Dual Problem: Construction and Its Meaning
Associated with each (primal) LP problem is a companion problem called the dual. The
following classification of the decision variable constraints is useful and easy to
remember in construction of the dual.
The Dual Problem Construction
Objective: Max (e.g. Profit)
Constraint types:
£ a Sensible constraint
= a Restricted constraint
³ an Unusual const.
Variables types:
 0 a Sensible condition
... un-Restricted in sign
 0 an Unusual
condition
Objective: Min (e.g. Cost)
Constraint types
³ a Sensible constraint
= a Restricted const.
£ an Unusual const.
---------------------------------------------------------------------------

24
A one-to-one correspondence between the constraint type and the variable type exists
using this classification of constraints and variables for both the primal and the dual
problems.
Dual Problem Construction:
- If the primal is a maximization problem, then its dual is a minimization problem (and
vice versa).
- Use the variable type of one problem to find the constraint type of the other problem.
- Use the constraint type of one problem to find the variable type of the other problem.
- The RHS elements of one problem become the objective function coefficients of the
other problem (and vice versa).
- The matrix coefficients of the constraints of one problem is the transpose of the matrix
coefficients of the constraints for the other problem. That is, rows of the matrix becomes
columns and vice versa.
You may check your dual constructions rules by using your WinQSB package.
These results imply the only possible combinations of primal and dual properties
as shown in the following table:
Possible Combinations of Primal and Dual Properties
Primal Problem Condition Implies Dual Problem
Feasible; bounded objective  Feasible; bounded objective
Feasible; unbounded objective  Infeasible
Infeasible  Feasible; unbounded objective
Infeasible  Infeasible
Multiple solutions  Degenerate solution
Degenerate solution  Multiple solutions
Agriculture:
The active production of useful plants or animals in ecosystems that have been created by
people. Agriculture may include cultivating the soil, growing and harvesting crops, and raising
livestock.

25
Agriculture is the art and science of crop and livestock production. In its broadest sense,
agriculture comprises the entire range of technologies associated with the production of useful
products from plants and animals, including soil cultivation, crop and livestock management, and
the activities of processing and marketing.
Agriculture was independently developed in many places, including the Middle East, East Asia,
South Asia, and the Americas. The earliest evidence for agriculture has been found in the Middle
East and dates to between 14,500 and 12,000 BP. Early cultivars include wild barley (Middle
East), millet (China), and squash (the Americas). The domestication of many animals now
considered to be livestock occurred during roughly the same period, although dogs were
domesticated considerably earlier. Slash-and-burn land-clearing methods and crop rotation were
early agricultural techniques. Steady improvements in tools and methods over the centuries
increased agricultural output, as did mechanization, selective breeding and hybridization, and,
beginning in the 20th century, the use of herbicides and insecticides.
Category of Agriculture:
The primary agricultural products consist of crop plants for human food and animal feed and
livestock products. The crop plants can be divided into 10 categories:
1. Grain crops (wheat, for flour to make bread, many bakery products, and breakfast
cereals; rice, for food; maize, for livestock feed, syrup, meal, and oil; sorghum
grain, for livestock feed; and oats, barley, and rye, for food and livestock feed);
2. Food grain legumes (beans, peas, lima beans, and cowpeas, for food; and peanuts,
for food and oil);
3. Oil seed crops (soybeans, for oil and high-protein meal; and linseed, for oil and
high-protein meal);
4. Root and tuber crops (principally potatoes and sweet potatoes); sugar crops (sugar
beets and sugarcane);
5. Fiber crops (principally cotton, for fiber to make textiles and for seed to produce
oil and high-protein meal);
6. Tree and small fruits; nut crops; vegetables; and forages (for support of livestock
pastures and range grazing lands and for hay and silage crops).
7. The forages are dominated by a wide range of grasses and legumes, suited to
different conditions of soil and climate.

26
8. Livestock products include cattle, for beef, tallow, and hides; dairy cattle, for
milk, butter, cheese, ice cream, and other products;
9. Sheep, for mutton (lamb) and wool; pigs, for pork and lard;
10. Poultry (chiefly chickens but also turkeys and ducks) for meat and eggs; and
horses, primarily for recreation.
Factors of Agriculture:
Many different factors influence the kind of agriculture practiced in a particular area. Among
these are climate, soil, water availability, topography, nearness to markets, transportation
facilities, land costs, and general economic level. Climate, soil, water availability, and
topography vary widely throughout the world. This variation brings about a wide range in
agricultural production enterprises. Certain areas tend toward a specialized agriculture, whereas
other areas engage in a more diversified agriculture. As new technology is introduced and
adopted, environmental factors are less important in influencing agricultural production patterns.
Continued growth in the world's population makes critical the continuing ability of agriculture to
provide needed food and fiber.
Agriculture in Bangladesh:
Bangladesh has a primarily agrarian economy. Agriculture is the single largest producing sector
of the economy since it comprises about 30% of the country's GDP and employs around 60% of
the total labour force. The performance of this sector has an overwhelming impact on major
macroeconomic objectives like employment generation, poverty alleviation, human resources
development and food security.
Most Bangladeshis earn their living from agriculture. Although rice and jute are the primary
crops, wheat is assuming greater importance. Tea is grown in the northeast. Because of
Bangladesh's fertile soil and normally ample water supply, rice can be grown and harvested three
times a year in many areas. Due to a number of factors, Bangladesh's labor-intensive agriculture
has achieved steady increases in food grain production despite the often unfavorable weather
conditions. These include better flood control and irrigation, a generally more efficient use of
fertilizers, and the establishment of better distribution and rural credit networks. With 35.8
million metric tons produced in 2000, rice is Bangladesh's principal crop. National sales of the
classes of insecticide used on rice, including granular carbofuran, synthetic pyrethroids, and
Malathion exceeded 13,000 tons of formulated product in 2003.The insecticides not only
represent an environmental threat, but are a significant expenditure to poor rice farmers. The
Bangladesh Rice Research Institute is working with various NGOs and international
organizations to reduce insecticide use in rice.

27
In comparison to rice, wheat output in 1999 was 1.9 million metric tons. Population pressure
continues to place a severe burden on productive capacity, creating a food deficit, especially of
wheat. Foreign assistance and commercial imports fill the gap. Underemployment remains a
serious problem, and a growing concern for Bangladesh's agricultural sector will be its ability to
absorb additional manpower. Finding alternative sources of employment will continue to be a
daunting problem for future governments, particularly with the increasing numbers of landless
peasants who already account for about half the rural labor force.
Figure 2: Map showing the growing
areas of major agricultural products.
Food crops:
Although rice and jute are the primary crops, maize and vegetables are assuming greater
importance. Due to the expansion of irrigation networks, some wheat producers have switched to
cultivation of maize which is used mostly as poultry feed. Tea is grown in the northeast. Because
of Bangladesh's fertile soil and normally ample water supply, rice can be grown and harvested
three times a year in many areas. Due to a number of factors, Bangladesh's labor-intensive
agriculture has achieved steady increases in food grain production despite the often unfavorable
weather conditions. These include better flood control and irrigation, a generally more efficient
use of fertilizers, and the establishment of better distribution and rural credit networks. With 28.8
million metric tons produced in 2005-2006 (July–June), rice is Bangladesh's principal crop. By
comparison, wheat output in 2005-2006 was 9 million metric tons. Population pressure continues
to place a severe burden on productive capacity, creating a food deficit, especially of wheat.
Foreign assistance and commercial imports fill the gap. Underemployment remains a serious
problem, and a growing concern for Bangladesh's agricultural sector will be its ability to absorb
additional manpower. Finding alternative sources of employment will continue to be a daunting

28
problem for future governments, particularly with the increasing numbers of landless peasants
who already account for about half the rural labor force.
Bangladesh is the fourth largest rice producing country in the world. National sales of the classes
of insecticide used on rice, including granular carbofuran, synthetic pyrethroids, and Malathion
exceeded 13,000 tons of formulated products in 2003. The insecticides not only represent an
environmental threat, but are a significant expenditure to poor rice farmers. The Bangladesh
Rice Research Institute is working with various NGOs and international organizations to
reduce insecticide use in rice.
Wheat is not a traditional crop in Bangladesh, and in the late 1980s little was consumed in rural
areas. During the 1960s and early 1970s, however, it was the only commodity for which local
consumption increased because external food aid was most often provided in the form of wheat.
In the first half of the 1980s, domestic wheat production rose to more than 1 million tons per
year but was still only 7 to 9 percent of total food grain production. Record production of nearly
1.5 million tons was achieved in FY 1985, but the following year saw a decrease to just over 1
million tons. About half the wheat is grown on irrigated land. The proportion of land devoted to
wheat remained essentially unchanged between 1980 and 1986, at a little less than 6 percent of
total planted area.
Food grains are cultivated primarily for subsistence. Only a small percentage of total production
makes its way into commercial channels. Other Bangladeshi food crops, however, are grown
chiefly for the domestic market. They include potatoes and sweet potatoes, with a combined
record production of 1.9 million tons in FY 1984; oilseeds, with an annual average production of
250,000 tons; and fruits such as bananas, jackfruit, mangoes, and pineapples. Estimates of
sugarcane production put annual production at more than 7 million tons per year, most of it
processed into a coarse, unrefined sugar known as gur, and sold domestically.
Linear Programming in Agriculture:
Linear Programming (LP) offers significant advantages over Agriculture. There are a variety of
techniques available the most commonly used techniques have been; comparison of annual
returns using equivalent annuities, Intertemporal Linear Programming, and Dynamic
Programming.
Linear Programming (LP) offers significant advantages over other methods in terms of the
Information provided for analysis. Analysis of the dual solution provides Marginal Value of
Products which gives the value of adding or subtracting (Maximizing or Minimizing) one
additional unit.
Sensitivity analysis is also an option in most Linear Programming packages. This gives the range
Over which the optimal solution remains valid. Although LP is a powerful technique, it has a
number of features that may limit suitability in terms of accuracy and consistency. The
assumptions that must be valid for LP to be a viable technique and the impact on orchard
replacement decisions are

29
(a) Linear
A problem must be linear in both constraints and objective function for a linear programming
Approach to be valid. The property of linearity is usually discussed as two components
(I ) Additive
Use of resources and output must be the sum of activities. There cannot be any cross
Products between resources or products. There can be no economies of scale or synergistic
Effects for example nor can price be affected by increasing supply.
This assumption is fulfilled provided the size of the property is constrained to an individual
Orchards existing area. Individual varieties variable income and expenses can be added
together without any cross products.
Provided the size of the property is not increased, there are no additional capital costs needed
to handle increased crop. Overheads will remain constant at a similar scale of production.
An individual orchard can not affect market prices by increasing or decreasing supply. It is a
price taker.
(ii) Proportion (or Multiplication)
The proportionality constraint requires that an activity can be added to or subtracted from the
Objective functions at will without incurring any start up or close down costs.
This implies in particular that there are no increasing or decreasing returns to scale - i.e. one
Has of Braeburn will be as productive as 10 ha on a per hectare basis. Another area where this
Could be potentially limiting in this case is in packing facilities. The cost per unit can vary
Considerably with scale.
In practice these can be avoided as limits by using upper and lower bounds on these factors.
Area can be restricted to a maximum preferably close to the original orchard area. Packing
Capacity can be limited to a minimum and maximum through put that matches the realistic
Capacity and working requirements of the orchard.
(b) Continuous variables
In the case of pip fruit, it is possible to have fractions of a hectare.
(c) Certainty:
The model deals with events over time. As such the results are not certain. However,
While the result of the model is not perfect, some knowledge is better than none. If necessary
the model could be adapted to take risk into account through the MOTAD procedure.
Example:
Md: Mohidul Islam is a farmer who has 10 beghaes (locally called) of land in which he wants to
cultivate rice and/or wheat at the same season. Each begha of rice costs Tk.5, 000 (include
fertilizer, irrigation etc. costs), requires 8 man-days of the work and yields a profit of Tk.3, 500.
One begha of wheat costs Tk.3, 000 (include fertilizer, irrigation etc. costs), requires 12 man-

30
days of the work and yields a profit of Tk.2, 800. If the farmer has tk.45, 000 for growing and
can count on 160 man-days work, how many beghaes (locally called) should be allocated to each
crop to maximize total profit. To get the answer we can apply L.P. which is shown below:
For the sake of convenience, we tabulate the data in the following manner:
Products Decision variable Cost Man-days Profit
Rice R 5,000 8 3,500
Wheat W 3,000 12 2,,800
Available
capacity
45,000 160
Problem solve by using the graphical method
Objective Functions:
Z = 3,500R + 2,800W
Subject to constraints:
5,000R + 3,000W ≤ 45,000
8R + 12W ≤ 160
R + W ≤ 10
And R, W ≥ 0
The inequalities expressing constraints are converted into equalities by adding slack variable we
get,
5,000R + 3,000W + S1 = 45,000
Or, 5,000R + 3,000W + S1 +0S2 +0S3 = 45,000
8R + 12W + S2 = 160
Or, 8R + 12W + 0S1 + S2 +0S3= 160
R + W + S3 = 10

31
Or, R + W + 0S1 + 0S2 + S3 = 10
Now, Z= 3,500R + 2,800W + 0S1 + 0S2 + 0S3
By using the Simplex method we get:
Table-1
Solution
Mix
→
Cj↓
R W S1 S2 S3 R.H.S. Ratio
3,500 2,800 0 0 0
S1 0 5,000 3,000 1 0 0 45,000 9 ←
S2 0 8 12 0 1 0 160 20
S3 0 1 1 0 0 1 10 10
Zj 0 0 0 0 0 0
Cj-Zj 3,500
↑
2,800 0 0 0
So, the elements of entering variable(R) are 5,000/5,000, 3,000/5,000, 1/5,000, 0/5,000, 0/5,000
& 45,000/5,000 or 1, .6, .0002, 0, 0 & 9.
Calculation of new:
S2 S3
8 -8×1= 0 1-1×1=0
12-8×.6= 7.2 1-1×.6=.4
0-8×.0002= -.0016 0-1×.0002= -.0002
1-8×0=1 0-1×0= 0
0-8×0=0 1-1×0= 1
160-8×9= 88 10-1×9= 1

32
Table-2
Solution
Mix
→
Cj↓
3500 2800 0 0 0
R 3,500 1 .6 .0002 0 0 9 15
S2 0 0 7.2 -.0016 1 0 88 12.22
S3 0 0 .4 -.0002 0 1 1 2.5 ←
Zj 3,500 2,100 .7 0 0 31,500
Cj-Zj 0 700
↑
-.7 0 0
So, the elements of entering variable (W) are 0/.4, .4/.4, -.0002/.4, 0/.4, 1/.4 & 1/.4 or 0, 1, -
.0005, 0, 2.5 & 2.5.
Calculation of new:
R S2
1-.6×0=1 0-7.2×0=0
.6-.6×1=0 7.2-7.2×1=0
.0002-.6× (-.0005) =.0005 -.0016-7.2× (-.0005)=.002
0-.6×0=0 1-7.2×0=1
0-.6×2.5=-1.5 0-7.2×2.5=-18
9-.6×2.5=7.5 88-7.2×2.5=70

33
Table-3
Solution
Mix
→
Cj↓
3,500 2,800 0 0 0
R 3,500 1 0 .0005 0 -1.5 7.5
S2 0 0 0 .002 1 -18 70
W 2,800 0 1 -.0005 0 2.5 2.5
Zj 3,500 2,800 .35 0 1,750 33,250
Cj-Zj 0 0 -.65 0 -1750
So, the farmer should be allocated 7.5 beghaes rice and 2.5 beghaes wheat to maximize total
profit. The total maximize profit is Tk.33, 250. (Ans.).
Problem solve by using the graphical method
Objective Functions:
Z = 3,500R + 2,800W
Subject to constraints:
5,000R + 3,000W ≤ 45,000
8R + 12W ≤ 160
R + W ≤ 10
And R, W ≥ 0
The inequalities expressing constraints are converted into equalities we get,
5,000R + 3,000W = 45,000………………………………… (1)
8R + 12W = 160………………………………………………… (2)
R + W = 10……………………………………………………….. (3)
Decision:

34
From (1) equation we get,
5,000R + 3,000W = 45,000
Or, 5,000R/45,000 + 3,000W/45,000 = 45,000/45,000
Or, R/9 + W/15 = 1
So, equation (1) is passing by A (9, 0) and B (0, 15). [Let]
8R + 12W = 160
Or, 8R/160 + 12W/160 = 160/160
Or, R/20 + 3W/40 = 1
Or, R/20 + W/ (40/3) = 1
So, equation (2) is passing by C (20, 0) and D (0, 40/3). [Let]
R + W + = 10
Or, R/10 + W/10 = 10/10
Or, R/10 + W/10 = 1
So, equation (3) is passing by E (10, 0) and F (0, 10). [Let]
From equation (1) & (3) we get,
5,000R + 3,000W = 45,000
Or, 5,000R + 3,000W - 45,000 = 0
R + W = 10
Or, R + W – 10 = 0
R/ (3,000 × -10 -1 × -45,000) = W/(-45,000 × 1 –5,000 × 10 ) = 1/(5,000 × 1 – 3,000 × 1)
Or, R/ (-30,000 + 45,000) = W/(-45,000 + 50,000) = 1/(5,000 – 3,000)
Or, R/ 15,000 = W/5,000= 1/2,000
Or, R/ 15 = W/5 = 1/2

35
So, R = 15/2 & W = 5/2 or, R =7.5 & W = 2.5. So on equation (1) & (3) are passing through by
G (15/2, 5/2).
Now, put the points A (9, 0), B (0, 15), C (20, 0), D (0, 40/3), E (10, 0) and F (0, 10) in a graph
in which R is shown on the horizontal axis and W is shown on the vertical axis. And add A, B;
C, D; E, F. AB and EF are intersect by G (15/2, 5/2).
According to the condition of the question the area of FGAO is the possible solution area which
is shown by shadow. The corner points of the possible solution area are F (0, 10), G (15/2, 5/2),
A (9, 0) & O (0, 0).
Now the value of Z at
F (0, 10), Z = 3,500R + 2,800W or, Z = 3,500 × 0 + 2,800 × 10 or, Z = 0 + 28, 000 or, Z =
28,000

36
G (15/2, 5/2), Z = 3,500R + 2,800W or, Z = 3,500 × 15/2 + 2,800 × 5/2 or, Z =26,250 + 7, 000
or, Z = 33,250.
A (9, 0), Z = 3,500R + 2,800W or, Z = 3,500 × 9 + 2,800 × 0 or, Z =31,500 + 0 or, Z = 31,500.
O (0, 0), Z = 3,500R + 2,800W or, Z = 3,500 × 0 + 2,800 × 0 or, Z = 0 + 0 or, Z = 0.
The maximum value of Z is 33,250 where R = 15/2 & W = 5/2 or, R =7.5 & W = 2.5.
So, the farmer should be allocated 7.5 beghaes rice and 2.5 beghaes wheat to maximize total
profit. The total maximize profit is Tk.33, 250. (Ans.).
So, From above mentioned problem & solution we can say that Bangladeshi farmer should
cultivate their land by using Linear Programming so that they can get maximum profit. If they do
so, we belief that one day they might be able to remove their poverty problem.
1. Business Mathematics-D.C.Sancheti & V.K.Kapoor
2. Principles of Teaching – Bhatia & Bhatia
3. Teaching of Mathematics – Dr. M L Wangoo
4. Teaching of Mathematics – Kulbir Singh Sidhu
Comment:
REFERENCES
:
Decision:

37
5. Teaching of Mathematics -Kulshrestha.
6. Bellman, R. and Kalaba, R. Dynamic Programming and Modern Control Theory. New
York: Academic Press, 1965.
7. Dantzig, G. B. "Programming of Interdependent Activities. II. Mathematical Model."
Econometrical 17, 200-211, 1949.
8. Dantzig, G. B. Linear Programming and Extensions. Princeton, NJ: Princeton University
Press, 1963.Garey,
9. M. R. and Johnson, D. S. Computers and Intractability: A Guide to the Theory of NP-
Completeness. New York: W. H. Freeman, pp. 155-158, 287-288, and 339, 1983.
10. Greenberg, H. J. "Mathematical Programming Glossary."
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11. Karloff, H. Linear Programming. Boston, MA: Birkhäuser, 1991.Khachian, L. G. "A
Polynomial Algorithm in Linear Programming." Dokl. Akad. Nauk SSSR 244, 1093-1096,
1979. English translation in Soviet Math. Dokl. 20, 191-194, 1979.Karmarkar,
12. N. "A New Polynomial-Time Algorithm for Linear Programming." Combinatorica 4, 373-
395, 1984.Pappas, T. "Projective Geometry & Linear Programming." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 216-217, 1989.Press,
13. W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Linear Programming and
the Simplex Method." §10.8 in Numerical Recipes in FORTRAN: The Art of Scientific
Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 423-436,
1992.Sultan,
14. A. Linear Programming: An Introduction with Applications. San Diego, CA: Academic
Press, 1993.Tokhomirov,
15. V. M. "The Evolution of Methods of Convex Optimization." Amer. Math. Monthly 103,
65-71, 1996.Weisstein, E. W. "Books about Linear Programming."
http://www.ericweisstein.com/encyclopedias/books/LinearProgramming.html.
16. Wood, M. K. and Dantzig, G. B. "Programming of Interdependent Activities. I. General
Discussion." Econometrical 17, 193-199, 1949.Yudin, D. B. and Nemirovsky, A. S.
Problem Complexity and Method Efficiency in Optimization. New York: Wiley, 1983.

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