UNIT II
LINEAR PROGRAMMING
USING GRAPHICAL METHOD
INTRODUCTION
In a business organization, management has to make decisions on how to allocate
their resources to achieve its organization’s goal. A bank would like to allocate its funds
to achieve the highest possible return. It must operate within liquidity limits set by
regulatory agencies and it must maintain sufficient flexibility to meet the loan demands
of its customers.
Each organization wants to achieve some objective (maximize rate of return,
maximize profits, minimize costs) with constrained resources (deposits, available
machine time). To be able to find the best uses of an organization’s resources, a
mathematical technique called Linear Programming can be used. The adjective linear is
used to describe a relationship between two or more variables, a relationship which is
directly and precisely proportional. In a linear relationship between work hours and
output, for example, a 10 percent change in the number of productive hours used in the
operation will cause a 10 percent change in output.
COURSE OBJECTIVES



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To introduce students to the fundamental principles behind linear programming
To familiarize students with the basic structure, assumptions and limitations of
linear programming formulation
To discuss the graphical approach in solving linear programming problems and
illustrate how this approach could be used in handling maximization and
minimization problems
To highlight the differences between LP problems involving maximization and
minimization of the objective function
SUGGESTED TIMEFRAME:
6 hours
WHAT IS LINEAR PROGRAMMING?
Linear Programming (LP) is a mathematical optimization technique. By
optimization technique, it refers to a method which attempts to maximize or minimize
some objective, for example, maximize profits or minimize costs.
Linear programming is a subset of a larger area of mathematical optimization
procedures called mathematical programming, which is concerned with making an
optimal set of decisions.
In any LP problem, certain decisions need to be made. These decisions are
represented by decision variables which are used in the formulation of the LP model.
BASIC STRUCTURE OF A LINEAR PROGRAMMING PROBLEM
The basic structure of an LP problem is either to maximize or minimize an
objective function, while satisfying a set of constraining conditions called constraints.
Objective function. The objective function is a mathematical representation of the
overall goal stated in terms of the decision variables. The firm’s objective and its
limitations must be expressed as mathematical equations or inequalities, and these must
be linear equations and inequalities.
Constraints. Constraints are also stated in terms of the decision variables, and represent
conditions which must be satisfied in determining the values of the decision variables.
Most constraints in a linear programming problem are expressed as inequalities. They set
upper or lower limits, they do not express exact equalities; thus permit many possibilities.
Resources must be in limited supply. For example, a furniture plant has a limited
number of machine-hours available; consequently, the more hours it schedules for
furnitures, the fewer furnitures it can make.
There must be alternative courses of action, one of which will achieve the
objective.
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LINEAR PROGRAMMING ASSUMPTIONS AND GENERAL LIMITATIONS
When using Linear Programming to solve a real business problem, five
assumptions (also referred to as limitations) have to be made.
Linearity. The objective function and constraints are all linear functions; that is, every
term must be of the first degree. Linearity implies the next two assumptions.
Proportionality. For the entire range of the feasible output, the rate of substitution
between the variables is constant.
Additivity. All operations of the problem must be additive with respect to resource usage,
returns, and cost. This implies independence among the variables.
Divisibility. Non-integer solutions are permissible.
Certainty. All coefficients of the LP model are assumed to be known with certainty.
Remember, LP is a deterministic model.
Computational Techniques to Overcome the Limitations of LP
There are at least three other mathematical programming techniques that may be used
when the assumptions (limitations) above do not apply to the problem. These are:
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Integer programming
Dynamic programming
Quadratic programming
The Red Gadget-Blue Gadget Problem
We shall be using one particular example to illustrate most of the LP approaches
and concepts. We shall call this the red gadget-blue gadget problem.
A company produces gadgets which come in two colors: red and blue.
The red gadgets are made of steel and sell for 30 pesos each. The blue gadgets
are made of wood and sell for 50 pesos each. A unit of the red gadget requires 1
kilogram of steel, and 3 hours of labor to process. A unit of the blue gadget, on
the other hand, requires 2 board meters of wood and 2 hours of labor to
manufacture. There are 180 hours of labor, 120 board meters of wood, and 50
kilograms of steel available. How many units of the red and blue gadgets must
the company produce (and sell) if it wants to maximize revenue?
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The Graphical Approach
Step 1. Define all decision variables.
Let:
x1 = number of red gadgets to produce (and sell)
x2 = number of blue gadgets to produce (and sell)
Step 2. Define the objective function.
Maximize R = 30 x1 + 50 x2 (total revenue in pesos)
Step 3. Define all constraints.
(1)
x1
(2)
2 x2
(3) 3 x1 + 2 x2
x1 , x2
 50 (steel supply constraint in kilograms)
 120 (wood supply constraint in board meters)
 180 (labor supply constraint in man hours)

0 (non-negativity requirement)
Step 4. Graph all constraints.
X1
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Step 5. Determine the area of feasible solutions.
X1
Step 6. Determine the optimal solution.
The shot-gun approach
List all corners (identify the corresponding coordinates), and pick the best in terms of the
resulting value of the objective function.
(1) x1 = 0
x2 = 0
R = 30 (0) + 50 (0) = 0
(2) x1 = 50
x2 = 0
R = 30 (50) + 50 (0) = 1500
(3) x1 = 0
x2 = 60
R = 30 (0) + 50 (60) = 3000
(4) x1 = 20
x2 = 60
R = 30 (20) + 50 (60) = 3600 (the optimal solution)
(5) x1 = 50
x2 = 15
R = 30 (50) + 50 (15) = 2250
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The contour-line approach
Assume an arbitrary value for the objective function, then graph the resulting
contour line.
Let R = 3000
30 x1 + 50 x2 = 3000
x1 = 0 x2 = 60
x2 = 0 x1 = 100
If you are maximizing, draw contour lines parallel to the first contour line drawn,
such that it is to the right of (and above) the latter, and touches the last point (points) on
the area of the feasible solutions.
If minimizing, go the opposite direction.
X1
Step 7. Determine the binding and non-binding constraints.
Observing graphically, we find that the steel supply constraint is non-binding,
implying that the steel supply will not be completely exhausted. On the other hand, both
the wood supply and labor supply constraints are binding. By producing 20 units of the
red gadget and 60 units of the blue gadget, wood and labor will be completely used.
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LINEAR PROGRAMMING: GRAPHICAL SOLUTION
(Minimization Problem)
Example: Mindoro Mines
To illustrate the use of the graphical approach to solving linear programming
minimization problems, we shall use the Mindoro Mines example. If you already have
basic understanding of the graphical method from the example in Session 03, you should
have little difficulty appreciating the following application.
Mindoro Mines operates 2 mines: one in Katibo and the other on Itim Na Uwak
Island. The ore from the mines is crushed at the site and then graded into highsulfur ore (ligmite), low-sulfur ore (pyrrite) and mixed ore. The graded ore is
then sold to a cement factory which requires, every year, at least 12,000 tons of
ligmite, at least 8,000 tons of pyrrite, and at least 24,000 tons of the mixed ore.
Each day, at a cost of P22,000 per day, the Katibo mine yields 60 tons of ligmite,
20 tons of pyrrite, and 30 tons of the mixed ore. In contrast, at the Itim Na Uwak
Island mine, at a cost of P25,000 per day, the mine yields 20 tons of ligmite, 20
tons of pyrrite, and 120 tons of the mixed ore.
The management of Mindoro Mines would like to determine how many days a
year it should operate the two mines to fill the demand from the cement plant at
minimum cost. What are the binding constraints?
The Graphical Approach
Step 1. Define all decision variables.
Let
x1 = number of days (in a year) to operate the Katibo mine
x2 = number of days (in a year) to operate the Itim Na Uwak Island mine
Step 2. Define the objective function.
Minimize C = 22000 x1 + 25000 x2 (total cost of operating the mines in pesos)
Step 3. Define all constraints.
(1)
(2)
(3)
(4)
(5)
60 x1 + 20 x2
20 x1 + 20 x2
30 x2 + 120 x2
x1
x2
x1 , x2
 12000 (ligmite demand in tons)
 8000 (pyrrite demand in tons)
 24000 (mixed ore demand in tons)

365 (maximum number of days in a year)
 365 (maximum number of days in a year)

0 (non-negativity requirement)
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Step 4. Graph all constraints.
Step 5. Determine the area of feasible solutions.
For Steps 4 and 5, please refer to the following graph.
Step 6. Determine the optimal solution.
Using the shot-gun approach, we list down the following corners or extreme points (with
their respective coordinates):
(1) x1 = 365
x2 = 365
C = 22000 (365) + 25000 (365) = 17,155,000
(2) x1 = 365
x2 = 108.75
C = 22000 (365) + 25000 (108.75) = 10,748,750
(3) x1 = 78.33333
x2 = 365
C = 22000 (78.33333) + 25000 (365) = 10,848,333
(4) x1 = 100
x2 = 300
C = 22000 (100) + 25000 (300) = 9,700,000
(5) x1 = 266.66667
x2 = 133.3333 C = 22000 (266.667)+25000 (133.333) = 9,200,000
The Optimal Solution
Step 7. Determine the binding and non-binding constraints.
Observing graphically, we find that binding constraints are those associated with pyrrite
and mixed ore production. Total production of these two items will exactly match the
requirements of the cement factory. Meanwhile, ligmite production will be in excess of
the minimum requirement of 12000 tons by about 6,667 tons. Likewise, total available
operating days will not be completely utilized.
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YOUR ACTIVITY
1. Luzon Timber Corporation
The Luzon Timber Corporation cuts raw timber, lauan, and tanguile, into standard
size planks. Two steps are required to produce these planks from raw timber:
debarking and cutting. Each hundred meters of lauan takes 1.0 hour to debark 1.2
hours to cut. Each hundred meters of tanguile takes 1.5 hours to debark and 0.6 hour
to cut. The bark removing machines can operate up to 600 hours per week but the
cutting machines are limited to 480 hours per week.
Luzon Timber can buy a maximum of 36,000 meters of raw lauan and 32,000 meters
of raw tanguile. If the profit per hundred meters of processed logs is P1,800 for lauan
and P2,000 for tanguile, how much lauan and tanguile should be bought and
processed by the corporation in order to maximize total profits?
Suppose next that the finished planks must go through kiln-drying as well and the kiln
can only process a combined total of 45,000 meters of planks. What product
combinations are now feasible? Which combination will maximize combined profits
under the new conditions?
2. Small Refinery
A small refinery produces only two products: lubricants and sealants. These are
produced by processing crude oil through 3 processors: a cracker, a splitter, and a
separator. These processors have limited capacities. For the cracker, at most 1000
hours; for the splitter, at most 4200 hours; and for the separator, at most 2400 hours
per week. Similarly, there is a limit on the supply of crude oil: at most 700 barrels
per week.
To produce one barrel of lubricant, we need one hour at the cracker, 6 hours at the
splitter, and 4 hours at the separator. To produce a barrel of sealant, we need 2 hours
at the cracker, 7 hours at the splitter, and 3 hours at the separator.
Under these conditions, what product combinations of lubricants and sealants are
feasible? If a barrel of lubricant nets P2000 and a barrel of sealant nets P2500, which
product combination will maximize combined profits?
Next, suppose it were now possible to expand the splitter capacity from 4200 hours
up to 4374 hours per week. If this expansion will mean added costs (along with more
production), what is the most money the small refinery should pay to finance the
expansion? Assume all other conditions of the problem remain the same except for
the added capacity on the splitter.
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REFERENCES
Andersen, D.R., D.J. Sweeney, & T.A. Williams, An Introduction to Management
Science: Quantitative Approaches to Decision Making
Andersen, D. R., Dennis J. Sweeny, and Thomas A. Williams. 2001. Quantitative
Methods for Business. Eighth edition. Cincinnati, Ohio: South-Western College
Publishing.
Anderson, Michael Q. 1982. Quantitative Management Decision Making, Belmont,
California: Brooks/ Cole Publishing Co.,
Dunn, Robert A. and K. D. Ramsing. 1981. Management Science: A Practical Approach
to Decision Making. New York, New York: MacMillan Publishing Co.,
Krajewski, Lee C. and H. E. Thompson. 1981. Management Science: Quantitative
Methods in Context, New York: John Wiley & Sons,
Levin, R.I.., C.A. Kirkpatrich & D.S. Rubin. 1982. Quantitative Approaches to
Management. 5th ed. McGraw-Hill, Inc.
Zamora, Elvira A. 2004. Basic Quantitative Methods for Business Decisions. UP College
of Business Administration.
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