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Management Science Module

BS Accountancy (Batangas State University)

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Chapter I: INTRODUCTION TO MANAGEMENT SCIENCE

Introduction
Management science is applying a scientific approach to problem-solving management to
help managers make better choices. As suggested by this concept, management science includes
a variety of mathematically based strategies that have either been developed within the area of
management science or adopted from other disciplines such as mathematics, natural sciences,
statistics, and engineering. This module offers an introduction to the methods that make up the
philosophy of management and explains their contributions to crisis management.
Management science is a proven and recognized discipline in industry. Management
science techniques implementations are common, and they have also been credited with growing
the company firms' performance and profitability. Several suggest that they use management
science methods in different organization surveys, and the majority consider the findings to be
quite good. Management science (also referred to as operations research, systematic processes,
predictive study and decision-making sciences) is part of the core curriculum in most business
programs.

Learning Objectives
After studying this module, you should be able to:
1. Define management science and describe its importance in business.
2. Explain the major characteristics of management science and how it is applied in problem
solving.
3. Explain the importance of models in decision making process.
4. Explain and identify the appropriate management science technique in managerial
problems.

1.1 Definition and Characteristics of Management Science

Definition
Management science can be defined briefly as applying the scientific method to the
analysis and solution of problems with managerial decision. According to Turban and Meridith,
management science is applying analytical principles, procedures, and instruments to problems
concerning device processes in order to offer optimal answers to the problems for those in charge
of the processes.

Major Characteristics of Management Science

1. A main focus on decision making by the managers.
2. Usage of the analytical model to the decision-making phase.
3. Examining the condition of the judgment from a specific viewpoint
4. Using methods and expertise from multiple disciplines.
5. A dependency on the mathematical models of shape.
6. Computer widespread use.

Decision-making is a method of deciding between two or more possible courses of action

necessary to accomplish a particular goal. Decision-making is a systematic process and can be
described simply by defining the problem, searching for alternative courses of action, evaluating
alternatives and choosing one alternative.

System configuration comprises of three associated components: inputs, procedures, and

outputs. The inputs (people, raw materials, money) enter the system. Processes transform inputs
into outputs (processes may use resources, operating procedures, workers, machines). Outputs exit
the machine (Products, Customers served). Systems approach involves finding and defining all the
elements mentioned and the real world relationships between them. This method is the

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presumption required to use management science methods, since they allow the precise definition
of a problem. System approach allows mathematical formulation of the problem.

1.2 Models and Modeling

Model is a generalized reflection of the natural world, and only significant and appropriate
objects or properties can be included in the model, whereas modeling is the method of treating and
explaining real-life problems in mathematical terms.

Management science follows a rational, structured approach to problem solving that

strongly resembles what is known as the analytical process of problem-solving. This approach, as
illustrated in a generally recognized and ordered series of steps follows: (1) observation, (2)
problem definition, (3) model construction, (4) model solution, and (5) solution implementation.

Observation
The first phase in the cycle of handling management science process is to recognize a
problem that occurs within the organization. The system must be monitored continuously and
closely so as to identify problems as soon as they arise.

Definition of the Problem

When a problem has been established, the problem must be described explicitly and in a
clear manner. Defining a problem incorrectly can easily lead to either no solution or inappropriate
solution. Hence the limitations of the problem and the extent to which it pervades other units of
the organization should be included in the definition of the problem. It's evident that we need to
know the production process details and all the appropriate information.

Model Construction
A concept in management science is a theoretical representation of an actual issue scenario.
It can be in the form of a graph or diagram, but most commonly a model in management science
consists of a series of mathematical links. The numbers and symbols form these mathematical
relationships. Two types of models exist: determinist and probabilistic. All aspects are known with
certainty in deterministic models. There is no question that deterministic models are ideal,
however they can give a fairly decent estimation of truth. There is a particular degree of ambiguity
in the probabilistic models. This is tempting to neglect the small degree of ambiguity and to use
deterministic models instead of probabilistic models. In case of high uncertainty we should
consider random variables rather than constants.

Model Solution
When models have been developed in management science, the methods described in this
text are used to solve them. A technique for a management science solution usually applies to a
particular type of model. Therefore, the form of concept and the process of solution are also part
of the scientific management research. We also mean problem solution when we refer to model
solution. The typical goal of the most problems is to find an optimal solution, that is, the best of
all feasible solutions (solutions that meet all the constraints).

Implementation of Solution Result

Implementation is the actual use of the model once it has been developed or the solution
to the problem the model was developed to solve. This is actually the main goal of management
and the original purpose of the whole process - not the model itself, but adjustment of reality
according to the recommendations ensuing from the results of the modeling process. In case we
do not use the results in the real production process, all our effort was absolutely vain. On the
contrary, if we constructed the model in a wrong way and we did not validate it, the applied results
could seriously harm the real system. In order to achieve the best management results, each step
must be carefully considered and cannot be skipped.

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1.3 Management Science Techniques

Linear Programming
This is one of management science's best-known tools. This strategy describes the problem as
maximizing a linear function (minimizing) with reference to the set of linear constraints.

Integer Linear Programming

Additional variable values criteria are applied to the initial linear process ( i.e., process of
linear function and linear constraints). All or any of the values must be an integer. The particular
type of these variables is the binary variable (0-1 variable) with a value of 0 or 1. We think about
binary integer linear programming in such a situation. When only those variables are described as
integer (binary) in the model, then we are thinking about mixed integer linear programming.

Goal Programming
When several conflicting priorities need to be addressed concurrently, it needs more
effective resource. Goal programming is a specific methodology for coping with such situations,
usually in linear programming.

Distribution Models
A distribution problem is a specific category of problem in linear programming. There are
two major categories of distribution problems: the problem of transportation, and the problem of
assignment. The problem of transportation deals with shipping from a variety of suppliers to a
number of destinations while the problem of assignment deals with determining the right one-to-
one option for any of a number of prospective "candidates" to a number of possible "positions."

Nonlinear Programming
Models used in this field of management science are close to linear programming models;
but there is a major distinction between them: nonlinear models include nonlinear objective
function and/or other nonlinear constraints. Techniques employed in this field of management
science to solve problems are somewhat distinct from linear programming approaches.

Network Models
Many problems can be defined as a network (the collection of nodes and arcs) in graphic
terms. Typical circumstance is a network of transports: cities (nodes) are linked by roads (arcs) to
each other. When we evaluate the network (in this case we are involved, for example, in distances
among all the cities), the goal is always to locate the shortest distance from one city to all other
cities. Some forms of networks, instead of lengths, can be measured by capacities and then the
dilemma of maximal flow can be resolved. The most critical factor of several problems, addressed
by the help of network models, is unit cost and the aim is to find the lowest overall cost.

Project Management
Managers are responsible for organizing, managing and overseeing projects in certain
cases that consist of several different jobs or activities undertaken by a number of teams or persons.
It needs different time for a work to be completed. These problems are overcome by two simple
methods: CPM (Critical Path Method) and PERT (Program Analysis Review Technique). All
approaches allow the issue to be expressed on a network.

Inventory Models
Inventory control is one of the most common strategies, helping managers decide what to
buy and how much to buy. The primary purpose is generally to strike an accurate balance between
the expense of keeping inventory and the cost of implementing an order. Owing to the very various
actual distribution structures, there are several specific inventory models. We are considering two
distinct groups of models: deterministic and probabilistic. The demand rate in deterministic
models is constant over time, while the demand fluctuates over time in probabilistic inventory
models and can only be represented in probabilistic terms.

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Waiting Line Models

This field of management science approaches deals with the circumstances in which a
variety of networks (e.g. vendors) ought to represent units ( e.g. customers). Since the number of
vendors is constrained, some units must wait in queue for the service. Thus the alternative name
for the models of the waiting line: Queuing Models. In actual life, the process of arrival of units
as well as the service times are irregular and the probabilistic approach is essential. Simple models
of waiting line can be obtained analytically (exact solution using derivative formulas), while the
simulation technique is needed for complex queuing systems.

Simulation
As management challenges are more complicated, it is always difficult to address them
using traditional methods (or not successful because of the time and expense spent). Simulation
solution is useful for this reason and is in many situations the best way to handle the issue.
Simulation is a programming experience with a simulation model intended to explain and measure
the actions of the real system-the machine software simulates the actual system. Complex waiting-
line models and inventory models are common conditions for effective simulation use.

Decision Analysis
Such methods may be used to choose appropriate approaches from a variety of alternatives
to decisions. According to the knowledge received by the manager, management challenges and
relevant methods are classified into three types: decisions under certainty (deterministic),
decisions under risk (probabilistic) and decisions under uncertainty. Of this function, we find
unique instruments: decision tables and trees.

Theory of Games
This field is an expansion of the decision making of two or more decision-makers to the
circumstances. All managers take simultaneous decisions (selected strategies) to execute an acts
that influences all decision-makers (players), i.e. their profits, costs etc. For certain disputes, two
or three decision-makers will collaborate, while battling with the others. We will consider a
standard case of a strategic game in economic theory-the oligopoly model.

Forecasting
Methods of forecasting support the manager predicting future elements of the business
operation. Statistics and econometrics deliver several time series and regression analysis methods
based. The key management task is to predict potential developments relative to the system's
previous behaviour. The well-known are the Moving Averages, Least Squares, Exponential
Smoothing processes, etc. Because the statistical significance for such models is very significant,
the manager's familiarity with validity hypotheses and statistical testing is relevant.

Multicriteria Decision Making

The decision maker needs to consider multiple criteria for many managerial problems. If
we find a solution which improves one criterion, it mostly aggravates some other criteria. It's
practically difficult to modify all the criteria at the same time. The Management's fair incentive is
to consider an acceptable solution. If the list of alternatives is small, we use alternative assessment
approaches. Few problems with set of constraints and set of objective functions are defined. In
this scenario multi-objective programming approaches have the remedy. Goal programming is the
unique type of this methodology in management science.

Markov Analysis
This approach can be used for explaining a system 's actions in a complex scenario (system
progression over time). If-at a given time point-the device is in one of the potential states, the
device may remain in the current state or switch into some other state at the following time point.
The transition probabilities are set for staying in the current state or moving to another state. The
manager will be confident in the possibility of the system being in the correct state at the present

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time. Markov Analysis is a really strong management science method with a number of practical
applications.

Dynamic Programming
Management also needs to recognize a series of decisions where each judgment impacts
potential decisions in a major way. Dynamic programming lets administrators overcome certain
forms of challenges with these complex decision making. There is no common paradigm to solve
dynamic programming problems and, thus, the problems are divided into several classes. One
probable classification takes the deterministic and probabilistic models into consideration. Models
also use the sequential problems to describe the network. The Markov analysis may be viewed as
a complex programming probabilistic process.

Exercises and Activities

Discuss the following.

1. Differentiate management science from other mathematical discipline.

2. How do the goal of management science pay to the attainment of business objectives?

3. How can management science be useful in practice?

4. Explain which management science technique is appropriate in the following managerial

problems:
a. Inventory control
b. Facility design
c. Allocation of scarce resources
d. Investment decisions

5. Assess the limitations of management science in dropping risk of industry failures.

True or False
1. Decision making is a process of choosing between two or more available alternative
courses of action for the purpose of attaining a specific goal.
2. A solution that respects all the constraints is goal programming.
3. The application of the scientific method to the study and analysis of problems involving
large and complex systems, organizations or activities is called operations research.
4. Programming is an abstraction of reality.
5. Management science seeks for optimization of objective function, which looks for the
highest objective value.
6. Linear programming is an approach when several objective functions are considered
simultaneously.
7. The objective of goal programming is to minimize the undesirable deviations from the
goals.
8. Optimization of objective function, which looks for the lowest objective value is
minimization.
9. Optimal solution is a mathematical function expressed in terms of decision variables,
which is to be optimized.
10. A model that incorporates uncertainty in its functional relations and uncontrollable
variables is probabilistic model.
11. Implementation is the main goal of management and the original purpose of the whole
management science process according to the recommendations ensuing from the results
of the modelling process.
12. If we want to examine the impact of changes in inputs on changes in outputs, we have to
use sensitivity analysis.
13. Validation of the model is its comparison with the real system.

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14. In dynamic programming, problems are classified into many groups.

15. In case of high level of uncertainty in model construction we should consider random
variables instead of constants.
16. The simpler the model, the easier its manipulation and solution will be.
17. Management science is an art.
18. A management science technique usually applies to a specific model type.
19. A model is a functional relationship that includes variables, parameters, and equations. T
20. Management science encompasses logical approach to problem solving. T

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Chapter II: LINEAR PROGRAMMING

Introduction

Many big choices made by a company rely on the best path to accomplish the firm's
objectives, according to the operational climate constraints imposed on the manager. Such
constraints that take the form of finite capital, such as time, labour, energy, content, or money; or
they may be in the form of stringent instructions, such as a cereal or engineering design manual.
One of the company firms' most important objectives is to get as much income as possible or, in
other terms, to increase profit. The aim of specific business structures within a company (such as
a production or packaging department) is frequently to minimize costs.

Learning Objectives
After studying this module, you should be able to:
1. Understand and define linear programming.
2. Identify all necessary items that must be included in a model.
3. Write a verbal statement of the objective function and each constraint.
4. Define the decision variables.
5. Write the objective function in terms of the decision variables.
6. Write the constraints in terms of the decision variables.
7. Makes a decision about the solution of the problem.

Linear Programming

Linear Programming is a computer science technique to address issues of optimization

process. The word "linear" implies the linearity of all mathematical relationships within a model.
The standard model is the set of linear equations and/or inequalities (called constraints) and the
linear objective function (to be maximized or minimised). The constraints of non-negativity
(variables are negative or positive) are also included in the model. The manager's aim is to find
the right solution for the highest objective function value.2.1 Formulation of the Mathematical
Model

Example

ABC, Inc. creates 2 types of toys: car and boat. The car is priced at P550, and the boat at P700.
The cost of the car is P50, while P70 for the boat. The car needs 1 hour of woodwork labor and 1
hour of painting and assembling labor. The boat requires 2 hours of woodwork labor and1 hour of
painting and assembling labor. Cost of woodwork labor is P30 per hour, worth of painting and
assembling labor is P20 per hour. Monthly, ABC has 5000 existing hours of woodwork labor and
3000 hours of painting and assembling labor. There is an unlimited demand for boat, while an
average demand for car is at most 2000. ABC wants to get the best out of monthly profit (total
revenue - total cost).

Formulation

1. Decision variables

The variables would explain thoroughly the management's decisions to be made. To

optimize the income, the manager needs to determine how many cars and how many boats will be
generated per month. In this scenario, the variables to the decision are:

x1= number of cars produce each month;

x2= number of boats produce each month

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2. Objective function

This function reflects the criteria of the management which must be maximized or minimized. The
management aims to optimize gross monthly income in the circumstance of the ABC as the
disparity between total monthly revenues and total monthly costs. Revenue and cost can be
represented as the function of the variables x1 and x2 for decision.

1. Total revenue (TR) = revenue from sold car + revenues from sold boat. The price of one car is
P550 and ABC produces x1 of car, the revenue resulting from the cars is 550x1. Likewise, the
revenue from sold boats is 700x2.

The total monthly revenue from production is expressed as:

TR= 500x1 + 700x2

2. Monthly wood cost (WC) = WC of produced cars + WC of produced boats. The wood cost of
production of one car (50 ) and the total number of produced car is x1, the monthly wood cost of
all produced cars is 50x1. Likewise, the monthly wood cost of all boats is 70x2.

Total monthly wood cost is :

WC= 50x1 + 70x2

3. Woodworrk labor cost (WLC) = WLC of produced cars + WLC of produced boats. One car
needs 1 hour of woodwork labor and cost of 1 hour of this labor is 30 , the unit cost is 30 . The
monthly cost of carpentry labor used for all produced cars is 30x1. One boat requires 2 hours, the
monthly cost of woodwork labor used for boats is 60x2.

Total monthly cost of woodwork labor:

WLC= 30x1 + 60x2

4. Painting and assembling labor cost (PALC) = PALC of produced cars + PALC of produced
boats. Both a piece of car and boat need 1 hour of painting and assembling labor. Cost of this labor
is 20 per hour. Therefore the total monthly cost of painting and assembling labor:

PALC= 20x1 +20x2

5. The total monthly cost can be expressed as:

TC= WC + WLC + PALC= (50x1 + 70x2) + (30x1 + 60x2) + (20x1 +20x2)
TC= 100x1 + 150x2

6. The total profit:

TP = TR − TC = (550x1 + 700x2) − (100x1 + 150x2 ) = 450x1 + 550x2

the objective function in the linear programming model is:

Maximize z = 450x1 + 550x2

450 and 550 in the function are termed objective function coefficients.

3. Constraints
If no restrictions are imposed, objective function (profit) can expand to infinity. There are three
limits (called constraints) to the development of toys, though:

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1. ABC, Inc. has just 5000 hours of woodwork labor accessible per month.
2. No more than 3000 hours of finished labor can be used each month.
3. Because of minimal demand, there will be a maximum of 2000 cars delivered each month.

Expressing the constraints in mathematical way:

1. One car needs 1 hour of woodwork labor. If ABC manufactures monthly x1 of cars, x1 hours
of labor are consumed. Since one boat requires 2 hours and the manufacturing quantity equals x2,
the monthly consumption of woodwork labor is 2x2 hours.
Total consumption of woodwork labor for both products can be expressed as x1 + 2x2.
This is actual use of labor (in hours) that cannot be greater than available number of hours (5000).
With this, the constraint can be:
x1 + 2x2 ≤ 5000

2. The creation of the second constraint regarding painting and assembling labor is like to the
foregoing:

x1 + x2 ≤ 3000

3. The last constraint is easy to be construct. The number of produced cars x1 must be less than or
equal to 2000:
x1 ≤ 2000

Technological coefficients are coefficients of the decision variables in the constraints while
numbers 5000, 3000 and 2000 are termed right-hand side values.

4. Non-negativity Constraints
Both decision variables have rational sign restrictions: because the values of variables
reflect numbers of toys made, we would expect them not to be negative:

x1, x2 ≥ 0

Summary of mathematical model in standard form:

Maximize z = 450x1 + 550x2

subject to
x1 + 2x2 ≤ 5000,
x1 + x2 ≤ 3000 ,
x1 ≤ 2000 ,
x1, x2 ≥ 0 .

2.2 Graphical Solution of Linear Programming Problems

When a linear programming problem only involves two decision variables, it can be graphically
overcome. The next phase proceeds to solution:

Step 1 - Graphing a feasible area

That restriction (including non-negativity constraints) can be drawn very easily; because
all constraints have to be met concurrently, the mixture of drawn constraints defines the area that
is feasible. Each point from the feasible area suits the feasible solution.

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There are 3 constraints and 2 nonnegativity constraints in the ABC’s problem.

Nonnegativity constraints are simple to graph, as all x1 and x2 variations must be in the first
quadrant of x1-x2 plane (Figure 2.1).

Linear inequality graphing (woodwork labor constraint)

x1 + 2x2 ≤ 5000

the borderline needs to be draw, corresponding to the linear equation

x1 + 2x2 = 5000 .

Coordinates of two points must be find to get the borderline. The simplest way is to look for the
borderline’s intersection points with axes x1 and x2. If x1 is set to zero (i.e. no cars are
manufactured), then x2 = 5000/2 = 2500 (2500 boats can be manufactured). This condition refers
to the point A (0, 2500) in Figure 2.2.

Also, if x2 is set to zero (no boats are manufactured), then x1 = 5000 (5000 cars can be
manufactured). It links to the point B (5000, 0) in Figure 2.2.

Linking points A and B by a straight get a graphical illustration of the equation x1 + 2x2 = 5000
We must now decide which portion of the first quadrant (divided by line AB) conforms to the x1
+ 2x2 5000 inequality.
For this reason the general approach is to pick an arbitrary point (of course not from the line) and
to test if it is feasible. When the point is feasible so the whole area is always feasible.

The best way is to take origin, (0 , 0) in our case. After these coordinates have been substituted by
the constraints, we get 5000 0 (the restriction is met). Therefore, the area of origin included is
feasible (shaded in Figure 2.2).

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Figure 2.2 Woodwork Labor

Note: We omitted negative values of x1 and x2 due to nonnegativity constraints (Figure 2.1) in
graphing the feasible field.

Drawn in Figure 2.3 is the second inequality 3000 x1 + x2 ≤ (painting and assembling labor
constraint), the last constraint 2000x1 ≤ is comprehended in Figure 2.4. Feasible area can be seen
on the left side of the line while 2000x1 = is emphasized by a line parallel to the axis x2.

Figure 2.3 Painting and Assembling Labor

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Step 2 - Combining the constraints

If all of the restrictions are drawn independently, they must be placed together in a single
graph. Practically, this final graph may be built by systematic addition of the individual feasible
areas from the beginning. The ultimate graph of the feasible area in the example is seen in figure
2.5. Any solution in this area is feasible as regards all the constraints.

Step 3 – Graphing an objective function

In the example the goal of the management is represented as the function of profit z= 450x1
+ 550x2. Because of the endless number of equations this feature can not necessarily be viewed
as a single line. The basic equation depends on z value.

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Figure 2.6 introduces three of those equations of values 495,000, 1,485,000 and 2,475,000.
All the lines are parallel to each other and can be constructed in the same way as the constraints
borderline in Step 1. Both points on a specific line produce the same profit, and hence the lines
are typically called isoprofit lines. The arrow expressed income growth.

Step 4 – Finding the optimal solution

Integrating the feasible region with the isoprofit lines, we will consider the isoprofit line that is as
far from the origin as feasible, which also crosses the feasible field.

The dotted isoprofit line in Figure 2.7 correlates to the maximum profit z = 1,550 000. This line
intersects a viable area corner point, which is considered an optimal solution. It is point X (1000,
2000)

There are two potential ways of seeking the best approach using the model's graphical
representation:

1. The first approach has only been illustrated. We can determine the "best" point of the feasible
area, by finding a slope and the growth direction of the isoprofit line. This level is the optimal
solution, and the isoprofit line suits the ideal objective function optimal value. Within this section

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of the text it is important to notice that a convex set is the feasible field of all linear programming
models.

A set of points S is a convex set if the section of the line connecting each pair of points in
S is entirely enclosed in S.

The disparity between convex and non-convex sets is shown in Figure 2.8. Although the first three
examples are convex sets (a), (b ) and ( c), the group (d) is non-convex.

The set (a) varies significantly from the sets (b ) and ( c), due to nonlinear border. If the
convex set boundary consists only of linear segments, as seen in cases (b ) and ( c), the set is called
a convex polyhedron, which reflects the standard feasible field of linear programming models. It
is the feature which is useful in linear programming methods.

The key problem is how to measure optimum solution of coordinates X(1000, 2000). As it
is apparent from Figure 2.6, this point is the borderline intersection of two constraints.
Determining the co-ordinates is therefore very simple by solving the series of two linear equations.
The borderlines compare in our situation are:
x1 + 2x2 = 5000 ,
x1 + x2 = 3000 .

Pretty simple solution: x1 = 1000 and x2 = 2000. Those values are point X coordinates.
Incorporating them into objective function z = 450x1 + 550x2 = 1 550 000.

2. It is important to identify a corner point before beginning to explain the second approach used
to find the best solution in the graph. With respect to the convex polyhedron definition it is possible
to describe the corner point clearly as follows:

A point P in convex polyhedron S is a corner point if it does not lie on any line
joining any pair of other (than P) points in S.

Keystone of the second approach is the basic linear programming theorem:

The optimal feasible solution, if it exists, will occur at one or more of the corner points.

In this theorem, what we need is to define the feasible area's corner points and determine the
objective values for all of those points. The optimal solution is the corner point and has the highest
objective value.

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The feasible area in the example contains 5 corner points (Figure 2.9), of which the
coordinates can be observed in Table 2.1. All the coordinates were calculated in the manner
described above: after classification of the intersected lines, the set of equations is solved and the
x1 and x2 co-ordinates provide the solution. The objective value z is then calculated for each
solution by incorporating the co-ordinates into the objective function. From the table it is evident
that the optimal solution corresponds to the maximum objective value z = 1,550,000.

Corner point x1 x2 z
A 0 0 0
B 2000 0 900 000
C 2000 1000 1 450 000
D 1000 2000 1 550 000
E 0 2500 1 375 000

Simplex method is a general method used for solving linear programming problems. It is
an iterative algorithm for efficient searching for the optimal solution. The method uses the Gauss-
Jordan method of solving simultaneous equations. In addition, the method is based on the basic
linear programming theorem (the search can concentrate only on the corner points of the feasible
area). The run of algorithm starts in one of the corner points (usually in the origin) and moves to
adjacent corner that improves the value of the objective function. The process of movement
continues until no further improvement is possible. The simplex method and its variations have
been programmed and therefore even large linear programming problems can be easy solved.

2.3 Understanding the Optimal Solution

The appropriate and reliable analysis of the findings is almost as critical as the model itself.
After the solution has been obtained, it is necessary to go back to the start of the modeling process
and compare the variables with their real representations. In the case, x1 represents the number of
cars created each month, and x2 is the number of boats generated every month. Therefore it is
optimal to produce 1000 cars and 2000 boats every month as the optimal solution we have got x1
= 1000 and x2 = 2000. When the management maintains this output plan, ABC, Inc. generates
income of 1,550,000 per month (see the target role description in the model).
When the production varied from the measured approach but nevertheless followed all
product constraints, the profit would've been lower. By violating at least one limit, the
management can achieve greater income than 1 550 000. That will be an infeasible approach.

Analyzing the findings ABC would be involved in practical usage of available resources
(working hours) and meeting limited demand. If we integrate the optimal values of x1 and x2 into

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all the restrictions, we can see whether they are fulfilled exactly as equations or as inequalities. As
for woodwork labor, the limit's left-hand side value is 1000 + 2(2000) = 5000, which is almost the
same as the right-hand side value 5000 (the constraint is obligatory). Since the right-hand side
relates to the hours of work available, equality means that no hour stays or all available hours are
being used for optimal production.

The painting and assembling labor condition is similar. The constraint's left-hand side
value is 1000 + 2000 = 3000, the right-hand side value is always equal to 3000; therefore, all
required painting and assembling labor hours are used. The last constraint relates to demand
limitation (the amount of cars created should be less than or equivalent to 2000). Because output
of 1000 cars is optimal (the constraint is non-binding), this is not the restriction that affects ABC,
Inc's production. The organization could increase cars production as it succeeded in having more
hours of work or developing the technologies ( i.e. reducing unit labor consumption).

Note: Slack variable is defined for the inequality of type “≤”. This variable assesses the
difference between the left-hand side and the right-hand side of the constraint. In both labor
constraints, zero is the slack variable in the example, as both sides of the constraints equal to each
other. In demand constraint, 1000 is the slack variable, as the right-hand side value is 2000 and
left-hand side value is 1000. In inequality of type “≥”, we express a surplus variable assessing the
difference between the left-hand side and the right-hand side of the constraint. For example, if
ABC produce at least 600 cars because of demand requirements (x1 ≥ 600) then ABC will actually
produce 1000 cars, therefore, 400 is the surplus variable.

The estimation of the effect on the optimal solution of the modifications in a particular
environment is the concept of a post-optimal (sensitivity) analysis. We have forecast future
increases in resource stocks (available labor hours) in technology, so it is important to examine
how shifts in commodity prices impact profit.

Figure 2.10 we compares the original objective function z=450x1+550x2 with the function
z=750x1 + 550x2 (the price of the cars changes from 550 to 850 ), getting a new optimal solution
(2000, 1000) with the profit z = 2,050,000.

Sensitivity analysis defines, typically for coefficients of objective function and right-hand
side values, the array of their potential changes that do not have the cardinal effect on the optimal
solution. Changing the car’s price from 550 to 649 (i.e. the objective function coefficient was
improved from 450 to 549), the optimal solution would be the original corner point (1000, 2000).

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Only modification would be higher profit, while the production assembly would be alike. If the
price increased to 651, the optimal solution (2000, 1000) would be attained, and resulting in this
recommendation the business should change the production assembly from the base. The price
650 provides the business the option to manufacture either 1000 cars (and 2000 boats) or 2000
cars (and 1000 boats).

Modification in coefficients of objective function impacts only its slope, whereas the
feasible area remains static. The best corner point was determined by the slope of the function. If
right-hand side value is modified the feasible area is also modified.

Figure 2.11 Change in Painting and Assembling Labor

The constraint’s borderline’s movement was due to change (the new borderline is parallel
with the original). Smaller change in the right-hand side value means smaller change in the feasible
area while big changes in the feasible area makes the cardinal change as shown by Figure 2.11
(decrease in available painting and assembling labor from 3000 to 2000 hours). Corner points
original number of 5 decreases to 3. In this situation, new optimal solution is presented by the
corner point (0, 2000).

As the optimal corner point is out of the constraint of woodwork labor, there are several
idle hours of this labor, i.e. in this inequality the slack variable now is nonzero. The precise sum
remaining can be calculated simply by inserting the coordinates (0, 2000) into the constraint's left-
hand side. When 0 + 2(2000) = 4000, the amount of hours remaining unused is 5000-4000 = 1000.

Note: These modifications can be analyzed independently, i.e. only one parameter is modified (the
other is fixed), so we are evaluating the effect on the optimal solution and the profit. The basic
aspects of the sensitivity analysis are the lowered costs and the shadow prices along with the levels
of objective function coefficients and the ranges on right-hand hands.

2.4 Special Cases of Linear Programming Models

Only one optimal solution has been attained in the production problem of ABC, Inc. It is
a distinctive in most cases linear programming problems. Figure 2.12 express the feasible area
together with the objective function z (the arrow specifies the improvement direction’s of the
objective value). Unique optimal solution is the corner point A.

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In the previous discussion, the price of the car is 650 (instead of 550), the management of
ABC, Inc. has two basic possibilities how to optimize the profit. Producing 1000 cars (and 2000
boats) or 2000 cars (and 1000 boats) would bring the identical optimal profit 1,650,000.

A linear programming problem with two or more optimal solutions is said to have
alternative (or multiple) optimal solutions.

This condition occurs in graphical depiction of the model when the objective function line
is parallel to the borderline of one constraint, as can be shown in Figure 2.13. Because the objective
function reflects the isoprofit line (in case of maximization) or the isocost line (in case of
minimization), all the solutions at the edge of the feasible area have the same objective value
(optimal). There are two optimal corner points (B and C) and the infinite number of optimal points
on line segment BC.

Figure 2.14 shows an interesting alternative of just described case. The difference is
evident – the feasible area is unbounded and there is only one optimal corner point D; the other
optimal points lie on the borderline running to infinity.

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There is no optimal solution if the feasible solution area is unbounded and the objective
value is being enhanced in the direction of unboundedness, therefore, the optimal solution is
endless (see Figure 2.15). Since these instances are uncommon in reality, this finding typically
indicates mistake in formulation.

Infeasibility exists when no solution meets any of the constraints. There is no common
feasible area (no feasible solution) in case of two constraints graphed in Figure 2.16. In reality this
scenario sometimes occurs, particularly if the manager wants to be too accurate in model
formulation. In order to remove the infeasibility, the model must be simplify by for example
eliminating any vain restrictions.

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