Quantitative Decision Techniques

Basic Concepts02/02/2009

Dilay Çelebi

1. MS/OR is the application of scientific methods, techniques and tools to problems involving the operations of systems so as to provide those in control of the operations with optimum solutions to the problems.

2. MS/OR is the application of the scientific method to the study of the operations of large, complex organizations or activities.

3. MS/OR is the application of the scientific method to the analysis and solution of managerial decision problems.

OPERATIONS RESEARCH

OR deals with making decisions based on modeling. Its origins date back to the second world war!

MATHEMATICAL MODELING

Mathematical Programming(Modeling), MP, is the use of mathematical models, particularly optimizing models, to assist in taking decisions

A model is a selective abstraction of reality*A model is a selective abstraction of reality*

A model is a representation of a situation**

A model is a representation of a situation**

*Introductory Management Science, F.J. Gould, G.D. Eppen, C.P. Schmidt, 1993.  **Quantitative Analysis for Management, 9th Edition, Barry Render, Ralph M. Stair, M. Hanna, 2006. 

A mathematical model is an abstract mathematical representation of a

problem situation.

A mathematical model is an abstract mathematical representation of a

problem situation.

Quantitative Models

• QM uses mathematics to represent the relationship between data of interest.

• Data should be quantifiable!

How long does it take to get to

city B from city A?

City A City B

V=30 km/hour

120 km

A model usually simplifies realityA model usually simplifies reality

Incorporate enough detail into your model so that1. The result meets your needs2. You can solve it in the time you have to devote

to the process

Incorporate enough detail into your model so that1. The result meets your needs2. You can solve it in the time you have to devote

to the process

Introductory Management Science, F.J. Gould, G.D. Eppen, C.P. Schmidt, 1993.  

Decision Models

V=?

City A City B120 km

Stop at Restaurant? Constraints

Max speed of carNumber of hours in the rest.

Decision Models• Selectively describe the environment• Designate decision variables• Designate objectives• Defined by constraints

Decision Models• Selectively describe the environment• Designate decision variables• Designate objectives• Defined by constraints

Decision VariablesSpeed, stop points, road

ObjectivesMinimize total travel timeBe in City B around 9

Introductory Management Science, F.J. Gould, G.D. Eppen, C.P. Schmidt, 1993.  

Optimization

Models

The goal of an optimization model is to make some function of the decision variables as large or as

small as possible.*

The goal of an optimization model is to make some function of the decision variables as large or as

small as possible.*

EXAMPLESProfit maximizationCost minimizationMinimization of waiting timesMaximum use of capacityMinimum working hours

Optimization means "the action of finding the best solution". Optimization modeling, is a branch of mathematical modeling which is concerned with finding the best solution to a problem

*Introductory Management Science, F.J. Gould, G.D. Eppen, C.P. Schmidt, 1993.  

QUANTITATIVE ANALYSIS

Problem Definition

Model Construction

Developing a Solution

Testing the Solution

Analyzing the Results

Problem Definition

Search for Alternatives

Evaluation

Choice

Decision Making Process Quantitative Analysis Conflicting Viewpoints

Assumptions and Boundaries

Decision Variables

Mathematical Relationships

Constraints

Observations

Selection of Best Alternative

Presentation

Implementation

Introductory Management Science, F.J. Gould, G.D. Eppen, C.P. Schmidt, 1993.  Fundamentals of Management Science, Efraim Turban, Jack R. Meredith, 1981

THE SEVEN - STEP MODEL BUILDING PROCESS

When quantitative approach is used to solve a problem of an organization, the following seven step model building procedure should be followed.

Step 1. Formulate the ProblemFirst define the organization's problem!Defining the problem includes specifying the organization's objectives and the parts of the organization (or system) that must be studied before the problem can be solved.

Step 2. Observe the SystemCollect data to estimate the value of parameters that affect the organization's problem. These estimates are used to develop (in Step 3) and evaluate (in Step 4) a mathematical model of the organization's problem.

Introduction to Management Science, 9th Edition, Taylor B.W., 2007.  

THE SEVEN - STEP MODEL BUILDING PROCESS

Step 3. Formulate a Mathematical Model of the ProblemDevelop a mathematical model (in other words an  idealized representation) of the problem.

Step 4. Verify the Model and Use the Model for PredictionTry to determine if the mathematical model developed in Step 3 is an accurate representation of reality. Even if a model is valid for the current situation, we must be aware of blindly applying it. Step 5. Select a Suitable AlternativeGiven a model and a set of alternatives, choose the alternative that best meets the organization's objectives.There may be more than one!

Introduction to Management Science, 9th Edition, Taylor B.W., 2007.  

THE SEVEN - STEP MODEL BUILDING PROCESS

Step 6. Present the Results and Conclusions of the Study to the OrganizationPresent the model and the recommendations from Step 5 to the decision making individual or group. After presenting the results, you may find that the organization does not approve of the recommendations. This may result from incorrect definition of the organization’s problem or from failure to involve decision maker from the start of the project. In this case, you should return to Step 1, 2, or 3.

Step 7. Implement and Evaluate RecommendationImplement the study. The system must be constantly monitored (and updated dynamically as the environment changes) to ensure that the recommendations enable the organization to meet its objectives.

Introduction to Management Science, 9th Edition, Taylor B.W., 2007.  

Linear Programming linear objective function – min/max linear constraints

Integer LP, Binary LP, Mixed Integer LP

Nonlinear Programming nonlinear objective function and/or nonlinear constraints

MANAGEMENT SCIENCE TECHNIQUES

Deterministic models are models which do not contain the element of probability. These are primarily optimization models.

special type of LP problems (special structure of model) transportation problem assignment problem

Distribution Models

MANAGEMENT SCIENCE TECHNIQUES

multiple criteria compromise limited/unlimited number of alternatives goal programming

Multicriteria Decision Making

when to order? how much to order?

servers, customers goal – optimal number of server

Inventory Models

Waiting Line Models (Queuing Models)

MANAGEMENT SCIENCE TECHNIQUES

Stochastic models are models which contain the element of probability.

computer experiments with models complex systems

Computer Simulation

___________________________________________________________________________ Operations Research Jan Fábry

Finding a proper balance between the level of simplification of the model and the good

representation of reality.

Reality Model

MATHEMATICAL MODELS

LINEAR PROGRAMMING

baxy

The word linear comes from the Latin word linearis, which means created by lines

c

ba

yx

yx

4323693

102

12

LINEAR FUNCTION:Each term is either a constant or the product of a constant times the first power of a variable.

LINEAR FUNCTION:Each term is either a constant or the product of a constant times the first power of a variable.

These are linear functions:

c

cb

a

xy

yx

4323

10

12 2

These are NOT linear functions:

LINEAR PROGRAMMING

A linear programming problem (LP) is an optimization problem for which we do the following:

1. We attempt to maximize (or minimize) a linear function of the decision variables. The function that is to be maximized or minimized is called the objective function.

For example, f (x1, x2)= 2x1 + x2 is a linear function of x1, x2, but f (x1, x2)= 2x1x2 is not a linear function of x1, x2.

A function f (x1, x2, …….xn) of x1, x2,…….xn is a linear function if and only if

for some set of constants c1, c2, …..cn , f (x1, x2,…..xn) = c1x1 + c2x2 + c3x3+……..cnxn.

A function f (x1, x2, …….xn) of x1, x2,…….xn is a linear function if and only if

for some set of constants c1, c2, …..cn , f (x1, x2,…..xn) = c1x1 + c2x2 + c3x3+……..cnxn.

2. The values of decision variables must satisfy a set of constraints. Each constraint must be a linear equation or a linear inequality.

For any linear function f (x1, x2, …….xn) and any number b,

the inequalities f (x1, x2, …….xn) b and

f (x1, x2, …….xn) ) b are linear inequalities.

For any linear function f (x1, x2, …….xn) and any number b,

the inequalities f (x1, x2, …….xn) b and

f (x1, x2, …….xn) ) b are linear inequalities.

For example, 2x1 + x2 = 100 is a linear equation, 2x1 + x2 ≤ 20 or x1 + 5x2 ≥40 are linear inequalities,but x1x2 ≥ 40 or x1

2x2 ≤ 20 are not linear inequalities.

LINEAR PROGRAMMING

LINEAR PROGRAMMING

3. A sign restriction is associated with each variable.

For any variable xi , the sign restriction specifies either that xi must be nonnegative (xi ≥ 0) or that xi may be unrestricted in sign (urs).

For any variable xi , the sign restriction specifies either that xi must be nonnegative (xi ≥ 0) or that xi may be unrestricted in sign (urs).

(An LP with unrestricted-in-sign variables is transformed into an LP in which all variables are required to be nonnegative because nonnegativity restrictions are essential for the development of the solution algorithm of the LP).

LINEAR PROGRAMMING

Render et al., pp. 236

The steps in formulating a linear program follow*:1. Completely understand the managerial problem

being faced2. Identify the objective and the constraints3. Define the decision variables4. Use the decision the variables to write mathematical

expressions for the objective function and the constraints.

The steps in formulating a linear program follow*:1. Completely understand the managerial problem

being faced2. Identify the objective and the constraints3. Define the decision variables4. Use the decision the variables to write mathematical

expressions for the objective function and the constraints.

• Decision Variables: In any linear programming model the decision variables should completely describe the decisions to be made (xi) .

• Objective Function: In any linear programming problem, the decision maker wants to maximize (usually revenue or profit) or minimize (usually costs) some function of the decision variables. The function to be maximized or minimized is called the objective function (Zmin or Zmax).

• The coefficient of a variable in the objective function is called the objective function coefficient of the variable (cj). The coefficients represent the contribution of one unit of variable to the company’s goal.

ELEMENTS OF LINEAR PROGRAMMING

Complete mathematical statement of the LP problem

Zmax = 3 x1+ 2x2

Subject to (s.t.)

2x1 + x2 ≤ 100  

x1 + x2 ≤ 80  

x1 ≥40  

x1 ≥ 0

x2 ≥ 0

Decision Variables

Objective Function

ConstraintsRestrictions on the values of decision variables

Right hand side of a constraints (bi) represent the quantity of a resource that is available. For a constraint to be reasonable, all term in the constraint must have the same units. Otherwise, the constraint will not have any meaning.

The coefficients of the decision variables in the constraints are called technological coefficients (aij). Technological coefficients often reflect the technology used to produce different product.

ELEMENTS OF LINEAR PROGRAMMING

Complete mathematical statement of the LP problem

Zmax = 3 x1+ 2x2

Subject to (s.t.)

2x1 + x2 ≤ 100

x1 + x2 ≤ 80

x1 ≥40

x1 ≥ 0

x2 ≥ 0

Decision Variables

Objective Function

Constraints

General Form of LP Problems

ursxxx

bxaxaxaxa

bxaxaxaxa

bxaxaxaxa

ts

xcxcxcxcZ

n

mnmnmmm

nn

nn

nn

,0,...,

,,....

,,....

,,....

..

....

21

332211

22323222121

11313212111

332211minmax,

Decision Variables

Objective Function

Constraints

General Form of LP Problems

ursx

bxa

ts

xcZ

j

i

n

jjij

n

jjj

,0

,,

..

1

1minmax,

Formulating Linear Programs

1. Choose the variables for the LP by considering the fundamental processes that the modeler can control.

2. Determine the objective function bya) Finding how much each variable contributes to the objective

per unit of that variable used

b) Adding the contributions of each of the variables to obtain the (linear) objective.

c) Determining whether the objective is to be maximized or minimized.

3. Determine the constraints of the problem, by formalizing the restrictions on the variables imposed by the modeler. Again, for each constraint:a) determine the demand of each variable for the resource

associated with that constraintb) add the demands of each of the variables to obtain the total

demand for that resourcec) determine whether that resource constitutes an upper bound

on the total demand (≤ constraint) a lower bound on the total demand (≥constraint) or an exact requirement on the total demand (= constraint).

4. Be sure to include the nonnegativity constraints if the variables require this.

Formulating Linear Programs

PRODUCTION MIX PROBLEM

A clothier makes coats and slacks.

The two resources required are wool cloth and labor.The clothier has 150 square yards of wool and 200 hours of labor available. Each coat requires 3 square yards of wool and 10 hours of labor, whereas each pair of slacks requires 5 square yards of wool and 4 hours of labor.

The profit for a coat is $50, and the profit for slacks is $40. The clothier wants to determine the number of coats and pairs of slacks to make so that profit will be maximized.

A jewelry store makes necklaces and bracelets from gold and platinum.

The store has 18 ounces of gold and 20 ounces of platinum. Each necklace requires 3 ounces of gold and 2 ounces of platinum, whereas each bracelet requires 2 ounces of gold and 4 ounces of platinum.

The demand for bracelet is no more than four.

A necklace earns $300 in profit and a bracelet, $400. The store wants to determine the number of necklaces and bracelets to make in order to maximize profit.

PRODUCTION MIX PROBLEM

PRODUCTION MIX PROBLEM - Giapetto's Woodcarving, Inc.

Assumptions of LP

Proportionality and Additivity Assumptions of Linear Programming

The fact that the objective function for an LP must be a linear function of the decision variables has two implications.

1.     The contribution of the objective function from each decision variable is proportional to the value of the decision variable. For example, the contribution of the objective function from making four soldiers (4 * 3 = $12) is exactly four times the contribution to the objective function from making one soldier ($3).

2.     The contribution to the objective function for any variable is independent of the values of the other decision variables. For example, no matter what the value of the X2, the manufacture of X1

soldiers will always contribute 3X1 to the objective function.

Analogously, the fact that each LP constraint must be a linear inequality or a linear equation has two implications.

1.     The contribution of each variable to the left-hand side of each constraint is proportional to the value of the variable. For example, it takes exactly three times as many finishing hours (2 * 3 = 6 finishing hours) to manufacture three soldiers as it takes to manufacture one soldier (2 finishing hours).

2.     The contribution of a variable to the left-hand side of each constraint is independent of the values of the variable. For example, no matter what the value of X1, the manufacture of X2

trains uses 1X2 finishing hours and 1X2 carpentry hours.

The first implication given in each list is called the Proportionality Assumption of Linear Programming.

 

Implication 2 of the first list implies that the value of the objective function is the sum of the contribution of from individual variables, and implication 2 of the second list implies that the left-hand side of each constraint is the sum of the contributions from each variable. For this reason, the second implication in each list is called the Additivity Assumption of Linear Programming.

The Divisibility Assumption

 

The Divisibility Assumption requires that each decision variable be allowed to assume fractional values. For example, in the Giapetto’s Problem, The Divisibility Assumption implies that it is acceptable to produce 1.5 soldiers or 1.63 trains. Since Giapetto cannot actually produce a fractional number of trains or soldiers, the Divisibility Assumption is not satisfied in the Giapetto Problem. A linear programming problem in which some or all of the variables must be nonnegative integers is called an integer programming problem.

The Certainty Assumption

 

The Certainty Assumption is that each parameter (objective function coefficient, right-hand side, and technological coefficient) is known with certainty. If we were unsure of the exact amount of carpentry and finishing hours required to build a train, the Certainty Assumption would be violated.

DIET PROBLEM

OTHER PROBLEMS

• Blending Problem - The Holiday Meal Turkey Ranch

• Media Selecting Problem - Dorian Auto

REFERENCES

• Introduction to Management Science, 9th Edition, Taylor B.W., Prentice Hall, New Jersey, 2007. ISBN: 0-13-1966133-0, ITU Library Number: T56.T39.1990/T56.T39 1986.

• Quantitative Analysis for Management, 9th Edition, Barry Render, Ralph M. Stair, M. Hanna, Prentice Hall, New Jersey, 2006. ISBN: 0-13-153688-5, ITU Library Number: T56.R46 2006.

• Fundamentals of Management Science, Efraim Turban, Jack R. Meredith, Plano, Tex. : Business Publications, 1981. ISBN: 025602393X, ITU Library Number: HD30.23.T87 1981

• Introductory Management Science, F.J. Gould, G.D. Eppen, C.P. Schmidt, Englewood Cliffs, N.J. : Prentice Hall, c1993. ISBN:0134864409, ITU Library Number: HD30.25.G68 1993.