Quantitative Decision Techniques Basic Concepts 02022009 Dilay elebi

Quantitative Decision Techniques Basic Concepts 02/02/2009 Dilay Çelebi

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

MATHEMATICAL MODELING A model is a selective abstraction of reality* A model is a representation of a situation** A mathematical model is an abstract mathematical representation of a problem situation. Mathematical Programming(Modeling), MP, is the use of mathematical models, particularly optimizing models, to assist in taking decisions *Introductory Management Science, F. J. Gould, G. D. Eppen, C. P. Schmidt, 1993. **Quantitative Analysis for Management, 9 th Edition, Barry Render, Ralph M. Stair, M. Hanna, 2006.

Quantitative Models • QM uses mathematics to represent the relationship between data of interest. • Data should be quantifiable! City A 120 km City B V=30 km/hour How long does it take to get to city B from city A? A model usually simplifies reality Incorporate enough detail into your model so that 1. The result meets your needs 2. 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 Decision Variables 120 km City A Speed, stop points, road City B V=? Stop at Restaurant? Objectives Minimize total travel time Be in City B around 9 Constraints Max speed of car Number of hours in the rest. Decision Models • Selectively describe the environment • Designate decision variables • Designate objectives • Defined by constraints Introductory Management Science, F. J. Gould, G. D. Eppen, C. P. Schmidt, 1993.

Optimization Models 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 The goal of an optimization model is to make some function of the decision variables as large or as small as possible. * EXAMPLES Profit maximization Cost minimization Minimization of waiting times Maximum use of capacity Minimum working hours *Introductory Management Science, F. J. Gould, G. D. Eppen, C. P. Schmidt, 1993.

QUANTITATIVE ANALYSIS Decision Making Process Problem Definition Quantitative Analysis Conflicting Viewpoints Problem Definition Assumptions and Boundaries Search for Alternatives Observations Decision Variables Mathematical Relationships Model Construction Evaluation Constraints Developing a Solution Testing the Solution Selection of Best Alternative Choice Analyzing the Results 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 Problem First 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 System Collect 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, 9 th Edition, Taylor B. W. , 2007.

THE SEVEN - STEP MODEL BUILDING PROCESS Step 3. Formulate a Mathematical Model of the Problem Develop a mathematical model (in other words an idealized representation) of the problem. Step 4. Verify the Model and Use the Model for Prediction Try 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 Alternative Given 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, 9 th Edition, Taylor B. W. , 2007.

THE SEVEN - STEP MODEL BUILDING PROCESS Step 6. Present the Results and Conclusions of the Study to the Organization Present 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 Recommendation Implement 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, 9 th Edition, Taylor B. W. , 2007.

MANAGEMENT SCIENCE TECHNIQUES Deterministic models are models which do not contain the element of probability. These are primarily optimization models. Ø Linear Programming linear objective function – min/max q linear constraints q Ø Integer LP, Binary LP, Mixed Integer LP Ø Nonlinear Programming nonlinear objective function and/or q nonlinear constraints q

MANAGEMENT SCIENCE TECHNIQUES Ø Distribution Models special type of LP problems (special structure of model) q transportation problem q assignment problem q Ø Multicriteria Decision Making multiple criteria q compromise q limited/unlimited number of alternatives q goal programming q

MANAGEMENT SCIENCE TECHNIQUES Stochastic models are models which contain the element of probability. Ø Inventory Models when to order? q how much to order? q Ø Waiting Line Models (Queuing Models) servers, customers q goal – optimal number of server q Ø Computer Simulation computer experiments with models q complex systems q

MATHEMATICAL MODELS Reality Model Finding a proper balance between the level of simplification of the model and the good representation of reality. ______________________________________ Operations Research Jan Fábry

LINEAR PROGRAMMING The word linear comes from the Latin word linearis, which means created by lines 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: 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. A function f (x 1, x 2, ……. xn) of x 1, x 2, ……. xn is a linear function if and only if for some set of constants c 1, c 2, …. . cn , f (x 1, x 2, …. . xn) = c 1 x 1 + c 2 x 2 + c 3 x 3+……. . cnxn. For example, f (x 1, x 2)= 2 x 1 + x 2 is a linear function of x 1, x 2, but f (x 1, x 2)= 2 x 1 x 2 is not a linear function of x 1, x 2.

LINEAR PROGRAMMING 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 (x 1, x 2, ……. xn) and any number b, the inequalities f (x 1, x 2, ……. xn) b and f (x 1, x 2, ……. xn) ) b are linear inequalities. For example, 2 x 1 + x 2 = 100 is a linear equation, 2 x 1 + x 2 ≤ 20 or x 1 + 5 x 2 ≥ 40 are linear inequalities, but x 1 x 2 ≥ 40 or x 12 x 2 ≤ 20 are not linear inequalities.

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). (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 The steps in formulating a linear program follow*: 1. Completely understand the managerial problem being faced 2. Identify the objective and the constraints 3. Define the decision variables 4. Use the decision the variables to write mathematical expressions for the objective function and the constraints. Render et al. , pp. 236

ELEMENTS OF LINEAR PROGRAMMING • 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.

Complete mathematical statement of the LP problem Decision Variables Zmax = 3 x 1+ 2 x 2 Objective Function Subject to (s. t. ) 2 x 1 + x 2 ≤ 100 x 1 + x 2 ≤ 80 x 1 ≥ 40 x 1 ≥ 0 x 2 ≥ 0

ELEMENTS OF LINEAR PROGRAMMING Constraints Restrictions 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.

Complete mathematical statement of the LP problem Zmax = 3 x 1+ 2 x 2 Subject to (s. t. ) 2 x 1 + x 2 ≤ 100 x 1 + x 2 ≤ 80 x 1 ≥ 40 x 1 ≥ 0 x 2 ≥ 0 Decision Variables Objective Function Constraints

General Form of LP Problems Decision Variables Objective Function Constraints

General Form of LP Problems

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 by a) 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.

Formulating Linear Programs 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 constraint b) add the demands of each of the variables to obtain the total demand for that resource c) 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.

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.

PRODUCTION MIX PROBLEM 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 - 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 X 2, the manufacture of X 1 soldiers will always contribute 3 X 1 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 X 1, the manufacture of X 2 trains uses 1 X 2 finishing hours and 1 X 2 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 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 is that each parameter (objective function coefficient, righthand 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 • Media Selecting Problem - Dorian Auto • Blending Problem - The Holiday Meal Turkey Ranch

REFERENCES • Introduction to Management Science, 9 th Edition, Taylor B. W. , Prentice Hall, New Jersey, 2007. ISBN: 0 -13 -1966133 -0, ITU Library Number: T 56. T 39. 1990/T 56. T 39 1986. • Quantitative Analysis for Management, 9 th Edition, Barry Render, Ralph M. Stair, M. Hanna, Prentice Hall, New Jersey, 2006. ISBN: 0 -13 -153688 -5, ITU Library Number: T 56. R 46 2006. • Fundamentals of Management Science, Efraim Turban, Jack R. Meredith, Plano, Tex. : Business Publications, 1981. ISBN: 025602393 X, ITU Library Number: HD 30. 23. T 87 1981 • Introductory Management Science, F. J. Gould, G. D. Eppen, C. P. Schmidt, Englewood Cliffs, N. J. : Prentice Hall, c 1993. ISBN: 0134864409, ITU Library Number: HD 30. 25. G 68 1993.