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Engineering Management

Lecture ByProf. Dr. Naseer Ahmed

Email: naseer@cecos.edu.pk

Department of Mechanical EngineeringCECOS University

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Management ScienceModel and Their analysis• A model is an abstraction or simplification of

reality, designed to include only the essential features that determine the behaviour of a real system

• For example, a three dimensional physical model of a chemical processing plant might include scale models of major equipment and large diameter pipes, but it would not normally include small piping or electrical wiring

• Most of the models of management science are mathematical models.

• These can be as simple as the common equation representing the financial operation of a company

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Management ScienceModel and Their analysis• Net income = revenue – expenses – taxes• On the other hand, they may involve a very

complex set of equations• As an example, the Urban Dynamics model was

created by Jay Forrester to simulate the growth and decay of cities

• This model consisted of 154 equations representing relationships between the factors that he believed were essential: three economic classes of workers (managerial/professional, skilled, and unemployed), three corresponding classes of housing, three types of industry (new, mature and declining), taxation and land use

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Management ScienceModel and Their analysis• The value of these factors evolved through 250

simulated years to model the changing characteristics of a city

• Even these 154 relationships still proved too simplistic to provide any reliable guide to urban development policies

• Management sciences uses a five step process that begins in the real world, moves into the model world to solve the problem, then returns to the real world for implementation

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Management ScienceModel and Their analysisReal World Simulated (model) world

Formulate the problem (defining objectives, variables and constraints)

Construct a mathematical model (a simplified yet realistic representation of the system)

Test the model’s ability to predict the present from the past, and revise until you are satisfied

Derive a solution from the model

Apply the model’s solution to the real system, document its effectiveness, and revise further as required

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Management ScienceThe Analyst and the Manager• To be effective, the management science analyst

cannot just create models in an “ivory tower”• The Problem-solving team must include

managers and others from the department or systems being studied – to establish objectives, explain system operations, review the model as it develops from an operative perspective, and help test the model

• The user who has been part of model development, has developed some understanding of it and confidence in it, and feels a sense of “ownership” of it is most likely to use it effectively

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Management ScienceThe Analyst and the Manager• The manager is not likely to have a detailed knowledge

of management science techniques, nor the time for model development

• Today’s manager should, however, understand the nature of management science tools and the types of management situations in which they might be useful

• Increasingly, management positions are being filled with graduates of management (or engineering management) programs that have included an introduction to the fundamentals of management science and statistics

• Regrettably, all too few operations research or management science programs require the introduction to organization and behavioural theory that would help close the manager/analyst gap from the opposite direction

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Management ScienceThe Analyst and the Manager• There is considerable discussion today of the effect

of computers and their applications (management science, decision support systems, expert systems, etc.) on managers and organizations

• Certainly, workers and managers whose jobs are so routine that their decisions can be reduced to mathematical equations have reason to worry about being replaced by computers

• For most managers, however, modern methods offer the chance to reduce the time one must spend on more trial matters, freeing up time for the types of work and decisions that only people can accomplish

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Tools for Decision MakingCategories of Decision Making• Decision making can be discussed conveniently

in three categories: decision making under certainty(confidence), under risk, and under uncertainty

• The payoff table, or decision matrix will help in this discussion

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Tools for Decision MakingCategories of Decision Making

Payoff Table

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Tools for Decision MakingCategories of Decision Making• Our discussion will be made among some

number of alternatives, identified as • There may be more that one future “State of

nature” (The model allows for different futures)• These future states of nature may not be equally

likely, but each state will have some (known or unknown) probability of occurrence . Since the future must take on one of n values of , the sum of values of must be 1.0.

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Tools for Decision MakingCategories of Decision Making• The outcome (or payoff, or benefit gained) will

depend on both the alternative chosen and the future state of nature that occurs

• For example, if you choose alternative and the state of nature takes place (as it will with probability ), the payoff will be outcome . A full payoff table will contain times possible outcomes.

• Let us consider what this model implies and the analytical tools we might choose to use under each of our three classes of decision making

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Decision making under certainty• Decision making under certainty implies that we

are certain of the future state of nature (or we assume that we are). In our model, this means that the probability of future is 1.0, and all other futures have zero probability

• The solution, naturally is to choose the alternative that gives us the most favourable outcome . Although this may seem like a trivial exercise, there are many problems that are so complex that sophisticated mathematical techniques are needed to find the best solution

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Decision making under certaintyLinear Programming• One common technique for decision making

under certainty is called linear programming• In this method, a desired benefit (such as profit)

can be expressed as a mathematical function (the value model or objective function) of several variables

• The solution is the set of values for the independent variables (decision variables) that serves to maximize the benefits (or, in many problems, to minimize the cost) subject to certain limits (constraints)

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Decision making under certaintyLinear Programming: Example• Consider a factory producing two products, product X

and Y• The problem is this: if you can realize $10 profit per unit

of product X and $14 per unit of product Y, what is the production level of x units of product X and y units of product Y that maximizes the profit P? that is you seek to

• You can get a profit of– $350 by selling 35 units of X or 25 units of Y– $700 by selling 70 units of X or 50 units of Y– $620 by selling 62 units of X or 44.3 units of Y; or (as

in the first two cases as well) any combination of X and Y on the isoprofit line connecting these two points

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Decision making under certaintyLinear Programming: Example

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Decision making under certaintyLinear Programming: Example• Your production, and therefore your profit, is

subject to resource limitations, or constraints• Assume in this example that you employ five

workers– Three machinists– Two assemblers

• And that each works only 40 hours a week• Product X and/or Y can be produced by these

workers subject to the following constraints:– Product X require three hours of machining and

one hour of assembly per unit– Product Y require two hours of machining and two

hours of assembly per unit

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Decision making under certaintyLinear Programming: Example• These constraints are expressed mathematically as

follows

• Since there are only two products, these limitations can be shown on a two-dimensional graph

• Since all relationships are linear, the solution to our problem will fall at one of the corners

• To find the solution, begin at some feasible solution (satisfying the given constraints) such as (x, y) = (0, 0), and proceed in the direction of “steepest ascent” of the profit function (in this case, by increasing production of Y at $ 14 profit per year) until some constraint is reached

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Decision making under certaintyLinear Programming: Example• Since assembly hours are limited to 80, no more than

80/2=40 units of Y can be made, earning 40X$14 = $560 profit

• Then proceed along the steepest allowable ascent from there (along the assembly constraint line) until another constraint (machining hours) is reached

• At that point, (x, y) = (20, 30) and profit

• Since there is no remaining edge along which profit increases, this is the optimum solution

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Decision making under certaintyLinear Programming: Example

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Decision making under certaintyLinear Programming: Computer Solution• About 50 years ago George Danzig of Stanford

University developed the simplex method, which expresses the foregoing technique in a mathematical algorithm that permits computer solution of linear programming problems with many variables (dimensions), not just the two (assembly and machining) of this example

• Now linear programs in a few thousand variables and constraints are viewed as “Small”

• Problems having tens or hundreds of thousands of continuous variables and constraints are regularly solved; tractable integer programs are necessarily smaller, but are still commonly in the hundreds or thousands of variables and constraints

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Decision making under certaintyLinear Programming: Computer Solution• Another classic linear programming application is the oil

refinery problem, where profit is maximized over a set of available crude oils, process equipment limitations, product with different unit profits, and other constraints

• Other applications include assignment of employees with differing aptitude to the jobs that need to be done to maximize the overall use of skills; selecting the quantities of items to be done to maximize the overall use of skills; selecting the quantities of items to be shipped from a number of warehouses to a variety of customers while minimizing transportation cost; and many more

• In each case there is one best answer, and the challenge is to express the problem properly so that it fits a known method of solution

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Assignment

• You operate a small wooden toy company making two products: alphabet blocks and wooden trucks. Your profit is $30 per box of blocks and $40 per box of trucks. Producing a box of blocks requires one hour of woodworking and two hours of painting; producing a box of trucks takes three hours of woodworking, but only one hour of painting. You employ three woodworkers and two painters, each working 40 hours a week. How many boxes of blocks (B) and trucks (T) should you make each week to maximize profit? Solve graphically as a linear program and confirm analytically.