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BUS 2420Management Science

• Instructor: Vincent WS Chow

• Office: WLB 818

• Ext: 7582

• E-mail: vwschow@hkbu.edu.hk• URL: http://ww.hkbu.edu.hk/~vwschow

• Office hours: (to p2)

• Refer to my website

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Subject outlineSubject outline (see handout)

– Textbook:• Bernard W. Taylor III, Introduction to Management Science, 10th

Edition, Prentice Hall, 2010

– Grading:– Topics:

– Refer to handout

– Tutorials• Start from 3rd hr of 3rd week lecture• Typically, we assign few questions in each lecture

and then taken them up for discussion in the next week session.

• How you are being graded?

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(lecture)

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Grading:

• Assignments 15%• Most likely be1-3 assignments• Group Memberships (refer to our web site)

• Class Participation 15%• Tutorial performance

• Test 20%• One mid-term exam

• Examination 50%• One final exam

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How you are being graded?

• Students will award marks if they show their works (by submission!) in the tutorial sessions

• Students are thus strongly encouraged to bring their works to show in tutorials or prepare materials for presentation ..… – Note: you may like to approach me later to

see how we could improve this process of grading!

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Lecture 1Introduction to Management Science

• What is Management Science?• How to apply Management Science

technique?• Types of Management Science

Models/techniques• We start with the most popular

Management Science technique:• Linear Programming

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Have we seen or used then before?

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Management Science

• Management science uses a scientific approach to solving management problems.

• It is used in a variety of organizations to solve many different types of problems.

• It encompasses a logical mathematical approach to problem solving.

• History of Management Science (to p8)

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History of Management Science

• It was originated from two sources:– Operational Research– Management Information Systems

• It is thus more emphasizing on the analysis of solution applications than learning their on how models were derived.

• Other names for management science: quantitative methods, quantitative analysis and decision sciences.

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Steps in applying Management Science teniques

(1)

(2)

(3)

(4)

(5)

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In practice, this step is critical

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Steps

1. Observation — Identification of a problem that exists in the system or organisation.

2. Definition of the Problem — Problem must be clearly and consistently defined showing its boundaries and interaction with the objectives of the organisation.

3. Model Construction — Development of the functional mathematical relationships that describe the decision variables, objective function and constraints of the problem.

4. Model Solution — Models solved using management science techniques.

5. Model Implementation — Actual use of the model or its solution.

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Models to be consideredin this subject

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*

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* Topics that will cover in this subject!

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Characteristics of Modeling Techniques

• Linear mathematical programming: clear objective; restrictions on resources and requirements; parameters known with certainty.

• Probabilistic techniques: results contain uncertainty.• Network techniques: model often formulated as

diagram; deterministic or probabilistic.• Forecasting and inventory analysis techniques:

probabilistic and deterministic methods in demand forecasting and inventory control.

• Other techniques: variety of deterministic and probabilistic methods for specific types of problems.

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Linear Programming

• Or denote as LP• Overview of LP• How does LP look like?• Components of LP• General LP format • Example 1: Maximizing Z• Example 2: Minimizing Z• We will talk about more LP formulations

and its solutions in next lecture

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Linear Programming - An Overview

• Objectives of business firms frequently include maximizing profit

or minimizing costs, or denote as Max Z or Min Z

• Linear programming is an analysis technique in which linear

algebraic relationships represent a firm’s decisions given a business objective and resource constraints.

• Steps in application:

1- Identify problem as solvable by linear programming.

2- Formulate a mathematical model of managerial problems.

3- Solve the model.

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4 Components of LP

1. Decision variables: mathematical symbols representing levels of activity of a firm.

2. Objective function: a linear mathematical relationship describing an objective of the firm, in terms of decision variables, that is maximized or minimized

3. Constraints: restrictions placed on the firm by the operating environment stated in linear relationships of the decision variables.

4. Parameters: numerical coefficients and constants used in the objective function and constraint equations.

5. Non-negativity (or necessary) constraints

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Example of Decision Variables

Decision Variables:• It is used to represent decision problem to be

solve

Let,

x1=number of bowls to produce/day x2= number of mugs to produce/day

How of them are needed is depended on the nature of the problem!

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Objective Functions

• It is used to represent the type of problems we are to solve

– In this subject, we only emphasize to either1. Maximizing a profit margin or

2. Minimizing a production cost

Example:

An Objective function

maximize Z = $40x1 + 50x2

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Constraints

• It is also referred to resource constraints

• They are to indicate how much resources made available in a firm

• Example:

Resource Constraints:

1x1 + 2x2 40 hours of labor

4x1 + 3x2 120 pounds of clay

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Non-negativity constraints

• We assumed that all decision variables are carried out positive values (why?)

• Example:

Non-negativity Constraints:

x10; x2 0

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Sample of LP

Let xi be denoted as xi product to be produced, andi = 1, 2or

Let x1 be numbers of product x1 to be produced and x2 be numbers of product 21 to be produced

Maximize Z=$40x1 + 50x2

subject to 1x1 + 2x2 40 hours of labor 4x2 + 3x2 120 pounds of clay x1, x2 0

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Objective function

Constraints

Decisionvariables

Cost

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Max/Min Z : Σ cixi

subject to

Σ aij xij (=, ≤, ≥) bj , j = 1,…., n

xij ≥ 0, for i=1,…,m, j=1,…,n

General LP format

It means there are total of m decision variables n resource constraints

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General steps for LP formulation(to p22)

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Steps for LP formulation

• Step 1: define decision variables

• Step 2: define the objective function

• Step 3: state all the resource constraints

• Step 4: define non-negativity constraints

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Example 1: Max Problem

• A Maximisation Model• Example The Beaver Creek Pottery Company produces bowls and

mugs. The two primary resources used are special pottery clay and skilled labour. The two products have the following resource requirements for production and profit per item produced (that is, the model parameters).

• Resource available: 40 hours of labour per day and 120 pounds of clay per day. How many bowls and mugs should be produced to maximizing profits give these labour resources?

• LP formulation

Resource RequirementsProduct Labor

(hr/unit)Clay

(lb/unit)Profit

($/unit)Bowl 1 4 40

Mug 2 3 50

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Max LP problem

• Step 1: define decision variables

Let x1=number of bowls to produce/day

x2= number of mugs to produce/day

Step 2: define the objective function

maximize Z = $40x1 + 50x2

where Z= profit per day

Step 3: state all the resource constraints

1x1 + 2x2 40 hours of labor ( resource constraint 1)

4x1 + 3x2 120 pounds of clay (resource constraint 2)

Step 4: define non-negativity constraints

x10; x2 0

Complete Linear Programming Model:

\ maximize Z=$40x1 + 50x2

subject to

1x1 + 2x2 40

4x2 + 3x2 120

x1, x2 0

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Example 2: Min Z• A farmer is preparing to plant a crop in the spring. There are two brands of

fertilizer to choose from, Supper-gro and Crop-quick. Each brand yields a specific amount of nitrogen and phosphate, as follows:

• The farmer’s field requires at least 16 pounds of nitrogen and 24 pounds of phosphate. Super-gro costs $6 per bag and Crop-quick costs $3 per bag. The farmer wants to know how many bags of each brand to purchase in order to minimize the total cost of fertilizing.

• LP formulation

Chemical Contribution

BrandNitrogen(lb/bag)

Phosphate(lb/bag)

Super-gro 2 4

Crop-quick 4 3

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Min ZStep 1: define their decision variables

x1 number of bags of Super-gro,x2 number of bags of Crop-quick.

Step 2: define the objective functionMinimise Z 6x1 3x2

Step 3: state all the resource constraints

2x1 4x2 16, (resource 1)4x1 3x2 24 (resource 2)

Step 4: define the non-negativity constraints

x1 0, x2 0

Overall LP: Minimise Z 6x1 3x2

subject to 2x1 4x2 16, 4x1 3x2 24, x1 0, x2 0

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