CHAPTER 4: MODELING AND ANALYSIS

Chapter 4 in “DECISION SUPPORT AND BUSINESS INTELLIGENCE SYSTEMS”

Chapter 17 part5 in “OPERATION MANAGEMENT. HEIZER, RENDER”

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The Structure of MSS Mathematical Models

Include financial and engineering models

Link decision variables, uncontrollable variables, intermediate variables, and result variables together.

The modeling process involves identifying the variables and the relationships among them.

Solving a model determines the values of these variables.

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The Components of Mathematical Models

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The Components of Mathematical Models (cont.)

Result (Outcome) Variables They indicate how well the system

performs or attains its goal. They considered dependent variables Result variables are dependent on decision

variables and uncontrollable variables.• Decision variables:

Decision variables describe alternative choices.

The decision maker controls the decision variables.

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The Components of Mathematical Models (cont.)

Uncontrollable variables: They affect the result variables but are

outside decision-maker’s control. These factors can be fixed factors, called

parameters. They can vary, called variables

Intermediate Result variables: intermediate outcomes produce

intermediate result variables.

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Examples of Components of Models

Ex: Financial investment Result variable: total profit Decision variable: investment amounts. Uncontrollable variables: competition, tax Intermediate Result variable: cost, income

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The Structure of MSS Mathematical Models

The components are linked together by mathematical expression:

P=R-CP =profit, R= revenue (income) , C=cost

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Mathematical Programming Optimization

Mathematical programmingA family of tools designed to help solve managerial problems in which the decision maker must allocate scarce resources among competing activities to optimize a measurable goal

 Linear programming (LP) LP is best-known technique in the mathematical

programming family. It is used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints

It used to find the optimal solution of resource allocation problems.

All the relationships among the variables in this type of model are linear

Common Elements to LP

Decision variables Should completely describe the decisions to be made by

the decision maker Objective Function (OF)

A linear mathematical function that relates the decision variables to the goal, measures goal attainment, and to be optimized

Most frequent objective of business firms is to maximize profit

Most frequent objective of individual operational units (such as a production or packaging department) is to minimize cost

Objective function coefficients unit profit or cost coefficients indicating the contribution

to the objective of one unit of decision variable

Common Elements to LP

Constraints Restrictions on resources such as time,

money, labor, etc. Expressed in a form of linear inequalities

There are three types of constraints Upper limits where the amount used is ≤ the amount

of a resource Lower limits where the amount used is ≥ the amount

of the resource Equalities where the amount used is = the amount

of the resource Capacities

Upper or lower limits of constraints

Objective function and constraints must be linear

Certainty All coefficients are known with certainty We are dealing with a deterministic world

The return of any allocation is independent of the others

LP Assumptions

Examples of LP Problems and Applications

SCM: transportation models Product Mix Scheduling (Production and Labor)

The Shader Electronics Company produces two products, (1) the Shader Walkman, and (2) the Shader Watch-TV. Each Walkman takes four hours of electronic work and two hours in the assembly shop. Each Watch-TV requires three hours of electronics and one hour in assembly. During the current production period, 240 hours of electronic time and 100 hours of assembly department time are available. Each Walkman sold yields a profit of $7; each Watch-TV produced may be sold for a $5 profit. How many of each should be produced?

Formulating LP ProblemsThe product-mix problem at Shader Electronics

Formulating LP Problems (con.)

1. Decision variables:• Let X1 = ----------------• Let X2 = ----------------

 2. Subject to:

• First Constraint: ----------------• Second Constraint: ----------------• Third Constraint: ----------------

 3. Objective function coefficients:

• For x1 -------------• For x2--------------

 4. Objective Function: • Maximize Profit = ----------------

Walkman Watch-TVs Available HoursDepartment (X1) (X2) This Week

Hours Required to Produce 1 Unit

Electronic 4 3 240

Assembly 2 1 100

Profit per unit $7 $5

Decision Variables:X1 = number of Walkmans to be producedX2 = number of Watch-TVs to be produced

Table B.2

Excel Solver Solution

Simplex Solution

Before the simplex algorithm can be used to solve an LP, the LP must be converted into a LP problem in a standard form where:1. All the constraints (inequalities) are

converted to equations (equalities) 2. All variables are nonnegative. 

Standard Maximization Problems in Standard Form

A linear programming problem is said to be a standard maximization problem in standard form if its mathematical model is of the following form:

Maximize the objective function

Subject to problem constraints of the form

With non-negative constraints

max 1 1 2 2 ... n nZ P c x c x c x

1 1 2 2 ... , 0n na x a x a x b b

1 2, ,..., 0nx x x

Slack Variables

In real life problems, it’s unlikely that all resources will be used completely, so there usually are unused resources.

Slack variables represent the unused resources between the left-hand side and right-hand side of each inequality.

Used to express constraints inequalities as equalities

Sımplex Method

Step-1Write the standard

maximization problem in standard

form, introduce

slack variables to

form the initial system, and write the initial tableau.

Step-3Select

the pivot colum

n

Step-5Select

the pivot

element and perform the pivot

operation

STOPThe optimal solution has been

found.

STOPThe linear programming problem has no optimal

solution

Step 2Are

there any

negative indicators in the bottom row?

Step 4Are there

any positive

elements in the pivot

column above the

dashed line?

Simplex algorithm for standard maximization problems

To solve a LP problem in standard form1- Convert each inequality in the set of constraints to an equation by

adding slack variables. Create the initial simplex tableau.

2- Check are there any negative indicators in the bottom row

3- Select the pivot column. ( The column with the “most negative value” element in the last row.)

4- Select the pivot row. (The row with the smallest non-negative result when the last element in the row is divided by the corresponding in the pivot column.)

5-Use elementary row operations calculate new values for the pivot row so that the pivot is 1 (Divide every number in the row by the pivot number.)

6- Use elementary row operations to make all numbers in the pivot column equal to 0 except for the pivot number. If all entries in the bottom row are zero or positive, this the final tableau. If not, go back to step 2.

7- If you obtain a final tableau, then the linear programming problem has a maximum solution, which is given by the entry in the lower-right corner of the tableau.

Pivot

Pivot Column: The column of the tableau representing the variable to be entered into the solution mix.

Pivot Row: The row of the tableau representing the variable to be replaced in the solution mix.

Pivot Number: The element in both the pivot column and the pivot row.

The first step of the simplex method requires that each inequality be converted into an equation. ”less than or equal to” inequalities are converted to equations by including slack variables.

Suppose S1 electronic hours and S2 assembly hours remain unused in a week. The constraints become;

4X1 + 3X2 + s1 = 240

2X1 + 1X2 + s2 = 100

Step 1

The problem can now be considered as solving a system of 3 linear equations involving the 5 variables in such a way that P has the maximum value;

Now, the system of linear equations can be written in 3x6 matrix.

1 2 1 2, , , ,x x s s P

4X1 + 3X2 + s1 = 240

2X1 + 1X2 + s2 = 100

P - 7X1 + 5X2 = 0

Step 1(con.)

Basic Variables

x1 x2 S1 S2 PRight Hand Side

S1 4 3 1 0 0 240

S2 2 1 0 1 0 100

P -7 -5 0 0 1 0

The tableau represents the initial solution;

The slack variables S1 and S2 form the initial solution mix. The initial solution assumes that all avaliable hours are unused. i.e. The slack variables take the largest possible values.

1 2 1 20, 0, 240, 100, 0x x s s P

Step 1(con.)

The initial tableau is

Variables in the solution mix are called basic variables. Each basic variables has a column consisting of all 0’s except for a single 1. all variables not in the solution mix take the value 0.

The simplex process, a basic variable in the solution mix is replaced by another variable previously not in the solution mix. The value of the replaced variable is set to 0.

Step 1(con.)

Step 2

Yes , There are negative values in the last raw

We can go on step 3

Step 3

Select the pivot column (determine which variable to enter into the solution mix). Choose the column with the “most negative” element in the objective function row.

Basic Variable

sx1 x2 S1 S2 P

Right hand side

S1 4 3 1 0 0 240

S2 2 1 0 1 0 100

P -7 -5 0 0 1 0Pivot column

• x1 should enter into the solution mix because each unit

of x1 (a Walkmans ) contributes a profit of $7 compared with only $5 for each unit of x2 (a TV)

Step 4

No, There aren’t any positive elements in the pivot column above the dashed line.

We can go on step 5

Step 5

Select the pivot row (determine which variable to replace in the solution mix). Divide the last element in each row by the corresponding element in the pivot column. The pivot row is the row with the smallest non-negative result.

Basic Variable

sx1 x2 S1 S2 P

Right hand side

S1 4 3 1 0 0 240

S2 2 1 0 1 0 100

P -7 -5 0 0 1 0

240 / 4 60

100 / 2 50

Pivot columnPivot row

Enter

Exit

Pivot number

Step 5 (con)

S2 Should be replaced by x1 in the solution mix.

Now calculate new values for the pivot row. Divide every number in the row by the pivot number.

Basic Variables

x1 x2 S1 S2 PRight hand side

S1 4 3 1 0 0 240

x1 1 1/2 0 1/2 0 50

P -7 -5 0 0 1 0

2

2

R

Basic Variable

sx1 x2 S1 S2 P

Right hand side

S1 0 1 1 -2 0 40

x1 1 1/2 0 1/2 0 50

P 0 - 3/2 0 7/2 1 350

2 14.R R

Use row operations to make all numbers in the pivot column equal to 0 except for the pivot number which remains as 1.

If 50 Walkmans are made, then the unused electronic hours are reduced by 200 hours (4 h/Walkmans multiplied by 50 Walkmans ); the value changes from 240 hours to 40 hours. Making 50 Walkmans results in the profit being increased by $350; the value changes from $0 to $350.

Back to step 2

Step 5

7R2 + R3

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Multiple Goals, Sensitivity Analysis, What-If Analysis, and Goal Seeking

Multiple goalsRefers to a decision situation in which alternatives are evaluated with several and simultaneous, sometimes conflicting, goals.

Determining single measure of effectiveness is difficult.

Handling methods: Utility theory Goal programming Linear programming with goals as constraints Point system

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Multiple Goals, Sensitivity Analysis, What-If Analysis, and Goal Seeking

Sensitivity analysis Assesses impact of change in inputs or

parameters on solutions

Allows for adaptability and flexibility to changing condition.

Sensitivity analysis Can be:1. automatic2. trial and error

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Multiple Goals, Sensitivity Analysis, What-If Analysis, and Goal Seeking

Sensitivity analysis Can be:1. automatic:

Performed in standard quantitative model implementation such as LP.

It reports the range within which a certain input variable can vary without having any significant impact on the proposed solution.

It usually limited to one change at a time and only for certain value.

Lindo is used to do that 2. trial and error:

It determines the impact of changes in any variable or in several variables.

You change some input data and solve the problem again. Two approaches to do it: what-is analysis and goal seeking.

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Multiple Goals, Sensitivity Analysis, What-If Analysis, and Goal Seeking

What-If Analysis A process that involves asking a computer what the effect of changing some of the input data or parameters would be

Goal seeking Backwards approach, starts with goal Determines values of inputs needed to achieve

goal Example is break-even point determination:

Involves determining the value of the decision variables that generate zero profit

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Problem-Solving Search Methods

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Problem-Solving Search Methods Analytical techniques

Analytical techniques use mathematical formulas to derive an optimal solution directly or to predict a certain result

They are used for structured problems

Algorithms: An algorithm is a step-by-step search

process for obtaining an optimal solution Solutions are generated and tested for

possible improvements.

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Problem-Solving Search Methods Blind search Blind search techniques are arbitrary

search approaches that are not guided. There are two types:

In a complete enumeration all the alternatives are considered and therefore an optimal solution is discovered

In an incomplete enumeration (partial search) continues until a good-enough solution is found (a form of suboptimization)

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Problem-Solving Search Methods Heuristic searching

Informal, judgmental knowledge of an application area that constitutes the rules of good judgment in the field.

Heuristics also encompasses the knowledge of how to solve problems efficiently and effectively, how to plan steps in solving a complex problem, how to improve performance, and so forth