MME3271_Lecture02_IntroductionToTheQuantitativeAnalysis

Lecture #2Introduction to the Quantitative Analysis

International Islamic University Malaysia

Manufacturing and Materials Engineering 1

Introduction to the Quantitative Analysis



Problem Solving & Decision Making Quantitative Analysis & Decision Making Quantitative Analysis Process

1. Model Development• Mathematical Model• Example #1

2. Data Preparation3. Model Solution4. Report Generation

Example #2 Example #3 Group Assignment #2

Problem Solving & Decision Making (1)



1• Define the problem

2• Identify the alternative

3• Determine the criteria for

evaluating

4• Evaluate the alternative

5• Choose an alternative

6• Implement selected alternative

7• Evaluate the result

7 steps of problem solving

• 7 Steps of Problem Solving

Decision MakingProcess

Structuring the problem

Analyzing the

problem




• Types of analysis for decision making

• based largely on the manager’s judgment and experience

• includes the manager’s intuitive “feel” for the problem

• is more of an art than a science

Qualitative Analysis

• analyst will concentrate on the quantitative facts or data associated with the problem

• analyst will develop mathematical expressions that describe the objectives, constraints, and other relationships that exist in the problem

• analyst will use one or more quantitative methods to make a recommendation

Quantitative Analysis




• Our focus in this course would be;

• based largely on the manager’s judgment and experience

• includes the manager’s intuitive “feel” for the problem

• is more of an art than a science

Qualitative Analysis

• analyst will concentrate on the quantitative facts or data associated with the problem

• analyst will develop mathematical expressions that describe the objectives, constraints, and other relationships that exist in the problem

• analyst will use one or more quantitative methods to make a recommendation



Qualitative Analysis & Decision Making (1)



• Potential reasons for a ‘quantitative analysis’ approach to decision making

Reasons:When the problem;

Complex

Important

New

Repetitive

Qualitative Analysis & Decision Making (2)



• Quantitative Analysis Process

1• Model Development

2• Data Preparation

3• Model Solution

4• Report Generation

Model Development (1)






3• Model Solution


• Model Development




• Models are representation of real of real objects or real situations.

• Three form of models are:

• physical replicas (scalar representations) of real objectsIconic Model

• physical in form, but do not physically resemble the object being modeledAnalog Model

• represent real world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses

Mathematical Model




• Advantages of experimenting with models compares to real situation

* The more closely the model represents the real situation, the accurate the conclusions and predictions will be.

Advantages

Less time

Less riskLess expensive




• Our focus in this course would be;

• physical replicas (scalar representations) of real objectsIconic Model

• physical in form, but do not physically resemble the object being modeledAnalog Model

• represent real world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses

Mathematical ModelMathematical Model

Mathematical Model (1)



• Elements of Mathematical Model

• a mathematical expression that describes the problem’s objective, such as maximizing profit or minimizing cost

Objective Function

• a set of restrictions or limitations, such as production capacitiesConstraints

• environmental factors that are not under the control of the decision maker

Uncontrollable Inputs

• controllable inputs; decision alternatives specified by the decision maker, such as the number of units of Product X to produce

Decision Variables




• Type of Mathematical Model

• if all uncontrollable inputs to the model are known and cannot vary

Deterministic Model

• if any uncontrollable are uncertain and subject to variation

Stochastic *(or probabilistic)

* Stochastic models are often more difficult to analyze




• Cost/benefit consideration must be made in selecting an appropriate mathematical model

• Frequently a less complicated (and perhaps less precise) model is more appropriate than a more complex and accurate one due to cost and ease of solution considerations

Selecting an appropriate

model

• Selecting an appropriate model for a problem




• Transforming Model Input into Output

Controllable Input (Decision

Variables)

Mathematical Model

Output (Projected

Result)

Uncontrollable

Input

(Environmental

Factors)

Example 1: Project Scheduling (1)



Consider the construction of a 250-unit apartment complex. The project consists of hundreds of activities involving excavating, framing, wiring, plastering, painting, landscaping, and more. Some of the activities must be done sequentially and others can be done at the same time. Also, some of the activities can be completed faster than normal by purchasing additional resources (workers, equipment, etc.).




• What is the best schedule for the activities and for which activities should additional resources be purchased? How could management science be used to solve this problem?

Question

• Management science can provide a structured, quantitative approach for determining the minimum project completion time based on the activities' normal times and then based on the activities' expedited (reduced) times.

Answer




• What would be the uncontrollable inputs?Question

• Normal and expedited activity completion times• Activity expediting costs• Funds available for expediting• Precedence relationships of the activities

Answer




• What would be the decision variables of the mathematical model? The objective function? The constraints?

Question

• Decision variables: which activities to expedite and by how much, and when to start each activity

• Objective function: minimize project completion time• Constraints: do not violate any activity precedence

relationships and do not expedite in excess of the funds available

Answer




• Is the model deterministic or stochastic?Question

• Stochastic. Activity completion times, both normal and expedited, are uncertain and subject to variation. Activity expediting costs are uncertain. The number of activities and their precedence relationships might change before the project is completed due to a project design change

Answer




• Suggest assumptions that could be made to simplify the modelQuestion

• Make the model deterministic by assuming normal and expedited activity times are known with certainty and are constant. The same assumption might be made about the other stochastic, uncontrollable inputs

Answer

Data Preparation (1)






3• Model Solution


• Data Preparation

Data Preparation (2)



Data Preparation

• Careful with time constraint & data collection error

• A model with 50 decision variables and 25 constraints could have over 1300 data elements!

• Often, large database is needed• System specialist might be needed

Model Solution (1)






3• Model Solution


• Model Solution

Model Solution (2)



Ɛ = Solutions

FEASIBLE(Satisfied all constraints)

INFEASIBLE(Do not satisfied all constraints)

OPTIMAL SOLUTION(The best solution)

Model Solution (3)



• Approaches to Solution

• Might not provide the best solution• Inefficient (numerous calculation required)Trial & Error

• Some small models/problems can be solved by hand calculations

• Most practical applications require using a computer

Special Solution Procedure

(Math. Model)

Model Solution (4)



• A variety of software packages are available for solving mathematical models.

Microsoft Excel The Management Scientist LINGO

Model Solution (5)



Model Testing and Validation:

Generate Solution(Math. Model)

Asses goodness/accurac

y

(Test on part of the model which solution is known

or expected )

NO Corrective Actions(collect more accurate data,

modify the model)

Use the model on full-scale problem

YES

Report Generation (1)






3• Model Solution

4• Report Generation• Report Generation

Report Generation (2)



Result of the

model

Managerial Report Implementation & Follow-up

Easily understood Should includes• Recommended decision• Other pertinent

information such as sensitivity of the

assumption on the model

Successful implementation is critical

Maximize user involvement in modeling process

Continuous monitoring on model contribution

Refine or expand the model if needed

Example 2: Austin Auto Auction (1)



An auctioneer has developed a simple mathematical model for deciding the starting bid he will require when auctioning a used automobile.

Essentially, he sets the starting bid at seventy percent of what he predicts the final winning bid will (or should) be. He predicts the winning bid by starting with the car's original selling price and making two deductions, one based on the car's age and the other based on the car's mileage.

The age deduction is $800 per year and the mileage deduction is $.025 per mile.




Question:Develop the mathematical model that will give the starting bid (B ) for a car in terms of the car's original price (P ), current age (A) and mileage (M ).

Answer:The expected winning bid can be expressed as:P - 800(A) - .025(M)

The entire model is: B = .7 x (expected winning bid) B = .7 x ( P - 800(A) - .025(M) )

B = .7P - 560A - .0175M




Question:Suppose a four-year old car with 60,000 miles on the odometer is being auctioned. If its original price was $12,500, what starting bid should the auctioneer require?

Answer:B = .7P - 560A - .0175M

= .7(12,500) - 560(4) - .0175(60,000) = $5,460




Question:The model is based on what assumptions?

Answer:The model assumes that the only factors influencing the value ofa used car are the original price, age, and mileage (not condition,rarity, or other factors).

Also, it is assumed that age and mileage devalue a car in a linearmanner and without limit. (Note, the starting bid for a very oldcar might be negative!)

Example 3: Iron Works, Inc. (1)



Iron Works, Inc. manufactures two products made from steel and just received this month's allocation of b pounds of steel. It takes pounds of steel to make a unit of product 1 and pounds of steel to make a unit of product 2.

Let and denote this month's production level of product 1 and product 2, respectively. Denote by and the unit profits for products 1 and 2, respectively.

Iron Works, Inc. has a contract calling for at least m units of product 1 this month. The firm's facilities are such that at most n units of product 2 may be produced monthly.




Information for the Iron Works problem(Always make this table to formulate your model)

Resource Product 1 Product 2 Available

Steel b

Production

Unit Profit -




The total monthly profit = (profit per unit of product 1) x (monthly production of product 1) +

(profit per unit of product 2) x (monthly production of product 2) = p1x1 + p2x2

We want to maximize total monthly profit:Max p1x1 + p2x2

The total amount of steel used during monthly production equals= (steel required per unit of product 1)x (monthly production of product

1) + (steel required per unit of product 2) x (monthly production of

product 2) = a1x1 + a2x2

This quantity must be less than or equal to the allocated b pounds of steel: a1x1 + a2x2 < b




Other Constraints:

The monthly production level of product 1 must be greater than or equal to m :

x1 > m

The monthly production level of product 2 must be less than or equal to n :

x2 < n

However, the production level for product 2 cannot be negative:

x2 > 0




Max +

s.t. + ≤ b ≥ m ≤ n ≥ 0

Mathematical Model Summary

Objective Function :

To maximize the profit

Subject to: Several

constraints}




Question:Suppose b = 2000, a1 = 2, a2 = 3, m = 60, n = 720, p1 = 100, p2 = 200. Rewrite the model with these specific values for the uncontrollable inputs.

Answer:Substituting, the model is:

Max 100x1 + 200x2

s.t. 2x1 + 3x2 < 2000 x1 > 60

x2 < 720 x2 > 0




Question:The optimal solution to the current model is x1 = 60 and x2 = 626 . If the product were engines, explain why this is not a true optimal solution for the "real-life" problem.

Answer:One cannot produce and sell of an engine. Thus, the problem is further restricted by the fact that both x1 and x2 must be integers. (They could remain fractions if it is assumed these fractions are work in progress to be completed the next month.)




Controllable Input (Decision

Variables)

Mathematical Model

Output (Projected

Result)

Uncontrollable

Input

(Environmental

Factors)




60 units Prod. 1626.67 units Prod. 2

Max 100x1 + 200x2

s.t. 2x1 + 3x2 < 2000 x1 > 60

x2 < 720 x2 > 0

Profit = $131,333.33Steel Used = 2000)

$100 profit per unit Prod.

1

$200 profit per unit Prod.

2

2 lbs. steel per unit Prod.

1

3 lbs. Steel per unit Prod.

2

2000 lbs. steel allocated

60 units minimum Prod. 1

720 units maximum Prod.

2

0 units minimum Prod. 2

Group Assignment #2 (1)



Due on Monday 30/9/2013 Question 4 [20]• Your mother owns a Cake House shop. Every Friday, she

produces three special items – red velvet cupcake, french macaroon, and cream puffs. Design a problem (your own, be creative and creates your own reasonable values), and formulate a mathematical model to help your mother maximize her profit. Your model must contain three decision variables and seven constraints.




Recall: Information for the Iron Works problem

Resource Product 1 Product 2 Available

Steel b

Production

Unit Profit -




Information for the Cake House problem

Resource Cupcake Macaroon Cream-Puff Available

Decision Variable 1

Decision Variable 2

Decision Variable 3

Unit Profit -




Max +

s.t. + ≤ b ≥ m ≤ n ≥ 0

Recall: Mathematical Model Summary for Iron Works problem



Subject to: Several

constraints}




Max ………………………………….s.t. ………………

………………. ……………… ………………. ……………… ………………. ……………….

Mathematical Model Summary for Cake House problem



Subject to: 7 constraints}

Next Lecture



Decision Analysis

Problem formulation• Influence Diagram• Payoff Table• Decision Tree

Decision Making without Probabilities• Optimistic Approach• Conservative Approach• Minimax Regret Approach

MME3271_Lecture02_IntroductionToTheQuantitativeAnalysis

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