NEW MEXICO INSTITUTE OF MINING AND TECHNOLOGY Department of Management Management Science for Engineering Management (EMGT 501) Fall, 2005 Instructor :

NEW MEXICO INSTITUTE OF

MINING AND TECHNOLOGY

Department of Management

Management Science for Engineering

Management (EMGT 501)

Fall, 2005

Instructor : Toshi Sueyoshi (Ph.D.)HP address : www.nmt.edu/~toshiE-mail Address : [email protected] : Speare 143-A

1. Course Description:

The purpose of this course is to introduce

Management Science (MS) techniques for

manufacturing, services, and public sector.

MS includes a variety of techniques used in modeling

business applications for both better understanding

the system in question and making best decisions.

MS techniques have been applied in many

situations, ranging from inventory management

in manufacturing firms to capital budgeting in

large and small organizations.

Public and Private Sector Applications

The main objective of this graduate course is to

provide engineers with a variety of decisional tools

available for modeling and solving problems in a

real business and/or nonprofit context.

In this class, each individual will explore how to

make various business models and how to solve

them effectively.

2. Texts -- The texts for this course:

(1) Anderson, Sweeney and Williams

An Introduction to Management Science,

South-Western

(2) Chang Yih-Long, WinQSB , John Wiley&Sons

3. Grading:

In a course, like this class, homework problems are essential. We will have homework assignments. Homework has significant weight. The grade allocation is separated as follows:

Homework 20%

Mid-Term Exam 40%

Final Exam 40%

The usual scale (90-100=A, 80-89.99=B, 70-79.99=C, 60-69.99=D) will be used.

Please remember no makeup exam.

4. Course Outline:

Week Topic(s) Text(s)

1 Introduction and Overview Ch. 1&2

2 Linear Programming Ch. 3&4

3 Solving Linear Programming Ch. 5

4 Duality Theory Ch. 6

5 No Class

6 Project Scheduling: PERT-CPM Ch. 10

7 Inventory Models Ch. 11

8 Review for Mid-Term EXAM

Week Topic(s) Text(s)

9 Waiting Line Models Ch. 13

10 Waiting Line Models Ch. 13

11 Decision Analysis Ch. 14

12 Multi-criteria Decision Ch. 15

13 No Class

14 Forecasting Ch. 16

15 Markov Process Ch. 17

16 Review for FINAL EXAM

Linear Programming (LP):

A mathematical method that consists of an objective

function and many constraints.

LP involves the planning of activities to obtain an

optimal result, using a mathematical model, in

which all the functions are expressed by a linear

relation.

0,0

1823

1220

401

53

21

21

21

21

21

xx

xx

xx

xx

xxMaximize

subject to

A standard Linear Programming Problem

Applications: Man Power Design, Portfolio Analysis

Simplex method:

A remarkably efficient solution procedure for

solving various LP problems.

Extensions and variations of the simplex method

are used to perform postoptimality analysis

(including sensitivity analysis).

1x 2x 3x 4x 5xZ

3x4x5x

(0)(1)(2)(3)

21 53 xxZ 1x 3x

2x 4x21 23 xx 5x 18

12

4

0

(0)

(1)

(2)

(3)

(a) Algebraic Form

(b) Tabular Form

Coefficient of: RightSide

Basic VariableZ

Eq.

1 -3 -5 0 0 0 00 1 0 1 0 0 00 2 0 0 1 0 120 3 2 0 0 1 18

Duality Theory:

An important discovery in the early development

of LP is Duality Theory.

Each LP problem, referred to as ” a primal

problem” is associated with another LP problem

called “a dual problem”.

One of the key uses of duality theory lies in the

interpretation and implementation of sensitivity

analysis.

PERT (Program Evaluation and Review

Technique)-CPM (Critical Path Method):

PERT and CPM have been used extensively to

assist project managers in planning, scheduling,

and controlling their projects.

Applications: Project Management, Project

Scheduling

A 2

B

C

E

M N

START

FINISH

H

G

D

J

I

F

LK

4

10

4 76

7

9

8

54

62

5

0

0

Critical Path

2 + 4 + 10 + 4 + 5 + 8 + 5 + 6 = 44 weeks

Decision Analysis:

An important technique for decision making in

uncertainty.

It divides decision making between the cases

of without experimentation and with

experimentation.

Applications: Decision Making, Planning

Oil0.5 0.3Favorable

0.75Dry

0.85Dry

a

e

d

c

b

f

g

h

Drill

Sell

Drill

Sell

Sell

DrillOil0.14

Oil0.25

0.5DryDo

seism

ic

surv

ey

Unfavorable

0.7

No seismic survey

decision forkchance fork

Markov Chain Model:

A special kind of a stochastic process.

It has a special property that probabilities,

involving how a process will evolve in

future, depend only on the present state of

the process, and so are independent of events

in the past.

Applications: Inventory Control, Forecasting

Queueing Theory:

This theory studies queueing systems by

formulating mathematical models of their

operation and then using these models to derive

measures of performance.

This analysis provides vital information for

effectively designing queueing systems that

achieve an appropriate balance between the

cost of providing a service and the cost

associated with waiting for the service.

SS ServiceS facilityS

CCCC

Served customers

Served customers

C C C C C C

Queueing system

CustomersQueue

Applications: Waiting Line Design, Banking, Network Design

Inventory Theory:

This theory is used by both wholesalers and retailers

to maintain inventories of goods to be available for

purchase by customers.

The just-in-time inventory system is such an example

that emphasizes planning and scheduling so that the

needed materials arrive “just-in-time” for their use.

Applications: Inventory Analysis, Warehouse Design

Economic Order Quantity (EOQ) model

Q

Q

atQ

Time t

Inventory level

Batchsize

a

Q

a

Q20

Forecasting:

When historical sales data are available, statistical

forecasting methods have been developed for using

these data to forecast future demand.

Several judgmental forecasting methods use expert

judgment.

Applications: Future Prediction, Inventory Analysis

1/99 4/99 7/99 10/99 1/00 4/00 7/00

The evolution of the monthly sales of a product illustrates a time series

Mon

thly

sal

es (

unit

s so

ld)

10,000

8,000

6,000

4,000

2,000

0

Introduction to MS/OR

MS: Management Science

OR: Operations Research

Key components: (a) Modeling/Formulation

(b) Algorithm

(c) Application

Management Science (MS)

(1) A discipline that attempts to aid managerial

decision making by applying a scientific approach

to managerial problems that involve quantitative

factors.

(2) MS is based upon mathematics, computer

science and other social sciences like economics

and business.

General Steps of MS

Step 1: Define problem and gather data

Step 2: Formulate a mathematical model to

represent the problem

Step 3: Develop a computer based procedure

for deriving a solution(s) to the

problem

Step 4: Test the model and refine it as needed

Step 5: Apply the model to analyze the

problem and make recommendation

for management

Step 6: Help implementation

Linear Programming (LP)

[1] LP Formulation

(a) Decision Variables :

All the decision variables are non-negative.

(b) Objective Function : Min or Max

(c) Constraints

nxxx ,,, 21

21 32 xxMinimize

0,0

414

343..

21

21

21

xx

xx

xxts

s.t. : subject to

[2] Example

A company has three plants, Plant 1, Plant 2, Plant 3. Because of declining earnings, top management has decided to revamp the company’s product line.

Product 1: It requires some of production capacity

in Plants 1 and 3.

Product 2: It needs Plants 2 and 3.

The marketing division has concluded that the

company could sell as much as could be

produced by these plants.

However, because both products would be

competing for the same production capacity in

Plant 3, it is not clear which mix of the two

products would be most profitable.

The data needed to be gathered:

1. Number of hours of production time available per week in each plant for these new products. (The available capacity for the new products is quite limited.)

2. Production time used in each plant for each batch to yield each new product.

3. There is a profit per batch from a new product.

Production Timeper Batch, Hours

Production TimeAvailable

per Week, HoursPlant

Product

Profit per batch

1

2

3

4

12

18

1 2

1 0

0 2

3 2

$3,000 $5,000

: # of batches of product 1 produced per week : # of batches of product 2 produced per week : the total profit per week

Maximizesubject to

1x2x

Z

0x,0x

18x2x3

12x2x0

4x0x1

x5x3

21

21

21

21

21

1x0 2 4 6 8

2x

2

4

6

8

10

Graphic Solution

0,0 21 xx

Feasibleregion

1x0 2 4 6 8

2x

2

4

6

8

10

0,0 21 xx

41 x

Feasibleregion

1x0 2 4 6 8

2x

2

4

6

8

10

0,0 21 xx

122 2 x41 x

Feasibleregion

1x0 2 4 6 8

2x

2

4

6

8

10

0,0 21 xx

122 2 x41 x

1823 21 xx

Feasibleregion

1x0 2 4 6 8 10

2x

2

4

6

8

21 5310 xxZ

21 5320 xxZ

21 53 xx Maximize:

21 5336 xxZ

)6,2(

The optimal solution

The largest value

Slope-intercept form: 21 53 xxZ

Zxx

5

1

5

312

nn2211 xcxcxc

22222121

11212111

bxaxaxa

bxaxaxa

nn

nn

0,,0,0 21

2211

n

mnmnmm

xxx

bxaxaxa

Maximize

s.t.

[4] Standard Form of LP Model

[5] Other Forms

The other LP forms are the following:

1. Minimizing the objective function:

2. Greater-than-or-equal-to constraints:

.2211 nn xcxcxcZ

inin22i11i bxaxaxa

Minimize

3. Some functional constraints in equation form:

4. Deleting the nonnegativity constraints for

some decision variables:

ininii bxaxaxa 2211

jx : unrestricted in sign jjj xnxpx

[6] Key Terminology

(a) A feasible solution is a solution

for which all constraints are satisfied

(b) An infeasible solution is a solution

for which at least one constraint is violated

(c) A feasible region is a collection

of all feasible solutions

(d) An optimal solution is a feasible solution

that has the most favorable value of

the objective function

(e) Multiple optimal solutions have an infinite

number of solutions with the same

optimal objective value

,23 21 xxZ

1x

0,0

1823

21

21

xx

xx

122 2 x4

and

Maximize

Subject to

Example

Multiple optimal solutions:

21 2318 xxZ

1x0 2 4 6 8 10

2x

2

4

6

8

Feasibleregion

Every point on this red line

segment is optimal,

each with Z=18.

Multiple optimal solutions

(f) An unbounded solution occurs when

the constraints do not prevent improving

the value of the objective function.

2x

1x

Case Study - Personal Scheduling

UNION AIRWAYS needs to hire additional customer service agents.

Management recognizes the need for cost control while also consistently providing a satisfactory level of service to customers.

Based on the new schedule of flights, an analysis has been made of the minimum number of customer service agents that need to be on duty at different times of the day to provide a satisfactory level of service.

** ** ** * * * * * * * * * * * *

ShiftTime Period Covered Minimum #

of Agents needed

Time Period

6:00 am to 8:00 am8:00 am to10:00 am10:00 am to noon Noon to 2:00 pm2:00 pm to 4:00 pm4:00 pm to 6:00 pm6:00 pm to 8:00 pm8:00 pm to 10:00 pm10:00 pm to midnightMidnight to 6:00 am

1 2 3 4 548796587647382435215

170 160 175 180 195Daily cost per agent

The problem is to determine how many agents should be assigned to the respective shifts each day to minimize the total personnel cost for agents, while meeting (or surpassing) the service requirements.

Activities correspond to shifts, where the level of each activity is the number of agents assigned to that shift.

This problem involves finding the best mix of shift sizes.

1x2x3x

4x5x

: # of agents for shift 1 (6AM - 2PM)

: # of agents for shift 2 (8AM - 4PM)

: # of agents for shift 3 (Noon - 8PM)

: # of agents for shift 4 (4PM - Midnight)

: # of agents for shift 5 (10PM - 6AM)

The objective is to minimize the total cost of the agents assigned to the five shifts.

Min

s.t.54321 195180175160170 xxxxx

0ix )5~1( iall 15

52

43

82

73

64

87

65

79

48

5

54

4

43

43

32

321

21

21

1

x

xx

x

xx

xx

xx

xxx

xx

xx

x

15

52

43

82

64

87

79

48

5

54

4

43

32

321

21

1

x

xx

x

xx

xx

xxx

xx

x

)15,43,39,31,48(),,,,( 54321 xxxxx

Total Personal Cost = $30,610