1 Lecture 1 MGMT 650 Management Science and Decision Analysis

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Lecture1

MGMT 650Management Science and Decision Analysis

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Introduce YourselfIntroduce Yourself

Name Where do you work? What is your role? Have you taken a similar class earlier? What are your expectations of this

class?

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AgendaAgenda

Operations Management Management Science & relation to OM

• Quantitative Analysis and Decision Making

Cost, Revenue, and Profit Models Management Science Techniques

• Introduction to Linear Programming

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Operations ManagementOperations Management

The management of systems or processes that create goods and/or provide services

Organization

Finance Operations Marketing

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Production of Goods Production of Goods vs.vs. Delivery of Delivery of ServicesServices

Production of goods – tangible output Delivery of services – an act Service job categories

• Government

• Wholesale/retail

• Financial services

• Healthcare

• Personal services

• Business services

• Education

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Operations Management includes:• Forecasting• Capacity planning• Scheduling• Managing inventories• Assuring quality• Deciding where to locate facilities• And more . . .

The operations function• Consists of all activities directly related to producing goods

or providing services

Scope of Operations ManagementScope of Operations Management

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Key Decisions of Operations Key Decisions of Operations ManagersManagers

WhatWhat resources/what amounts

WhenNeeded/scheduled/ordered

WhereWork to be done

HowDesigned

WhoTo do the work

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Decision MakingDecision Making

System Design– capacity– location– arrangement of departments– product and service planning– acquisition and placement of equipment

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Decision MakingDecision Making

System operation– Management of personnel

– Inventory planning and control

– Scheduling

– Project Management

– Quality assurance

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Management ScienceManagement Science

The body of knowledge involving quantitative approaches to decision making is referred to as

• Management Science

• Operations research

• Decision science It had its early roots in World War II and is

flourishing in business and industry with the aid of computers

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Steps of Problem Solving

(First 5 steps are the process of decision making)

• Define the problem.

• Identify the set of alternative solutions.

• Determine the criteria for evaluating alternatives.

• Evaluate the alternatives.

• Choose an alternative (make a decision).

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• Implement the chosen alternative.

• Evaluate the results.

Problem Solving and Decision MakingProblem Solving and Decision Making

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Quantitative Analysis and Decision MakingQuantitative Analysis and Decision Making

Potential Reasons for a Quantitative Analysis Approach to Decision Making

• The problem is complex

• The problem is very important

• The problem is new

• The problem is repetitive

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ModelsModels

A model is an abstraction of reality.

– Physical– Schematic– Mathematical

What are the pros and cons of models?

Tradeoffs

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A Simulation ModelA Simulation Model

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Models Are BeneficialModels Are Beneficial

Easy to use, less expensive Require users to organize Systematic approach to problem solving Increase understanding of the problem Enable “what if” questions: simulation models Specific objectives Power of mathematics Standardized format

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Quantitative ApproachesQuantitative Approaches

• Linear programming: optimal allocation of

resources

• Queuing Techniques: analyze waiting lines

• Inventory models: management of inventory

• Project models: planning, coordinating and

controlling large scale projects

• Statistical models: forecasting

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Product Mix ExampleProduct Mix Example

Type 1 Type 2

Profit per unit $60 $50

Assembly time per unit

4 hrs 10 hrs

Inspection time per unit

2 hrs 1 hr

Storage space per unit

3 cubic ft 3 cubic ft

Resource Amount available

Assembly time 100 hours

Inspection time 22 hours

Storage space 39 cubic feet

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Objective – profit maximizationMaximize 60X1 + 50X2

Subject toAssembly 4X1 + 10X2 <= 100 hours

Inspection 2X1 + 1X2 <= 22 hours

Storage 3X1 + 3X2 <= 39 cubic feet

X1, X2 >= 0

A Linear Programming ModelA Linear Programming Model

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Cost, Revenue and Profit ModelsCost, Revenue and Profit Models

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Cost Classification CostVariable Costs: Standard miles per gallon Average fuel price per gallon Fuel and oil per mile $0.0689 Maintenance per mile $0.0360 Tires per mile $0.0141

Annual Fixed Costs: Insurance: $372 License & Registration $95

Mixed Costs: Depreciation Fixed portion per year $3,703 Variable portion per mile $0.04

References

20 miles/ gallon$1.34/ gallon

Cost Classification of Owning and Operating a Passenger Car

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Cost-Volume RelationshipCost-Volume Relationship

5,000 10,000 15,000 20,000

Variable costs ($0.1190/mile) $595 $1,190 $1,785 $2,380Mixed costs: Variable portion 200 400 600 800 Fixed portion 3,703 3,703 3,703 3,703Fixed costs: 467 467 467 467Total variable cost 795 1,590 2,385 3,180Total fixed cost 4,170 4,170 4,170 4,170

Total costs $4,965 $5,760 $6,555 $7,350Cost per mile $0.9930 $0.5760 $0.4370 $0.3675

Volume Index (miles)

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Cost-Volume RelationshipCost-Volume Relationship

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Cost-Volume RelationshipsCost-Volume Relationships A

mo

un

t ($

)

0Q (volume in units)

Total cost = VC + FC

Total variable cost (V

C)

Fixed cost (FC)

Am

ou

nt

($)

Q (volume in units)0

Total r

evenue

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Cost-Volume RelationshipsCost-Volume Relationships

Am

ou

nt

($)

Q (volume in units)0 BEP units

Profit

Total r

even

ue

Total cost

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Example: Ponderosa Development Corp.Example: Ponderosa Development Corp.

Ponderosa Development Corporation (PDC) is a small real estate developer that builds only one style house.

The selling price of the house is $115,000. Land for each house costs $55,000 and lumber, supplies, and

other materials run another $28,000 per house. Total labor costs are approximately $20,000 per house.

Ponderosa leases office space for $2,000 per month. The cost of supplies, utilities, and leased equipment runs another $3,000 per month.

The one salesperson of PDC is paid a commission of $2,000 on the sale of each house. PDC has seven permanent office employees whose monthly salaries are given on the next slide.

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Employee Monthly Salary

President $10,000

VP, Development 6,000

VP, Marketing 4,500

Project Manager 5,500

Controller 4,000

Office Manager 3,000

Receptionist 2,000

Example: Ponderosa Development Corp.Example: Ponderosa Development Corp.

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Identify all costs and denote the marginal cost and marginal revenue for each house.

Write the monthly cost function c (x), revenue function r (x), and profit function p (x).

What is the breakeven point for monthly sales of the houses? What is the monthly profit if 12 houses per month are built

and sold? Determine the BEP for monthly sale of houses graphically.

Example: Ponderosa Development Corp.Example: Ponderosa Development Corp.

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Example: Ponderosa Development Corp.Example: Ponderosa Development Corp.

00

200200

400400

600600

800800

10001000

12001200

00 11 22 33 44 55 66 77 88 99 1010Number of Houses Sold (x)Number of Houses Sold (x)

Th

ousa

nds

of

Dolla

rsTh

ousa

nds

of

Dolla

rs

Break-Even Point = 4 HousesBreak-Even Point = 4 Houses

Total Cost Total Cost = = 40,000 + 40,000 + 105,000x105,000x

Total Total Revenue =Revenue = 115,000x115,000x

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Example: Step Fixed CostsExample: Step Fixed Costs

A manager has the option of purchasing 1, 2 or 3 machines Fixed costs and potential volumes are as follows:

Variable cost = $10/unit and revenue = $40/unit? If the projected annual demand is between 580 and 630

units, how many machines should the manager purchase?

# of machines Total annual FC ($) Range of output

1 9600 0 – 300

2 15000 301 – 600

3 20000 601 – 900

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Break-Even Problem with Step Fixed Break-Even Problem with Step Fixed CostsCosts

Quantity

FC + VC = TC

FC + VC = TC

FC + VC =

TC

Step fixed costs and variable costs.

1 machine

2 machines

3 machines

Total RevenueBEVs

Total Cost

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1.One product is involved2.Everything produced can be sold3.Variable cost per unit is the same regardless

of volume4.Fixed costs do not change with volume5.Revenue per unit constant with volume6.Revenue per unit exceeds variable cost per

unit

Assumptions of Cost-Volume AnalysisAssumptions of Cost-Volume Analysis

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Linear ProgrammingLinear Programming

George Dantzig – 1914 -2005

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Concerned with optimal allocation of limited resources such as Materials Budgets Labor Machine time

among competitive activities under a set of constraints

Linear ProgrammingLinear Programming

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Maximize 60X1 + 50X2

Subject to

4X1 + 10X2 <= 100

2X1 + 1X2 <= 22

3X1 + 3X2 <= 39

X1, X2 >= 0

Linear Programming ExampleLinear Programming ExampleVariables

Objective function

Constraints

What is a Linear Program?

• A LP is an optimization model that has

• continuous variables

• a single linear objective function, and

• (almost always) several constraints (linear equalities or inequalities)

Non-negativity Constraints

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Decision variables unknowns, which is what model seeks to determine for example, amounts of either inputs or outputs

Objective Function goal, determines value of best (optimum) solution among all feasible (satisfy

constraints) values of the variables either maximization or minimization

Constraints restrictions, which limit variables of the model limitations that restrict the available alternatives

Parameters: numerical values (for example, RHS of constraints)

Feasible solution: is one particular set of values of the decision variables that satisfies the constraints Feasible solution space: the set of all feasible solutions

Optimal solution: is a feasible solution that maximizes or minimizes the objective function

There could be multiple optimal solutions

Linear Programming ModelLinear Programming Model

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Another Example of LP: Diet Another Example of LP: Diet ProblemProblem

Energy requirement : 2000 kcal Protein requirement : 55 g Calcium requirement : 800 mg

Food Energy (kcal) Protein(g) Calcium(mg) Price per serving($)

Oatmeal 110 4 2 3

Chicken 205 32 12 24

Eggs 160 13 54 13

Milk 160 8 285 9

Pie 420 4 22 24

Pork 260 14 80 13

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Example of LP : Diet ProblemExample of LP : Diet Problem

oatmeal: at most 4 servings/day chicken: at most 3 servings/day eggs: at most 2 servings/day milk: at most 8 servings/day pie: at most 2 servings/day pork: at most 2 servings/day

Design an optimal diet plan which minimizes the cost per

day

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Step 1: define decision variablesStep 1: define decision variables

x1 = # of oatmeal servings x2 = # of chicken servings x3 = # of eggs servings x4 = # of milk servings x5 = # of pie servings x6 = # of pork servings

Step 2: formulate objective function• In this case, minimize total cost

minimize z = 3x1 + 24x2 + 13x3 + 9x4 + 24x5 + 13x6

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Step 3: ConstraintsStep 3: Constraints

Meet energy requirement110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000 Meet protein requirement4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6 55 Meet calcium requirement2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6 800 Restriction on number of servings0x14, 0x23, 0x32, 0x48, 0x52, 0x62

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So, how does a LP look like?So, how does a LP look like?minimize 3x1 + 24x2 + 13x3 + 9x4 + 24x5 + 13x6

subject to

110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000

4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6 55

2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6 800

0x14

0x23

0x32

0x48

0x52

0x62

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Guidelines for Model FormulationGuidelines for Model Formulation

Understand the problem thoroughly. Describe the objective. Describe each constraint. Define the decision variables. Write the objective in terms of the decision

variables. Write the constraints in terms of the decision

variables Do not forget non-negativity constraints

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Transportation ProblemTransportation Problem Objective:

determination of a transportation plan of a single commodity from a number of sources to a number of destinations, such that total cost of transportation is minimized

Sources may be plants, destinations may be warehouses Question:

how many units to transport from source i to destination j such that supply and demand constraints are met, and total transportation cost is minimized

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A Transportation TableA Transportation Table

Warehouse

4 7 7 1100

12 3 8 8200

8 10 16 5150

450

45080 90 120 160

1 2 3 4

1

2

3

Factory Factory 1can supply 100units per period

Demand

Warehouse B’s demand is 90 units per period Total demand

per period

Total supplycapacity perperiod

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LP Formulation of Transportation ProblemLP Formulation of Transportation Problem

minimize 4x11+7x12+7x13+x14+12x21+3x22+8x23+8x24+8x31+10x32+16x33+5x34

Subject to x11+x12+x13+x14=100 x21+x22+x23+x24=200 x31+x32+x33+x34=150 x11+x21+x31=80 x12+x22+x32=90 x13+x23+x33=120 x14+x24+x34=160 xij>=0, i=1,2,3; j=1,2,3,4

Supply constraint for factories

Demand constraint of warehouses

Minimize total cost of transportation

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Assignment ProblemAssignment Problem

Special case of transportation problem When # of rows = # of columns in the

transportation tableau All supply and demands =1

Objective: Assign n jobs/workers to n machines such that the total cost of assignment is minimized

Plenty of practical applications Job shops Hospitals Airlines, etc.

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Cost Table for Assignment ProblemCost Table for Assignment Problem

1 2 3 4

1 1 4 6 3

2 9 7 10 9

3 4 5 11 7

4 8 7 8 5

Worker (i)

Machine (j)

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LP Formulation of Assignment ProblemLP Formulation of Assignment Problem

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Product Mix ProblemProduct Mix Problem• Floataway Tours has $420,000 that can be used to

purchase new rental boats for hire during the summer. • The boats can be purchased from two

different manufacturers.• Floataway Tours would like to purchase at least 50 boats.• They would also like to purchase the same number from

Sleekboat as from Racer to maintain goodwill. • At the same time, Floataway Tours wishes to have a total

seating capacity of at least 200.

• Formulate this problem as a linear program

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Maximum Expected Daily

Boat Builder Cost Seating Profit

Speedhawk Sleekboat $6000 3 $ 70

Silverbird Sleekboat $7000 5 $ 80

Catman Racer $5000 2 $ 50

Classy Racer $9000 6 $110

Product Mix ProblemProduct Mix Problem

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Define the decision variables

x1 = number of Speedhawks ordered

x2 = number of Silverbirds ordered

x3 = number of Catmans ordered

x4 = number of Classys ordered Define the objective function Maximize total expected daily profit: Max: (Expected daily profit per unit) x

(Number of units)

Max: 70x1 + 80x2 + 50x3 + 110x4

Product Mix ProblemProduct Mix Problem

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Define the constraints(1) Spend no more than $420,000:

6000x1 + 7000x2 + 5000x3 + 9000x4 < 420,000 (2) Purchase at least 50 boats: x1 + x2 + x3 + x4 > 50 (3) Number of boats from Sleekboat equals number

of boats from Racer: x1 + x2 = x3 + x4 or x1 + x2 - x3 - x4 = 0

(4) Capacity at least 200: 3x1 + 5x2 + 2x3 + 6x4 > 200

Nonnegativity of variables: xj > 0, for j = 1,2,3,4

Product Mix ProblemProduct Mix Problem

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Max 70x1 + 80x2 + 50x3 + 110x4

s.t.

6000x1 + 7000x2 + 5000x3 + 9000x4 < 420,000

x1 + x2 + x3 + x4 > 50

x1 + x2 - x3 - x4 = 0

3x1 + 5x2 + 2x3 + 6x4 > 200

x1, x2, x3, x4 > 0

Product Mix Problem - Complete FormulationProduct Mix Problem - Complete Formulation

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Applications of LPApplications of LP

Product mix planning Distribution networks Truck routing Staff scheduling Financial portfolios Capacity planning Media selection: marketing

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Graphical Solution of LPsGraphical Solution of LPs

Consider a Maximization ProblemConsider a Maximization Problem

Max 5x1 + 7x2

s.t. x1 < 6

2x1 + 3x2 < 19

x1 + x2 < 8

x1, x2 > 0

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Graphical Solution ExampleGraphical Solution Example

Constraint #1 GraphedConstraint #1 Graphed

xx22

xx11

xx11 << 6 6

(6, 0)(6, 0)

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Graphical Solution ExampleGraphical Solution Example

Constraint #2 GraphedConstraint #2 Graphed

22xx11 + 3 + 3xx22 << 19 19

xx22

xx11

(0, 6 (0, 6 1/31/3))

(9 (9 1/21/2, 0), 0)

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Graphical Solution ExampleGraphical Solution Example

Constraint #3 GraphedConstraint #3 Graphed

xx22

xx11

xx11 + + xx22 << 8 8

(0, 8)(0, 8)

(8, 0)(8, 0)

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Graphical Solution ExampleGraphical Solution Example

Combined-Constraint GraphCombined-Constraint Graph

22xx11 + 3 + 3xx22 << 19 19

xx22

xx11

xx11 + + xx22 << 8 8

xx11 << 6 6

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Graphical Solution ExampleGraphical Solution Example

Feasible Solution RegionFeasible Solution Region

xx11

FeasibleFeasibleRegionRegion

xx22

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Graphical Solution ExampleGraphical Solution Example

Objective Function LineObjective Function Line

xx11

xx22

(7, 0)(7, 0)

(0, 5)(0, 5)Objective FunctionObjective Function55xx11 + + 7x7x2 2 = 35= 35Objective FunctionObjective Function55xx11 + + 7x7x2 2 = 35= 35

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Graphical Solution ExampleGraphical Solution Example

Optimal SolutionOptimal Solution

xx11

xx22

Objective FunctionObjective Function55xx11 + + 7x7x2 2 = 46= 46Objective FunctionObjective Function55xx11 + + 7x7x2 2 = 46= 46

Optimal SolutionOptimal Solution((xx11 = 5, = 5, xx22 = 3) = 3)Optimal SolutionOptimal Solution((xx11 = 5, = 5, xx22 = 3) = 3)

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1.Set up objective function and constraints in mathematical format

2.Plot the constraints

3.Identify the feasible solution space

4.Plot the objective function

5.Determine the optimum solution

Graphical Linear ProgrammingGraphical Linear Programming