Transcript decision analysis - University of Redlands

Lecture
1
MGMT 650
Management Science
and Decision Analysis
1
Agenda
Operations Management
Chapter 1
• 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 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 vs. Delivery of
Services
 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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Scope of Operations Management
 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
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Management 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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Problem Solving and Decision Making
 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).
--------------------------------------------------------------------• Implement the chosen alternative.
• Evaluate the results.
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Quantitative 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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Models
A model is an abstraction of reality.
– Physical
– Schematic
– Mathematical
Tradeoffs
What are the pros and cons of models?
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A Simulation Model
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Models 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 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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Analysis of Trade-offs


How many more jeans would Levi need to sell to justify the
cost of additional robotic tailors?
Cost of additional robotic tailors vs Inventory Holding Cost
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Product 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
Storage space per
unit
3 cubic ft
Resource
Amount available
Assembly time
100 hours
Inspection time
22 hours
Storage space
39 cubic feet
1 hr
3 cubic ft
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A Linear Programming Model

Objective – profit maximization
Maximize 60X1 + 50X2

Subject to
Assembly
4X1 + 10X2 <= 100 hours
Inspection
2X1 + 1X2 <= 22 hours
Storage 3X1 + 3X2 <= 39 cubic feet
X1, X2 >= 0
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Cost, Revenue and Profit Models
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Cost Classification of Owning and Operating a Passenger Car
Cost Classification
Variable Costs:
Standard miles per gallon
Average fuel price per gallon
Fuel and oil per mile
Maintenance per mile
Tires per mile
References
Cost
20 miles/ gallon
$1.34/ gallon
$0.0689
$0.0360
$0.0141
Annual Fixed Costs:
Insurance:
License & Registration
$372
$95
Mixed Costs: Depreciation
Fixed portion per year
Variable portion per mile
$3,703
$0.04
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Cost-Volume Relationship
Volume Index (miles)
Variable costs ($0.1190/mile)
Mixed costs:
Variable portion
Fixed portion
Fixed costs:
Total variable cost
Total fixed cost
Total costs
Cost per mile
5,000
10,000
15,000
20,000
$595
$1,190
$1,785
$2,380
200
3,703
467
795
4,170
400
3,703
467
1,590
4,170
600
3,703
467
2,385
4,170
800
3,703
467
3,180
4,170
$4,965
$0.9930
$5,760
$0.5760
$6,555
$0.4370
$7,350
$0.3675
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Cost-Volume Relationship
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Amount ($)
Amount ($)
Cost-Volume Relationships
Fixed cost (FC)
0
0
Q (volume in units)
Q (volume in units)
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Amount ($)
Cost-Volume Relationships
0
BEP units
Q (volume in units)
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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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Example: Ponderosa Development Corp.
Employee
Monthly Salary
President
$10,000
VP, Development
6,000
VP, Marketing
4,500
Project Manager
5,500
Controller
4,000
Office Manager
3,000
Receptionist
2,000
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Example: Ponderosa Development Corp.
 Identify all costs and denote the marginal cost and marginal
revenue for each house.
 Write the monthly cost function c (x), revenue function r (x),
and profit function p (x).
 What is the breakeven point for monthly sales of the houses?
 What is the monthly profit if 12 houses per month are built
and sold?
 Determine the BEP for monthly sale of houses graphically.
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Graph of costs and Break-even point
Break-even Analysis
700000
600000
500000
Fixed cost(FC)
400000
($)
Variable cost(VC)
Total cost(TC)
300000
Total revenue
200000
100000
0
1
2
3
4
5
6
Number of houses
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Example: Step Fixed Costs
 A manager has the option of purchasing 1, 2 or 3 machines
 Fixed costs and potential volumes are as follows:
# of machines
Total annual FC ($)
Range of output
1
9600
0 – 300
2
15000
301 – 600
3
20000
601 – 900
 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?
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Break-Even Problem with Step Fixed Costs
Total Cost
Total Revenue
BEVs
3 machines
2 machines
1 machine
Quantity
Step fixed costs and variable costs.
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Assumptions of Cost-Volume Analysis
1. One
product is involved
2. Everything produced can be sold
3. Variable cost per unit is the same regardless of
volume
4. Fixed costs do not change with volume
5. Revenue per unit constant with volume
6. Revenue per unit exceeds variable cost per unit
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Cost-Volume & Locational Break-Even
Analysis
Three locations:
City
Akron
Bowling Green
Chicago
Fixed Variable
Cost
Cost
$30,000
$75
$60,000
$45
$110,000
$25
Total
Cost
$180,000
$150,000
$160,000
Total Cost = Fixed Cost + Variable Cost x Volume
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Locational Break-Even Analysis
Annual cost
Graph of Break-Even Points
–
$180,000 –
–
$160,000 –
$150,000 –
–
$130,000 –
–
$110,000 –
–
–
$80,000 –
–
$60,000 –
–
–
$30,000 –
–
$10,000 –
|
–
0
Akron
lowest
cost
Chicago
lowest
cost
Bowling Green
lowest cost
|
|
|
|
|
|
500
1,000
1,500
2,000
2,500
3,000
Volume
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Linear Programming – Chapter 2
George Dantzig – 1914 -2005
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Linear Programming

Concerned with optimal allocation of limited
resources such as

Materials
 Budgets
 Labor
 Machine time

among competitive activities
 under a set of constraints
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Linear Programming Example
Variables
Maximize 60X1 + 50X2
Subject to
Objective function
4X1 + 10X2 <= 100
2X1 + 1X2 <= 22
Constraints
3X1 + 3X2 <= 39
Non-negativity Constraints
X1, X2 >= 0
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)
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Linear Programming Model

Decision variables



Objective Function



unknowns, which is what model seeks to determine
for example, amounts of either inputs or outputs
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
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Another Example of LP: Diet Problem

Energy requirement : 2000 kcal
 Protein requirement : 55 g
 Calcium requirement : 800 mg
Food
Energy (kcal)
Protein(g)
Calcium(mg)
Price per
serving($)
Oatmeal
Chicken
Eggs
Milk
110
205
160
160
4
32
13
8
2
12
54
285
3
24
13
9
Pie
Pork
420
260
4
14
22
80
24
13
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Example 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 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: Constraints

Meet energy requirement
110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000

Meet protein requirement
4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6  55

Meet calcium requirement
2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6  800

Restriction on number of servings
0x14, 0x23, 0x32, 0x48, 0x52, 0x62
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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
0x14
0x23
0x32
0x48
0x52
0x62
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Optimal Solution – Diet Problem
Using LINDO 6.1

Food
Oatmeal
# of servings
4
Chicken
Eggs
Milk
Pie
0
0
6.5
0
Pork
2
Cost of diet = $96.50 per day
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Optimal Solution – Diet Problem
Using Management Scientist

Food
Oatmeal
# of servings
4
Chicken
Eggs
Milk
Pie
0
0
6.5
0
Pork
2
Cost of diet = $96.50 per day
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Guidelines 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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Graphical Solution of LPs
 Consider
a Maximization Problem
Max
s.t.
5x1 + 7x2
x1
< 6
2x1 + 3x2 < 19
x1 + x2 < 8
x 1, x 2 > 0
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Graphical Solution Example

Constraint #1 Graphed
x2
8
7
x1 < 6
6
5
4
3
2
(6, 0)
1
1
2
3
4
5
© 2005 Thomson/South-Western
6
7
8
9
10
x1
Slide 44
Graphical Solution Example

Constraint #2 Graphed
x2
(0, 6 1/3)
8
7
6
5
2x1 + 3x2 < 19
4
3
(9 1/2, 0)
2
1
1
2
3
4
5
© 2005 Thomson/South-Western
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7
8
9
10
x1
Slide 45
Graphical Solution Example

Constraint #3 Graphed
x2
(0, 8)
8
x1 + x2 < 8
7
6
5
4
3
2
(8, 0)
1
1
2
3
4
5
© 2005 Thomson/South-Western
6
7
8
9
10
x1
Slide 46
Graphical Solution Example

Combined-Constraint Graph
x2
x1 + x2 < 8
8
7
x1 < 6
6
5
4
2x1 + 3x2 < 19
3
2
1
1
2
3
4
5
© 2005 Thomson/South-Western
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7
8
9
10
x1
Slide 47
Graphical Solution Example

Feasible Solution Region
x2
8
7
6
5
4
3
Feasible
Region
2
1
1
2
3
4
5
© 2005 Thomson/South-Western
6
7
8
9
10
x1
Slide 48
Graphical Solution Example

Objective Function Line
x2
8
7
(0, 5)
Objective Function
5x1 + 7x2 = 35
6
5
4
3
2
(7, 0)
1
1
2
3
4
5
© 2005 Thomson/South-Western
6
7
8
9
10
x1
Slide 49
Graphical Solution Example

Optimal Solution
x2
Objective Function
5x1 + 7x2 = 46
8
7
Optimal Solution
(x1 = 5, x2 = 3)
6
5
4
3
2
1
1
2
3
4
5
© 2005 Thomson/South-Western
6
7
8
9
10
x1
Slide 50
Graphical Linear Programming
1. Set
up objective function and constraints
in mathematical format
2. Plot
the constraints
3. Identify
4. Plot
the feasible solution space
the objective function
5. Determine
the optimum solution
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Product 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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Product Mix Problem
Boat
Maximum
Builder
Cost
Speedhawk Sleekboat
Silverbird Sleekboat
Catman
Racer
Classy
Racer
$6000
$7000
$5000
$9000
Expected
Seating
3
5
2
6
Daily
Profit
$ 70
$ 80
$ 50
$110
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Product Mix Problem

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
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Product Mix Problem

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
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Product Mix Problem - Complete Formulation
Max 70x1 + 80x2 + 50x3 + 110x4
s.t.
6000x1 + 7000x2 + 5000x3 + 9000x4 < 420,000
x1 + x2 + x3 + x4 > 50
Boat
# purchased
x1 + x2 - x3 - x4 = 0
Speedhawk
28
3x1 + 5x2 + 2x3 + 6x4 > 200
Silverbird
0
x1, x2, x3, x4 > 0
Catman
0
Classy

28
Daily profit = $5040
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Applications of LP

Product mix planning
 Distribution networks
 Truck routing
 Staff scheduling
 Financial portfolios
 Capacity planning
 Media selection: marketing
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