Spring, 2007
ISE 102
Capacity Decisions
Mahmut Ali GÖKÇE
Industrial Systems Engineering Dept.
İzmir University of Economics
Week 1
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
www.izmirekonomi.edu.tr
1
Spring, 2007
Review of Past Four Weeks
 Overview of Manufacturing (Service) Systems
 Problems and Methods
 Our job is (generally speaking) making the best
decision
 i.e., optimization;
 Parameters of the model - acquire data
 Constraints – rules to be satisfied
 Objective function – define the best
 Solution Techniques – methodology to be used
 Forecasting – Generate data (it is a problem as well!)
 LP – A Simple solution technique
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This Week
 Capacity decisions
Developing Alternatives - Considerations
Using Cost-Volume Analysis to make decisions
Make or buy decisions
Decision Trees as a methodology to help making
decision under uncertainty
 Location decisions
Things to be considered
Using Cost-Volume Analysis to make decisions
Transportation Model
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Hierarchy of Decision Problems
 Strategic Decisions
Location, capacity, new product, layout design,
etc.
 Horizon: years
 Tactical Decisions
 Allocation, production planning, inventory, etc.
 Horizon: months-year
 Operational Decisions
 Scheduling, vehicle routing, assignment, etc.
 Minutes-hours-days
 Note that the “design” decisions will be an input for
“operational” problems
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Capacity Planning
 Capacity can be defined as is the productive capability of a
facility, usually measured as a quantity of output per unit of
time
 Strategic capacity planning is an approach for determining the
overall capacity level of capital intensive resources, including
facilities, equipment, and overall labor force size.
 The basic questions in capacity handling are:
 What kind of capacity is needed?
 How much is needed?
 When is it needed?
 Management must review product and service choices
periodically
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Three Levels of Capacity Planning
 Long-term
Physical
 Medium-term
Workforce planning
 Short-term
detailed planning &
control
Our focus now is longterm
.
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Capacity Decisions are Important
•
•
•
•
•
•
Impacts ability to meet future demand – under capacity
Affects operating costs – excess number of workers
Major determinant of initial costs – fixed cost
Involves long-term commitment – design decision
Affects competitiveness – delivery speed
Affects ease of management – over capacity
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Performance Measures
Think of Your Typical Daily Schedule!
• Design capacity
– maximum obtainable output
• Effective capacity
– maximum capacity given product mix, scheduling
difficulties, and other doses of reality.
• Actual output
– rate of output actually achieved--cannot exceed
effective capacity
 Efficiency = Actual Output
Effective capacity
 Utilization = Actual Output
Design capacity
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Efficiency/Utilization Example
Design capacity = 50 trucks/day
Effective capacity = 40 trucks/day
Actual output = 36 units/day
Efficiency = Actual output
= 36 units/day
Effective capacity = 40 units/ day
= 90%
Utilization = Actual output
= 36 units/day
Design capacity = 500 units/ day
= 72%
Note that higher utilization is not always desired! Why?
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Capacity – Developing Alternatives
• Before making strategic capacity decisions managers must
consider;
• Design flexibility into systems
• Provisions for future expansions – adjacent land
• Take a “big picture” approach to capacity changes
• Extra rooms for the hotel requires extra parking lots
• Prepare to deal with capacity “chunks”
• Can’t buy buses with 35 seats; must be 48!
• Attempt to smooth out capacity requirements
• Complementary products – heating and AC.
• Identify the optimal operating level
• Economies of Scale and Diseconomies of Scale
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Economies & Diseconomies of Scale
Average cost per unit
Production units have an optimal rate of output for minimal cost.
Minimum
cost
0
Figure 5-3
Rate of output
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Economies & Diseconomies of Scale
Average cost per unit
Figure 5-4
Minimum cost & optimal operating rate are
functions of size of production unit.
0
Small
plant
Medium
plant
Output rate
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
Week 1
Large
plant
Q* (What about Demand?)
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Capacity – Evaluating Alternatives
• One must examine different alternatives from different
perspectives. Most obvious ones are economical
consideration. Will it be feasible? Also one must consider
public opinion.
• A number of techniques are going to be discussed from an
economical stand point
• Cost-Volume analysis
• Decision analysis
• Financial analysis
• Waiting line analysis.
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Cost-Volume Analysis
• Focuses on relationships between cost, revenue and the
volume of the output.
• The idea is to estimate the income of the company under
different operating conditions and select the most
appropriate one.
• First step is identifying the all costs
• Fixed costs (FC) : Tend to remain constant regardless of
the volume of output; cost of land, heating, property tax,
etc.
• Variable cost (VC) : Vary directly with volume of output;
raw materials, labor, etc. {VC=Q*v – where v is variable
cost per unit)
• Total Cost (TC) =FC+VC
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Cost-Volume Analysis
• Revenue per unit(R) is the price of the product.
• Total Revenue (TR) = R*Q
• Profit (P) = TR – TC = R*Q – (FC + Q*v)
• Rearranging the terms:
P = Q(R-v) – FC
• Q where we have no loss or no profit is known as
breakeven point.
• A figure worth hundred words…
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Cost-Volume Relationships
Amount ($)
Figure 5-5a
Fixed cost (FC)
0
Q (volume in units)
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Cost-Volume Relationships
Amount ($)
Figure 5-5b
0
Q (volume in units)
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Cost-Volume Relationships
Amount ($)
Figure 5-5c
0
BEP units
Q (volume in units)
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Cost-Volume Analysis
• Capacity alternatives may involve step costs, e.g., a firm may
have the options of purchasing one, two or three machines, with
each machine fixed costs increases, perhaps not linearly.
• Potential volume would depend on the number of machines
purchased: implication is multiple break-even quantities.
• Example:
• A manager has the option of purchasing one, two or three
machines. Fixed costs and potential volumes are as follows:
MC#
1
2
FC
Capaci
$9600 ty
$1500 301
0 to 0
600
300
Asst. Prof. Dr.
Izmir University
of Economics
3Mahmut Ali GÖKÇE,
$2000
601
Week 1
0
900
Variable cost is $10
Revenue is $ 40
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Break-Even Problem with Step Fixed
Costs
Spring, 2007
Figure 5-6a
3 machines
2 machines
1 machine
Quantity
Step fixed costs and variable costs.
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Break-Even Problem with Step Fixed
Costs
Spring, 2007
Figure 5-6b
Number of
Machines
to be
bought
should be
decided
based on
sales
forecast.
BEP
$
BEP
TC
TC
TC
3
2
1
Quantity
Step fixed costs and variable costs.
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Make or Buy?
 Sometimes a company requires “temporary” increase in the
capacity.
 Such situations are handled with over time working, hiring
more workers, etc. if the extra capacity needed (or constrained
resource – or bottleneck resource) is labor force.
 Another possibility is outsourcing the production to external
companies. This might be helpful particularly if extra machining
hours is needed. Rather than investment on new machines it
might be better to “rent” it.
 Cost volume analysis is helpful for these decisions.
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Cost-Volume-Profit Analysis Example
Alto, Inc. is planning to introduce a new model of
student saxophone which will be sold for $500
each. If they produce it themselves, they will incur
annual fixed costs of $1,000,000 and the variable
cost of production will be $300 per unit. Another
option is to subcontract production to Tenor, Inc. If
they subcontract, then Tenor, Inc. will charge them
$400 per unit.
a. If Alto, Inc. decides to produce the saxophones
themselves, what quantity is required for them to
break even for the year?
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Cost-Volume-Profit Analysis Example
b. What is the “indifference point” for these two
options? Over what range of volumes would each
option be preferred?
c. If Alto, Inc. decides to produce the saxophones
themselves, by how much would they have to
reduce annual fixed costs in order to reduce their
break-even volume to 3,000 units? Assume that
revenues and variable costs remain the same.
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Decision Making Under Uncertainty
 Sometimes the decision maker may not be certain
about the future.
 The decision must be done under uncertainty
 For our purposes suppose that the uncertainty is
in future demand and represented with certain
probabilities associated with different possible
scenarios.
 An example is as follows;
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Example of a Decision Tree Problem
A glass factory specializing in crystal is experiencing a
substantial backlog, and the firm's management is
considering three courses of action:
A) Arrange for subcontracting,
B) Construct new facilities.
C) Do nothing (no change)
The correct choice depends largely upon demand, which
may be low, medium, or high. By consensus,
management estimates the respective demand
probabilities as .10, .50, and .40.
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Decision Trees
 Decision trees are methodologies that help us to
handle these type of problems.
 One must first identify alternatives (course of
actions), secondly possible sources of
uncertainties (state of natures) and finally the
payoffs (outcomes).
 Note that for each possible course of action and
state of nature there will be a certain outcome.
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Decision Trees
 In this problem we have three possible alternatives
 Arrange for subcontracting,
 Construct new facilities.
 Do nothing (no change)
 We have three possible states of nature
 Medium Demand
 Low Demand
 High Demand
 Suppose that each demand scenario (low, medium, high) is
quantified with a number. Based on these demand values,
fixed cost of new facilities, variable costs with new facilities,
variable cost of subcontracting, and all other necessary data,
we have already calculated a profit for each possible scenario.
These profits are presented in the following “payoff table”.
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Example of a Decision Tree Problem:
Step 1. The Payoff Table
The management estimates the profits when choosing
from the three alternatives (A, B, and C) under the
differing probable levels of demand. These costs, in
thousands of dollars are presented in the table below:
A
B
C
0.1
Low
10
-120
20
0.5
Medium
50
25
40
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0.4
High
90
200
60
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Decision Trees
 Next step is constructing the decision trees;
Each decision is represented with a rectangle.
Each alternative is an arc originated from this
rectangle to the right side.
Each chance event is represented with a circle.
Each state of nature is an arc originated from
this circle to the right side.
All relevant data, probabilities, outcomes, etc.
should be presented accordingly.
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Step 2. Draw The Decision Tree
High demand (.4)
Medium demand (.5)
Low demand (.1)
A
High demand (.4)
B
Medium demand (.5)
Low demand (.1)
$90k
$50k
$10k
$200k
$25k
-$120k
C
High demand (.4)
Medium demand (.5)
Low demand (.1)
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$60k
$40k
$20k
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Step 3. Calculate the Exp. Monetary
Values
 Third step is known as rolling back the tree.
 For each chance event one must determine the
expected monetary values that the chance event
will yield
 For each decision node one must select the
decision that yields the highest expected
monetary return.
 In our simple example there are three chance
events and after calculating expected return of
these chance events there is only one decision to
be made.
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Step 2. Draw The Decision Tree
High demand (.4)
Medium demand (.5)
$62k
A
B
$80.5k
Low demand (.1)
High demand (.4)
Medium demand (.5)
Low demand (.1)
$90k
$50k
$10k
$200k
$25k
-$120k
C
High demand (.4)
$46k
Medium demand (.5)
Low demand (.1)
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$60k
$40k
$20k
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Step 4. Make decision
High demand (.4)
Medium demand (.5)
$62k
A
B
$80.5k
Low demand (.1)
High demand (.4)
Medium demand (.5)
Low demand (.1)
$90k
$50k
$10k
$200k
$25k
-$120k
C
High demand (.4)
$46k
Medium demand (.5)
Low demand (.1)
$60k
$40k
$20k
Alternative B generates the greatest expected profit, so our
choice is B or to construct a new facility.
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Location Decisions
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Spring, 2007
Location Problem
 As the capacity decision, another important design parameter
used as an input in operational decisions (transportationallocation, vehicle routing, inventory, etc.) is the location
decisions.
 Some objectives;
 Maximize population coverage within an x-mile radius
 Minimize or limit the maximum travel distance for a
customer (or raw materials)
 Minimize average transportation cost (or customer
travel distance)
 Options include, expand existing facilities, add new facilities,
move to a new location or do nothing!
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Location Decision Factors
Regional Factors
Community
Considerations
Site-related
Factors
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Regional Factors
 Location of raw materials – perishability, transportation
 Canning fresh fruits, paper industry near forests, etc.
 Location of markets – convenience, responsiveness
 Retail stores, restaurants, etc.
 Labor factors – cost, availability
 High tech firms near metropolitans,
 Climate and taxes
 Nobody goes to Siberia!
 Foreign locations
 Less expensive areas, time zone differences in case
of call centers!
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Community Considerations
Desirability of the community;
 Quality of life
 Services
 Attitudes
 Taxes
 Environmental regulations
 Developer support
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Site Related Factors
What about the construction sites?
 Land
 Transportation
 Environmental
 Legal
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Multiple Plant Strategies
 If the company has several sites may use;
Product plant strategy
Market area plant strategy
Process plant strategy
Multiple Plant
Strategies
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Manufacturing vs. Service
Considerations
Manufacturing/Distribution
Service/Retail
Cost Focus
Revenue focus
Transportation modes/costs
Demographics:
age,income,etc
Energy availability, costs
Population/drawing area
Labor cost/availability/skills
Competition
Building/leasing costs
Traffic volume/patterns
Customer access/parking
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Evaluating Locations
 Cost-Profit-Volume Analysis
Determine fixed and variable costs for each
alternative
Plot total costs
Determine lowest total costs
Note that the fixed and variable costs
includes all costs, land, labor, transportation, etc.
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Example 1: Cost-Volume Analysis
Fixed and variable costs for four potential
locations
L o c a tio n
A
B
C
D
F ix e d
C ost
$ 2 5 0 ,0 0
1 0 0 ,0 0
1 5 0 ,0 0
2 0 0 ,0 0
0
0
0
0
V a r ia b le
C ost
$11
30
20
35
How did you come up with these costs in the first
place?
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Example 1: Solution for
Output=10,000
Spring, 2007
Fixed
Costs
A
B
C
D
$250,000
100,000
150,000
200,000
Variable
Costs
$11(10,000)
30(10,000)
20(10,000)
35(10,000)
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Total
Costs
$360,000
400,000
350,000
550,000
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Example 1: Solution
$(000)
800
700
600
500
400
300
200
100
0
0
D
B
C
A
A Superior
C Superior
B Superior
2
4
6
8
10
12
14
16
Annual Output (000)
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Evaluating Locations (cont.)
 Transportation Model
Decision based on movement costs of raw
materials or finished goods
 Factor Rating
Decision based on quantitative and
qualitative inputs
 Center of Gravity Method
Decision based on minimum distribution
costs
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The Transportation Model
 While evaluating the best location alternative we use
costs (fixed and variable). Particularly for multiple plant
cases these costs can be identified after an allocation
decision is made. That is to say we can’t calculate the
cost of transportation from an alternative site to demand
centers before we decide which demand centers are
going to be served by which production site.
 This problem is known as allocation problem.
 The allocation problem with objective of minimizing the
transportation cost is known as “transportation model”.
 Also after a location decision is made transportation
problem must be solved to reduce the distribution costs!
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The Transportation Model
 Requirements:
List of origins and each one’s capacity
List of destinations and each one’s demand
Unit cost of shipping
 Assumptions
Items to be shipped are homogeneous
Shipping cost per unit is the same
Only one route between origin and destination
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The Transportation Problem: An
Example
D
(demand)
S
(supply)
D
(demand)
D
(demand)
S
(supply)
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Transportation Problem
 Network Representation (m = 2, n = 3)
1
d1
2
d2
3
d3
c11
s1
c12
1
c13
c21
s2
2
c22
c23
SOURCES
DESTINATIONS
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
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Spring, 2007
Example: BBC
Building Brick Company (BBC) has orders for 80
tons of bricks at three suburban locations as follows:
Northwood -- 25 tons, Westwood -- 45 tons, and
Eastwood -- 10 tons. BBC has two plants, each of
which can produce 50 tons per week.
How should end of week shipments be made to fill
the above orders given the following delivery cost
per ton:
Plant 1
Plant 2
Northwood
24
30
Westwood
30
40
Eastwood
40
42
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Spring, 2007
Example: BBC
 LP Formulation
Decision Variables Defined
xij = amount shipped from plant i to suburb j
where i = 1 (Plant 1) and 2 (Plant 2)
j = 1 (Northwood), 2 (Westwood),
and 3 (Eastwood)
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
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Example: BBC
 LP Formulation
 Objective Function
Minimize total shipping cost per week:
Min 24x11 + 30x12 + 40x13 + 30x21 + 40x22 + 42x23
 Constraints
s.t.
x11 + x12 + x13 < 50 (Plant 1 capacity)
x21 + x22 + x23 < 50 (Plant 2 capacity)
x11 + x21 = 25 (Northwood demand)
x12 + x22 = 45 (Westwood demand)
x13 + x23 = 10 (Eastwood demand)
all xij > 0
(Non-negativity)
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
www.izmirekonomi.edu.tr
Spring, 2007
Example: BBC
 Partial Spreadsheet Showing Optimal Solution
A
10
11 Dec.Var.Values
12
13
14
15
16
17
18
19
B
X11
C
X12
D
X13
E
X21
F
X22
G
X23
5
45
0
20
0
10
Minimized Total Shipping Cost
2490
Constraints
LHS
P1.Cap.
P2.Cap.
N.Dem.
W.Dem.
E.Dem.
50
30
25
45
10
RHS
<=
<=
=
=
=
50
50
25
45
10
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
www.izmirekonomi.edu.tr
Spring, 2007
Example: BBC
 Optimal Solution
From
Plant 1
Plant 1
Plant 2
Plant 2
To
Amount Cost
Northwood
5
120
Westwood
45
1,350
Northwood
20
600
Eastwood
10
420
Total Cost = $2,490
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
www.izmirekonomi.edu.tr
Spring, 2007
The Transportation Problem
 LP Formulation
The linear programming formulation in terms of
the amounts shipped from the origins to the
destinations, xij, can be written as:
Min SScijxij
ij
s.t.
Sxij < si for each origin i
j
Sxij = dj for each destination j
i
xij > 0 for all i and j
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
www.izmirekonomi.edu.tr
Spring, 2007
The Transportation Problem
 LP Formulation Special Cases
The following special-case modifications to the
linear programming formulation can be made:
Minimum shipping guarantees from i to j:
xij > Lij
Maximum route capacity from i to j:
xij < Lij
Unacceptable routes:
delete the variable
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
www.izmirekonomi.edu.tr