Inventory Management V
Finite Planning Horizon
Lecture 9
ESD.260 Fall 2003
Caplice
Assumptions: Basic FPH Model
•Demand

Constant vs Variable

Known vs Random

Continuous vs Discrete
•Lead time
• Instantaneous
• Constant or Variable
•
(deterministic/stochastic)
•Dependence of items
• Independent
• Correlated
• Indentured
•Review Time
• Continuous vs Periodic
•Number of Echelons
• One vs Many
•Capacity / Resources
•
Unlimited vs Limited
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•Discounts
• None
• All Units or Incremental
•Excess Demand
• None
• All orders are
backordered
• Lost orders
• Substitution
•Perishability
• None
• Uniform with time
•Planning Horizon
• Single Period
• Finite Period
• Infinite
•Number of Items
• One
• Many
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Example
When should I order and for how much?
Costs
D = 2000 items per year
Cp = $50.00 per item
Ch = 24% per item per year
Chp = ( Ch Cp )/ N
= $1 per month per item
Demand
Co = $500.00 per order
N = number of periods per year
More Assumptions
• Demand is required and consumed on first day of the period
Month
• Holding costs are not charged on items used in that period
• Holding costs are charged for inventory ordered in advance of need
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Five Basic Approaches
1.
2.
3.
4.
5.
The One-Time Buy
Lot For Lot
Simple EOQ
The Silver Meal Algorithm
Optimal Procedures
Wagner-Whitin (Dynamic Programming)
Mixed Integer Programming
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Approach: One-Time Buy
On
Hand
Inventory
Month
2000
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Approach: One-Time Buy
Order
Holding Ordering
Months Demand Quantity Cost
Cost
1
200
2000
$1800
$500
2
150
0
$1650
$0
3
100
0
$1550
$0
Period
Costs
2300
1650
1550
4
5
6
7
8
9
10
11
12
Totals:
$1500
$1450
$1300
$1200
$1000
$800
$550
$250
$0
$13600
50
50
100
150
200
200
250
300
250
2000
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0
0
0
0
0
0
0
0
0
2000
$1500
$1450
$1300
$1200
$1000
$800
$550
$250
$0
$13100
6
$0
$0
$0
$0
$0
$0
$0
$0
$0
$500
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Approach: Lot for Lot
On
Hand
Inventory
Month
200 150 100 50
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50 100 150 200 200 250 300 250
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Approach: Lot for Lot
Month
1
2
3
4
5
6
7
8
9
10
11
12
Yotals:
Demand
Ordering
Quantity
200
150
100
50
50
100
250
200
200
250
300
250
2000
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Holding
Cost
200
150
100
50
50
100
150
200
200
250
300
250
2000
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
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Ordering
Cost
$500
$500
$500
$500
$500
$500
$500
$500
$500
$500
$500
$500
$6000
Period
Costs
$500
$500
$500
$500
$500
$500
$500
$500
$500
$500
$500
$500
$6000
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Approach: EOQ
On
Hand
Inventory
Month
400
400
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400
9
400
400
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Approach: EOQ
Month
Demand
1
2
3
4
5
6
7
8
9
10
11
12
Totals:
200
150
100
50
50
100
150
200
200
250
300
250
2000
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Order
Quantity
400
0
400
0
0
0
0
400
0
400
400
0
2000
10
Holding
Cost
$200
$50
$350
$300
$250
$150
$0
$200
$0
$150
$250
$0
$1900
Ordering
Cost
$500
$0
$500
$0
$0
$0
$0
$500
$0
$500
$500
$0
$2500
Period
Costs
$700
$50
$850
$300
$250
$150
$0
$700
$0
$650
$750
$0
$4400
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Approach: Silver-Meal Algorithm
On
Hand
Inventory
Month
550
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250
11
400
550
250
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Approach: Silver-Meal Algorithm
Mon
Dmd
Lot
Qty
Order
Cost
Holding Cost
Lot
Cost
Mean Cost
1st
Buy:
1
200
200
$500
$0
$500
$500
2
150
350
$500
$150
$650
$325
3
100
450
$500
$150+$200
$850
$283
4
50
500
$500
$150+$200+$150
$1000
$250
5
50
550
$500
$150+$200+$150+$200
$1200
$240
6
100
650
$500
$1700
$283
2nd
Buy:
6
100
100
$500
0
$500
$500
7
150
250
$500
150
$650
$325
8
200
450
$500
$150+$400
$1050
$350
$150+$200+$150+$200+
$500
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Approach: Silver-Meal Algorithm
Mon Dmd
Lot
Qty
Order
Cost
Holding Cost Lot Cost Mean
Cost
3rd
Buy:
8
9
200
200
200 $500
400 $500
$0
$200
$500
$700
$500
$350
10
250
650 $500
$200+$500
$1200
$400
4th
Buy:
10
250
250 $500
$0
$500
$500
11
300
550 $500
$300
$800
$400
12
250
800 $500
$300+$500
$1300
$433
5th
Buy:
12
250
250 $500
$0
$500
$500
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Approach: Silver-Meal Algorithm
Months
Demand
Order
Quantity
Holding
Cost
Ordering
Cost
1
2
3
4
5
6
7
8
9
10
11
12
Totals:
200
150
100
50
50
100
150
200
200
250
300
250
2000
550
0
0
0
0
250
0
400
0
550
0
250
2000
$350
$200
$100
$50
$0
$150
$0
$200
$0
$300
$0
$0
$1350
$500
$0
$0
$0
$0
$500
$0
$500
$0
$500
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$500
$2500
Period
Costs
$850
$200
$100
$50
$0
$650
$0
$700
$0
$800
$0
$500
$3850
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Approach: Optimization (MILP)
On
Hand
Inventory
550
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450
15
450
550
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Approach: Optimization (MILP)
Decision Variables:
Qi = Quantity purchased in period i
Zi = Buy variable = 1 if Qi>0, =0 o.w.
Bi = Beginning inventory for period I
Ei = Ending inventory for period
Data:
Di = Demand per period, i = 1,,n
Co = Ordering Cost
Chp = Cost to Hold, $/unit/period
M = a very large number….
MILP Model
Objective Function:
• Minimize total relevant costs
Subject To:
• Beginning inventory for period 1 = 0
• Beginning and ending inventories must match
• Conservation of inventory within each period
• Nonnegativity for Q, B, E
• Binary for Z
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Objective Function
Beginning & Ending
Inventory Constraints
Conservation of
Inventory Constraints
Ensures buys occur
only if Q>0
Non-Negativity &
Binary Constraints
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Approach: Optimization (MILP)
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Comparison of Approaches
Month
1
2
3
4
5
6
7
8
9
10
11
12
Demand OTB
200
2000
150
100
50
50
100
150
200
200
250
300
250
Totals Cost
$13,600
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L4L
200
150
100
50
50
100
150
200
200
250
300
250
$6,000
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EOQ
400
S/M
550
OPT
550
250
450
400
400
400
450
400
400
550
550
250
$4,400 $3,850 $3,750
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