IEEM 517
Inventory Control –
Dynamic Lot Sizing
LEARNING OBJECTIVES
1.
Understand that heuristics based on the EOQ (Economic Order Quantity)
model may perform poorly for dynamic lot sizing
2.
Learn what optimal solution approaches are available for dynamic lot sizing
3.
Learn to choose the right approach for a given situation
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1
CONTENTS
2
• Modeling Assumptions and Formulation
• Heuristic Solution Approaches
• Optimal Solution Approaches
• Summary
FUNDMENTAL ASSUMPTION: DYNAMIC DEMAND
3
Demand changes over time horizon because:
•
Seasonality of demand (e.g., Sun’s hockey stick)
•
Pricing/promotions (e.g., 30% off at McDonald’s)
•
Product life cycles (e.g., IBM‘s PC production)
•
Lumpy order schedule (e.g., Solectron’s production)
2
4
MODELING ASSUMPTIONS
1.
Production is instantaneous
2.
No delivery leadtime
3.
Demand is deterministic and known
4.
Demand can change over time
5.
A production run incurs a fixed setup cost, regardless of production
quantity
6.
Consider the case of a single product
7.
No backorders allowed
8.
Finite planning horizon
Å Major change in assumptions
compared to the EOQ model
MODEL PARAMETERS AND DECISION VARIABLE
t
Index for time period, t = 1, …, T, where T represents the length of
planning horizon
Dt
Demand in period t (in units)
ct
Unit production cost in period t (in dollars per unit)
At
Setup cost to produce one lot in period t (in dollars)
ht
Holding cost to carry one unit of inventory from period t to period t+1 (in
dollars per unit per period)
zt
Binary decision variable
5
= 1 if production happens in period t
= 0 otherwise
Qt
Lot size in period t (in units)
It
Inventory left over at the end of period t (in units)
decision variables
3
6
COMMENTS AND ISSUES
• We adopt discrete time periods
• Besides the demand, some other factors (e.g., unit production cost, setup
cost, and unit inventory holding cost) can also change over time
• Given the sequence of demand over time, we basically need to determine
when and how much to produce
• The objective is still to minimize total setup, inventory holding, and production
costs for the entire planning horizon
• If the holding cost consists of only the interest on money tied up in inventory,
if i is the effective annual interest rate, and if periods are weeks, ht = ict/52
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FORMULATION: INTEGER LINEAR PROGRAMMING
Minimize
∑t=1T (Atzt + htIt + ctQt)
Subject to
It = It-1 + Qt - Dt
for t = 1, 2, …, T
Qt ≤ Mzt
for t = 1, 2, …, T
It, Qt ≥ 0, zt binary
for t = 1, 2, …, T
IT = 0
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CONTENTS
• Modeling Assumptions and Formulation
• Heuristic Solution Approaches
• Optimal Solution Approaches
• Summary
9
HEURISTICS
Heuristic
Rational from the EOQ Model
1. Lot-for-Lot
None
2. Period Order Quantity
Directly apply the EOQ model by estimating data
3. Part-Period Balancing
At optimality, setup cost equals inventory holding cost
4. Silver-Meal
We minimize the summation of the setup cost and the
inventory cost per unit time (e.g., per year)
5. Least Unit Cost
We minimize the summation of the setup cost and the
inventory cost allocated to each product unit
(since the demand rate is constant)
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EXAMPLE
A manufacturing company has given the demand data for the next five
months. Demand is deterministic but changes over time. Production runs can
be made only at the beginning of a month. The production cost, the setup
cost, and the inventory holding cost are all known and are actually constant
over time. No backorders are allowed.
t
1
2
3
4
5
Dt
27
22
13
19
12
ct
100
100
100
100
100
At
200
200
200
200
200
ht
10
10
10
10
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LOT-FOR-LOT HEURISTIC
Basic idea
Set up a production run in each period and produce the exact
quantity that is demanded for that period.
Algorithm
Set the production quantity Qt in period t the same as the
demand Dt in that period
Solution
D = (27, 22, 13, 19, 12), ct = 100, At = 200, and ht = 10
11
6
PERIOD ORDER QUANTITY HEURISTIC
Basic idea
Use the EOQ formula to determine a lot size. This lot size can be
translated to an integer cycle length T. This heuristic then produces
every T periods to meet demand in those periods (no more and no
less).
Algorithm
1. Transfer the data under the dynamic environment to a static
environment
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2. Apply the EOQ model to compute the lot size
3. Transfer this lot size to the corresponding cycle length and then
make the cycle length an integer number
PERIOD ORDER QUANTITY: SOLUTION
13
D = (27, 22, 13, 19, 12), ct = 100, At = 200, and ht = 10
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PART-PERIOD BALANCING HEURISTIC
Basic idea
Determine the cycle length by balancing the inventory holding cost
incurred during the cycle length with the setup cost incurred in the
current period
Algorithm
1.
Let the current period be period t = 1
2.
Compute Ht,τ the inventory holding cost if we produce in period t
and cover demand up to period τ, where
Ht, τ = ∑ mτ =t +1 ∑ mn=−t1hnDm
3.
Select τ for which Ht,τ is closest to the setup cost At
4.
Set period τ+1 as the current period and repeat until the end of
the planning horizon
PART-PERIOD BALANCING: SOLUTION
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15
D = (27, 22, 13, 19, 12), ct = 100, At = 200, and ht = 10
8
16
SILVER-MEAL HEURISTIC
Basic idea
Determine the cycle length such that average per-period cost of
setup and inventory holding is small
Algorithm
1.
Let the current period be period t = 1
2.
Compute Jt,τ, the average per-period setup plus inventory
holding cost if we produce in period t and cover demand up to
period τ , where
Jt,τ =
A t + ∑ mτ = t +1 ∑ mn=−t1hnDm
τ − t +1
3.
Select the nearest value of τ for which Jt,τ < Jt,τ+1
4.
Set period τ+1 as the current period and repeat until the end of
the planning horizon
SILVER-MEAL: SOLUTION
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D = (27, 22, 13, 19, 12), ct = 100, At = 200, and ht = 10
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LEAST UNIT COST HEURISTIC
Basic idea
Determine the cycle length such that average per-unit cost of setup
and inventory holding is small
Algorithm
1.
Let the current period be period t = 1
2.
Compute Kt,τ, the average per-unit setup plus inventory holding
cost if we produce in period t and cover demand up to period τ ,
where
A + ∑ mτ = t +1 ∑ mn=−t1hnDm
K t,τ = t
∑ mτ =tDm
3.
Select the nearest value of τ for which Kt,τ < Kt,τ+1
4.
Set period τ+1 as the current period and repeat until the end of
the planning horizon
LEAST UNIT COST: SOLUTION
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D = (27, 22, 13, 19, 12), ct = 100, At = 200, and ht = 10
10
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CONTENTS
• Modeling Assumptions and Formulation
• Heuristic Solution Approaches
• Optimal Solution Approaches
• Summary
OPTIMAL SOLUTION APPROACHES
Approach
Solution time
Software implementation
1. Integer LP
Exponential in problem scale
Easy to implement with existing
software packages
2. Wagner-Whitin
Algorithm
(Forward DP)
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(our focus in this lecture)
Polynomial in problem scale
Implementation may not be easy
3. DP (backward)
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INTEGER LP: SOLUTION
By using the previous formulation of integer LP and Microsoft Excel, we get
z* = (1, 1, 0, 1, 0), Q* = (27, 35, 0, 31, 0), and I* = (0, 13, 0, 12, 0),
which yields a minimum total cost as $850 + $9,300
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WAGNER-WHITIN: SETUP
Definitions
TCt*
Cost of an optimal production schedule over the first t time periods
Wt,τ
Cost of producing in period t and covering demand up to period τ
Yt,τ
Cost of a production schedule from period 1 to period τ by
- following an optimal production schedule for periods 1 through t-1
- producing in period t and covering demand up to period τ
1
2
“Optimal“
t-1
τ
t
Qt = Dt +... + Dτ
W t,τ
TCt-1*
Yt,τ = TC*t-1 + Wt,τ
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WAGNER-WHITIN: RECURSION
Forward recursion relationship
TCτ* = minimize Yt,τ = minimize (TC*t-1 + Wt,τ )
t = 1:
Y1,τ
t = 2:
Y2,τ
t = 3:
Y3,τ
over 1 ≤ t ≤ τ
W1,τ
W2,τ
TC1*
W3,τ
TC2*
TCτ*
t = τ:
Wτ-1,τ
TCτ-2*
t = τ-1: Yτ-1,τ
W τ,τ
TCτ-1*
Yτ,τ
1
2
3
τ-1
τ
WAGNER-WHITIN: PROPERTY ONE
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Boundary condition
It-1Qt = 0
Common sense
It is typically never optimal to both carry inventory into a period and produce
in that period simultaneously
Implications
1.
Produce only when the beginning inventory is zero
2.
Produce exactly the quantity that cover demand for an integral number
of time periods
13
WAGNER-WHITIN: PROPERTY TWO
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Planning Horizon Theorem
If in solving the τ-period problem, it is optimal to produce in period jτ* (jτ* ≤ τ) to
cover demand up to period τ, then in solving the s-period problem (s > τ), it is
optimal to meet the demand of period s by producing in period jτ* or later
jτ*
τ
s
Common sense
If it is optimal to produce in period 4 (= jτ*) to meet the demand in period 6 (= τ),
it is typically never optimal to produce in period 2 (< jτ*) and meet the demand
in period 8 (= s). To meet the demand in period 8 (= s), we should produce in
period 4 (= jτ*) or later
Implication
The recursion can be simplified to
TCτ* = minimize Yt,τ = minimize (TC*t-1 + Wt,τ )
over jτ−1* ≤ t ≤ τ
WAGNER-WHITIN: ALGORITHM
1.
The algorithm requires zero initial and ending inventory: Adjust demand such
that initial and ending inventory is zero
2.
Start with first period, i.e., τ = 1. Since all demand must be met, the solution to
the one-period problem is j1* = 1. Compute TC1* = Y1,1 = W1,1
3.
Set τ = τ + 1.
If τ > T, stop. The algorithm terminates with the optimal solution
If τ ≤ T,
a.
Compute Yt,τ = TC*t-1 + Wt,τ for all t satisfying jτ−1* ≤ t ≤ τ
b.
Compute TCτ* = minimize Yt,τ over jτ−1* ≤ t ≤ τ
c.
Go to Step 3
27
14
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WAGNER-WHITIN: SOLUTION
D = (27, 22, 13, 19, 12), ct = 100, At = 200, and ht = 10
Planning horizon τ
1
2
3
4
5
Y1,τ
Y2,τ
Y3,τ
Y4,τ
Y5,τ
TCτ*
jτ*
WAGNER-WHITIN: SOLUTION
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CONTENTS
• Modeling Assumptions and Formulation
• Heuristic Solution Approaches
• Optimal Solution Approaches
• Summary
31
COMPARISON OF SOLUTIONS
Algorithm
Production setup z
Optimal algorithms
Integer LP
Wagner-Whitin
(1, 1, 0, 1, 0)
(1, 1, 0, 1, 0)
Heuristics
Lot-for-lot
Period order quantity
Part-period balancing
Silver-Meal
Least unit cost
(1, 1, 1, 1, 1)
(1, 0, 1, 0, 1)
(1, 0, 1, 0, 1)
(1, 1, 0, 1, 0)
(1, 1, 1, 0, 1)
Setup plus inventory
holding cost
850
850
1,000
1,010
1,010
850 Å also optimal
990
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SUMMARY
• In dynamic-demand environments, algorithms that are constructed for
dynamic settings perform much better than algorithms that are designed for
constant-demand environments
• When choosing an algorithm, trade-offs must be made
High
WagnerWhitin
Part-Period Balancing
Silver-Meal
Least Unit Cost
Solution
quality
Low
Integer
LP
Lot-for-lot
Period Order Quantity
Short
Long
Solution time
ANNOUNCEMENTS
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• For the Dynamic Lot Sizing model, read Section 2.3 and Section 3.1.6
• Lab 3 will be held on Tuesday, March 15
- Time:
5:00pm – 6:50pm
- Location:
IS Lab of IEEM Dept (Room 3207)
- Content:
Solution approaches of Dynamic Lot Sizing
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