MS 401
Production and Service Systems Operations
Spring 2009-2010
Lot Sizing
Slide Set #11
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1
Lot Sizing (VBWJ Chapter 14)
• Issue: “How to group time-phased requirements data into a
schedule of replenishment orders that minimize the
combined costs of placing orders and holding inventory”
• Lot sizing techniques
– Lot-for-lot (L4L)
– Economic Order Quantity (EOQ)
– Periodic Order Quantity
– Part-Period Balancing (PPB)
– Wagner-Whitin Algorithm (finds the optimal schedule)
• No backordering allowed
• Holding cost might be based on either
– average inv. level = (beginning inv. + ending inv.) / 2
– or, to the ending inventory level only
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2
Lot-for-Lot (LFL, L4L)
• In each period, order that period’s requirements
• Example: Fixed ordering cost: 54
Holding cost/unit/week: 0.4, charged to avg. inventory level
Period
Net Requirements
Order Quantity
Beginning Inventory
Ending Inventory
Setup/Ordering Costs
Holding Costs
1
2
10
62
10
62
10
62
0
0
54
54
2 12.4
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3
12
12
12
0
54
2.4
4
5
6
7
8
9
130 154 129
88
52 124
130 154 129
88
52 124
130 154 129
88
52 124
0
0
0
0
0
0
54
54
54
54
54
54
26 30.8 25.8 17.6 10.4 24.8
10
11
160 238
160 238
160 238
0
0
54
54
32 47.6
12
41
41
41
0
54
8.2
3
EOQ
• Using the EOQ rule to find the order quantity (because of its simplicity)
– not optimal in this case (demand is not stationary, and other reasons…)
• Calculate the “average net requirement” to use in the EOQ formula
– 100/week in this example, resulting in EOQ=164
• Ignores the changes in demand (NR) by using a fixed order quantity
– may result in high inventory costs
Period
Net Requirements
Order Quantity
Beginning Inv.
Ending Inv.
Ordering Cost
Holding Cost
1
10
164
164
154
54
63.6
2
62
0
154
92
0
49.2
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3
12
0
92
80
0
34.4
4
130
164
244
114
54
71.6
5
154
164
278
124
54
80.4
6
129
164
288
159
54
89.4
7
88
0
159
71
0
46
8
52
0
71
19
0
18
9
124
164
183
59
54
48.4
10
160
164
223
63
54
57.2
11
238
175
238
0
54
47.6
12
41
164
164
123
54
57.4
4
Period Order Quantity (POQ)
• Use the EOQ formula to compute
• In our example, 164/100= 2 (by rounding)
– order exactly the requirements for a two-week interval
• POQ method improves the inventory cost performance by allowing the lot
sizes to vary
– this time, however, the order interval is fixed
Period
Net Requirements
Order Quantity
Beginning Inventory
Ending Inventory
Setup/Ordering Costs
Holding Costs
1
10
72
72
62
54
26.8
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2
62
62
0
0
12.4
3
12
142
142
130
54
54.4
4
130
130
0
0
26
5
154
283
283
129
54
82.4
6
129
129
0
0
25.8
7
88
140
140
52
54
38.4
8
52
52
0
0
10.4
9
124
284
284
160
54
88.8
10
160
160
0
0
32
11
238
279
279
41
54
64
12
41
0
41
0
0
8.2
5
Part Period Balancing (PPB)
• This method tries to equate the fixed ordering cost with the
inventory holding cost
• Procedure for period 1: Choose the alternative below in
which the inventory holding cost is the closest to the fixed
ordering cost (54 in our example)
–
–
–
–
order to cover the requirements of period 1
order to cover the requirements of period 1 and 2
order to cover the requirements of period 1, 2 and 3
….
• Remember: Inventory holding cost is charged to the
average inventory level in a period
– average inv = (beginning inv. + ending inv.) / 2
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Part Period Balancing (PPB)
This table illustrates the holding costs for different order scenarios
Calculations
period 1
periods 1+2
periods 1+2+3
periods 1+2+3+4
period 4
periods 4+5
period 5
periods 5+6
period 6
periods 6+7
period 8
periods 8+9
period 10
periods 10+11
period 11
periods 11+12
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Order 1 Order 2 Order 3 Order 4 Order 5 Order 6 Order 7
2
39.2
51.2
233.2
26
118.4
30.8
108.2
25.8
78.6
10.4
84.8
32
174.8
47.6
72.2
7
Part Period Balancing (PPB)
Period
Net Requirements
Order Quantity
Beginning Inventory
Ending Inventory
Setup/Ordering Costs
Holding Costs
1
10
84
84
74
54
31.6
2
62
3
12
74
12
0
17.2
12
0
0
2.4
4
130
130
130
0
54
26
5
154
154
154
0
54
30.8
6
129
217
217
88
54
61
7
88
88
0
0
17.6
8
52
176
176
124
54
60
9
124
124
0
0
24.8
10
160
160
160
0
54
32
11
238
238
238
0
54
47.6
12
41
41
41
0
54
8.2
• PPB permits both the lot size and the time between orders to vary
– when requirements are low, the size of the orders will be low and the orders
will be infrequent (periods 1-3, for example)
• However, PPB will not always yield the minimum cost ordering
plan because it does not evaluate all possible alternatives
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8
Wagner-Whitin (WW) Algorithm
• To find the optimal lot sizes (when planning horizon is finite)
• How many feasible policies are there?
– too many… we cannot search all of them to find the optimal one
• WW algorithm is based on the following observation
– An optimal policy has the property that in each period, the production
quantity is either “0”, or it is exactly the sum of some future
requirements. That is,
y1=r1, or
y1=r1+r2,
or
….. or
y1=r1 + r2 + r3 …rn
y2=0,
or
y2=r2, or y2=r2+r3, or …. or
yn=0,
or
yn=rn
y2=r2 + r3 + …+ rn
• Hence, the number of policies to consider to find the optimal
policy is not as large as the number of all feasible policies
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9
Wagner-Whitin Example
• 4 period problem. Requirements: (52, 87, 23, 56)
• h=$1, Setup cost=$75
– for simplicity, assume that the holding cost is only applied to
ending inventory in this example
• Define ctv= setup and holding cost of producing in period
t, to meet the requirements in periods t to v.
• We calculate the ctv values as follows:
t\v
1
2
3
4
75
131
1
2
3
4
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75
10
Wagner-Whitin Example
• Define F(t) = The total cost of the best replenishment
strategy that satisfies the requirements in periods (1, 2, … ,t)
• F(1)=75, simply the setup cost…
• F(2)=
That is, we choose between two options:
– option 1: Produce in period 1 to satisfy the requirements of periods 1
and 2. The cost will be c12
– option 2: Produce in period 2 (cost: c22). Assume that an optimal
replenishment policy was chosen to take care of period 1 (costs F(1)).
Hence, the total cost of this option is (F(1)+c22).
We have F(2)=min{c12, F(1)+c22}=min{162, 75+75}=150.
Hence, the optimal replenishment policy to meet the requirements in
periods 1 and 2 is to produce in periods 1 and 2. Policy cost=150
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11
Wagner-Whitin Example
• F(3)=min{c13, F(1)+c23 , F(2)+c33}. We choose between three options:
– option 1: Produce in period 1 to satisfy the requirements of periods 1,
2 and 3. The cost will be c13
– option 2: Produce in period 2 to meet the requirements of periods 2
and 3 (cost: c23). Using the optimal replenishment policy for period 1
(cost F(1)), the total cost of this option is (F(1)+c23).
– option 3: Produce in period 3 to meet the requirement of period 3
(cost: c33). Using the optimal replenishment policy for periods 1 and 2
(cost F(2)), the total cost of this option is (F(2)+c33).
F(3)=min{c13, F(1)+c23 , F(2)+c33}= min{208, 75+98 , 150+75}=173
Hence, the optimal replenishment policy to meet the requirements in
periods 1 to 3 is to produce in periods 1 (for period 1) and period 2 (for
periods 2 and 3). The cost of the policy is 173.
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12
Wagner-Whitin Example
• F(4) = min{c14, F(1)+c24 , F(2)+c34 , F(3)+c44}
= min{376, 75+210, 150+131, 173+75}=248
Hence, the optimal replenishment policy to meet the requirements
in periods 1 to 4 is to produce in period 1 (for period 1), in period
2 (for periods 2 and 3), and in period 4 (for period 4).
The cost of the policy is 248.
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13
The Table to Summarize the Solution
1
1
2
3
4
c11
c12
c13
c14
F(1)+c22
F(1)+c23
F(1)+c24
F(2)+c33
F(2)+c34
2
3
4
F(3)+c44
• This table summarizes the solution algorithm we discussed
• Column “t” shows the total cost of production alternatives
for periods 1 to “t”
• The chosen alternatives for each “t” are shown in bold red
– these are the F(t) values
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14
The Table to Summarize the Solution-2
1
1
2
3
4
2
3
4
376
285
281
248
• This table summarizes the solution algorithm we discussed
• Column “t” shows the total cost of production alternatives
for periods 1 to “t”
• The chosen alternatives for each “t” are shown in bold red
– these are the F(t) values
Murat Kaya, Sabancı Üniversitesi
15
Wagner-Whitin for the 12-period Example
• This is the example we used for the other methods
• Setup cost: $54, Holding cost: $0.4 charged to average inventory
Net Requirements
10
62
12
130
154
129
88
52
124
160
238
41
Period (t)
1
2
3
4
5
6
7
8
9
10
11
12
1
56
93.2
105.2
287.2
564.4
848.2
1077
122.4
129.6
259.6
475.2
707.4
149.6
227.6
381.6
562.2
720.6
185.2
277.6
406.6
529.8
623.4
896.2 1312.2 2026.2 2165.6
270
347.4
435.4
508.2
731.4 1083.4 1702.2 1825.2
349.8
402.6
454.6
628.2
916.2 1439.8 1546.4
419
450.2
574.2
798.2 1226.6 1316.8
467
541.4
701.4 1034.6 1108.4
2
3
4
5
6
7
8
9
10
11
12
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1233 1654.6 2262.6 3262.2 3450.8
901 1036.2 1408.2 1952.2 2856.6 3028.8
835 1157.4 1637.4 2446.6 2602.4
529
625
863
920.4
615
757.8
798.8
716.6
741.2
778.8
16
Wagner-Whitin for the 12-period Example
• We calculated that the optimal decision for the requirement at
period 4 was producing it at week 4 rather than carrying it
from earlier periods
• Given this information, do you think it is possible to produce
the requirement for period 5 at periods 1, 2 or 3?
• No… The only options are meeting period-5 requirement by
production in period 4 or in period 5. This observation further
reduces the computational requirements of the problem
– see the simplified table in the following slide
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17
Wagner-Whitin (WW) Algorithm
Net Req.
10
62
12
130
154
129
88
52
124
160
238
41
Period (t)
1
2
3
4
5
6
7
8
9
10
11
12
1
56
93.2
105.2
287.2
122.4
129.6
259.6
149.6
227.6
347.4
435.4
349.8
402.6
454.6
419
450.2
574.2
467
541.4
2
3
4
5
6
7
8
9
10
11
12
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185.2
277.6
270
529
625
615
757.8
716.6
741.2
778.8
18
Comparison of Methods (for the 12-Period Example)
Method
LFL
EOQ
POQ
PPB
WW
Ordering/Setup Costs
648
432
324
432
378
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Holding Costs
240
663.2
469.6
359.2
363.2
Total Costs
888
1095.2
793.6
791.2
741.2
19