ms 401 production and service systems operations spring 2009-2010

Murat KayaMurat Kaya, Sabancı Üniversitesi, Sabancı Üniversitesi 1

MS 401Production and Service Systems Operations

Spring 2009-2010

Lot SizingSlide Set #11


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


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 1 2 3 4 5 6 7 8 9 10 11 12Net Requirements 10 62 12 130 154 129 88 52 124 160 238 41Order Quantity 10 62 12 130 154 129 88 52 124 160 238 41Beginning Inventory 10 62 12 130 154 129 88 52 124 160 238 41Ending Inventory 0 0 0 0 0 0 0 0 0 0 0 0Setup/ Ordering Costs 54 54 54 54 54 54 54 54 54 54 54 54Holding Costs 2 12.4 2.4 26 30.8 25.8 17.6 10.4 24.8 32 47.6 8.2


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 1 2 3 4 5 6 7 8 9 10 11 12Net Requirements 10 62 12 130 154 129 88 52 124 160 238 41Order Quantity 164 0 0 164 164 164 0 0 164 164 175 164Beginning Inv. 164 154 92 244 278 288 159 71 183 223 238 164Ending Inv. 154 92 80 114 124 159 71 19 59 63 0 123Ordering Cost 54 0 0 54 54 54 0 0 54 54 54 54Holding Cost 63.6 49.2 34.4 71.6 80.4 89.4 46 18 48.4 57.2 47.6 57.4


Period Order Quantity (POQ)

Period 1 2 3 4 5 6 7 8 9 10 11 12Net Requirements 10 62 12 130 154 129 88 52 124 160 238 41Order Quantity 72 142 283 140 284 279 0Beginning Inventory 72 62 142 130 283 129 140 52 284 160 279 41Ending Inventory 62 0 130 0 129 0 52 0 160 0 41 0Setup/ Ordering Costs 54 0 54 0 54 0 54 0 54 0 54 0Holding Costs 26.8 12.4 54.4 26 82.4 25.8 38.4 10.4 88.8 32 64 8.2

• 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


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



Calculations Order 1 Order 2 Order 3 Order 4 Order 5 Order 6 Order 7period 1 2periods 1+2 39.2periods 1+2+3 51.2periods 1+2+3+4 233.2period 4 26periods 4+5 118.4period 5 30.8periods 5+6 108.2period 6 25.8periods 6+7 78.6period 8 10.4periods 8+9 84.8period 10 32periods 10+11 174.8period 11 47.6periods 11+12 72.2

This table illustrates the holding costs for different order scenarios



Period 1 2 3 4 5 6 7 8 9 10 11 12Net Requirements 10 62 12 130 154 129 88 52 124 160 238 41Order Quantity 84 130 154 217 176 160 238 41Beginning Inventory 84 74 12 130 154 217 88 176 124 160 238 41Ending Inventory 74 12 0 0 0 88 0 124 0 0 0 0Setup/ Ordering Costs 54 0 0 54 54 54 0 54 0 54 54 54Holding Costs 31.6 17.2 2.4 26 30.8 61 17.6 60 24.8 32 47.6 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


• 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 y2=r2 + r3 + …+ rn

yn=0, or yn=rn

• Hence, the number of policies to consider to find the optimal policy is not as large as the number of all feasible policies

Wagner-Whitin (WW) Algorithm


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

1

2

3 75 131

4 75



• 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



• 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.



• 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.


The Table to Summarize the Solution

1 2 3 4

1 c11 c12 c13 c14

2 F(1)+c22 F(1)+c23 F(1)+c24

3 F(2)+c33 F(2)+c34

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


The Table to Summarize the Solution-2

1 2 3 4

1 376

2 285

3 281

4 248

• 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


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 1233 1654.6 2262.6 3262.2 3450.8

2 122.4 129.6 259.6 475.2 707.4 901 1036.2 1408.2 1952.2 2856.6 3028.8

3 149.6 227.6 381.6 562.2 720.6 835 1157.4 1637.4 2446.6 2602.4

4 185.2 277.6 406.6 529.8 623.4 896.2 1312.2 2026.2 2165.6

5 270 347.4 435.4 508.2 731.4 1083.4 1702.2 1825.2

6 349.8 402.6 454.6 628.2 916.2 1439.8 1546.4

7 419 450.2 574.2 798.2 1226.6 1316.8

8 467 541.4 701.4 1034.6 1108.4

9 529 625 863 920.4

10 615 757.8 798.8

11 716.6 741.2

12 778.8

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


• 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

Wagner-Whitin for the 12-period Example


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

2 122.4 129.6 259.6

3 149.6 227.6

4 185.2 277.6

5 270 347.4 435.4

6 349.8 402.6 454.6

7 419 450.2 574.2

8 467 541.4

9 529 625

10 615 757.8

11 716.6 741.2

12 778.8

Wagner-Whitin (WW) Algorithm


Comparison of Methods (for the 12-Period Example)

Method Ordering/Setup Costs Holding Costs Total Costs

LFL 648 240 888EOQ 432 663.2 1095.2POQ 324 469.6 793.6PPB 432 359.2 791.2WW 378 363.2 741.2

ms 401 production and service systems operations spring 2009-2010

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