1. Murat Kaya, Sabancı Üniversitesi 1 MS 401 Production and Service Systems Operations Spring 2009-2010 Lot Sizing Slide Set #11

2. Murat Kaya, Sabancı Üniversitesi 2 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

3. Murat Kaya, Sabancı Üniversitesi 3 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

4. Murat Kaya, Sabancı Üniversitesi 4 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

5. Murat Kaya, Sabancı Üniversitesi 5 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

6. Murat Kaya, Sabancı Üniversitesi 6 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

7. Murat Kaya, Sabancı Üniversitesi 7 Part Period Balancing (PPB) This table illustrates the holding costs for different order scenarios

8. Murat Kaya, Sabancı Üniversitesi 8 Part Period Balancing (PPB) 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

9. Murat Kaya, Sabancı Üniversitesi 9 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, y 1 =r 1, or y 1 =r 1 +r 2, or ….. or y 1 =r 1 + r 2 + r 3 …r n y 2 =0, or y 2 =r 2, or y 2 =r 2 +r 3, or …. or y 2 =r 2 + r 3 + …+ r n y n =0, or y n =r n 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

10. Murat Kaya, Sabancı Üniversitesi 10 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 c tv = setup and holding cost of producing in period t, to meet the requirements in periods t to v. We calculate the c tv values as follows: t \ v 1234 1 2 3 75131 4 75

11. Murat Kaya, Sabancı Üniversitesi 11 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 c 12 –option 2: Produce in period 2 (cost: c 22 ). 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)+c 22 ). We have F(2)=min{c 12, F(1)+c 22 }=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

12. Murat Kaya, Sabancı Üniversitesi 12 Wagner-Whitin Example F(3)=min{c 13, F(1)+c 23, F(2)+c 33 }. 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 c 13 –option 2: Produce in period 2 to meet the requirements of periods 2 and 3 (cost: c 23 ). Using the optimal replenishment policy for period 1 (cost F(1)), the total cost of this option is (F(1)+c 23 ). –option 3: Produce in period 3 to meet the requirement of period 3 (cost: c 33 ). Using the optimal replenishment policy for periods 1 and 2 (cost F(2)), the total cost of this option is (F(2)+c 33 ). F(3)=min{c 13, F(1)+c 23, F(2)+c 33 }= 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.

13. Murat Kaya, Sabancı Üniversitesi 13 Wagner-Whitin Example F(4) = min{c 14, F(1)+c 24, F(2)+c 34, F(3)+c 44 } = 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.

14. Murat Kaya, Sabancı Üniversitesi 14 The Table to Summarize the Solution 1234 1 c 11 c 12 c 13 c 14 2 F(1)+c 22 F(1)+c 23 F(1)+c 24 3 F(2)+c 33 F(2)+c 34 4 F(3)+c 44 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

15. Murat Kaya, Sabancı Üniversitesi 15 The Table to Summarize the Solution-2 1234 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

16. Murat Kaya, Sabancı Üniversitesi 16 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

17. Murat Kaya, Sabancı Üniversitesi 17 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

18. Murat Kaya, Sabancı Üniversitesi 18 Wagner-Whitin (WW) Algorithm

19. Murat Kaya, Sabancı Üniversitesi 19 Comparison of Methods (for the 12-Period Example)