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MS 401 Production and Service Systems Operations Spring 2009-2010

MS 401 Production and Service Systems Operations Spring 2009-2010. Lot Sizing Slide 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”

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MS 401 Production and Service Systems Operations Spring 2009-2010

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  1. MS 401 Production and Service Systems Operations Spring 2009-2010 Lot Sizing Slide Set #11

  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. 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. 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. 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. 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. Part Period Balancing (PPB) This table illustrates the holding costs for different order scenarios

  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. 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 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

  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 inventoryin this example • Define ctv= setup and holdingcost of producing in period t, to meet the requirements in periods t to v. • We calculate the ctv values as follows:

  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 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

  12. 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.

  13. 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.

  14. The Table to Summarize the Solution • 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. The Table to Summarize the Solution-2 • 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. This is the example we used for the other methods Setup cost: $54, Holding cost: $0.4 charged to average inventory Wagner-Whitinfor the 12-period Example

  17. 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

  18. Wagner-Whitin (WW) Algorithm

  19. Comparison of Methods (for the 12-Period Example)

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