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LESSON 22: MATERIAL REQUIREMENTS PLANNING: LOT SIZING

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Outline

- Lot Sizing
- Lot Sizing Methods
- Lot-for-Lot (L4L)
- EOQ
- Silver-Meal Heuristic
- Least Unit Cost (LUC)
- Part Period Balancing

- In Lesson 21
- We employ lot for lot ordering policy and order production as much as it is needed.
- Exception are only the cases in which there are constraints on the order quantity.
- For example, in one case we assume that at least 50 units must be ordered. In another case we assume that the order quantity must be a multiple of 50.

- The motivation behind using lot for lot policy is minimizing inventory. If we order as much as it is needed, there will be no ending inventory at all!

- However, lot for lot policy requires that an order be placed each period. So, the number of orders and ordering cost are maximum.
- So, if the ordering cost is significant, one may naturally try to combine some lots into one in order to reduce the ordering cost. But then, inventory holding cost increases.
- Therefore, a question is what is the optimal size of the lot? How many periods will be covered by the first order, the second order, and so on until all the periods in the planning horizon are covered. This is the question of lot sizing. The next slide contains the statement of the lot sizing problem.

- The lot sizing problem is as follows: Given net requirements of an item over the next T periods, T >0, find order quantities that minimize the total holding and ordering costs over T periods.
- Note that this is a case of deterministic demand. However, the methods learnt in Lessons 11-15 are not appropriate because
- the demand is not necessarily the same over all periods and
- the inventory holding cost is only charged on ending inventory of each period

- Although we consider a deterministic model, keep in mind that in reality the demand is uncertain and subject to change.
- It has been observed that an optimal solution to the deterministic model may actually yield higher cost because of the changes in the demand. Some heuristic methods give lower cost in the long run.
- If the demand and/or costs change, the optimal solution may change significantly causing some managerial problems. The heuristic methods may not require such changes in the production plan.
- The heuristic methods require fewer computation steps and are easier to understand.
- In this lesson we shall discuss some heuristic methods. The optimization method is discussed in the text, Appendix 7-A, pp 406-410 (not included in the course).

- Some heuristic methods:
- Lot-for-Lot (L4L):
- Order as much as it is needed.
- L4Lminimizes inventory holding cost, but maximizes ordering cost.

- EOQ:
- Every time it is required to place an order, lot size equals EOQ.
- EOQ method may choose an order size that covers partial demand of a period. For example, suppose that EOQ is 15 units. If the demand is 12 units in period 1 and 10 units in period 2, then a lot size of 15 units covers all of period 1 and only (15-12)=3 units of period 2. So, one does not save the ordering cost of period 2, but carries some 3 units in

- Lot-for-Lot (L4L):

- Some heuristic methods:
the inventory when that 3 units are required in period 2. This is not a good idea because if an order size of 12 units is chosen, one saves on the holding cost without increasing the ordering cost!

- So, what’s the mistake? Generally, if the order quantity covers a period partially, one can save on the holding cost without increasing the ordering cost. The next three methods, Silver-Meal heuristic, least unit cost and part period balancing avoid order quantities that cover a period partially. These methods always choose an order quantity that covers some K periods, K >0.
- Be careful when you compute EOQ. Express both holding cost and demand over the same period. If the holding cost is annual, use annual demand. If the holding cost is weekly, use weekly demand.

- Some heuristic methods:
- Silver-Meal Heuristic
- As it is discussed in the previous slide, Silver-Meal heuristic chooses a lot size that equals the demand of some K periods in future, where K>0.
- If K =1, the lot size equals the demand of the next period.
- If K =2, the lot size equals the demand of the next 2 periods.
- If K =3, the lot size equals the demand of the next 3 periods, and so on.
- The average holding and ordering cost per period is computed for each K=1, 2, 3, etc. starting from K=1 and increasing K by 1 until the average cost per period starts increasing. The best K is the last one up to which the average cost per period decreases.

- Silver-Meal Heuristic

- Some heuristic methods:
- Least Unit Cost (LUC)
- As it is discussed before, least unit cost heuristic chooses a lot size that equals the demand of some K periods in future, where K>0.
- The average holding and ordering cost per unit is computed for each K=1, 2, 3, etc. starting from K=1 and increasing K by 1 until the average cost per unit starts increasing. The best K is the last one up to which the average cost per unit decreases.
- Observe how similar is Silver-Meal heuristic and least unit cost heuristic. The only difference is that Silver-Meal heuristic chooses K on the basis of average cost per period and least unit cost on average cost per unit.

- Least Unit Cost (LUC)

- Some heuristic methods:
- Part Period Balancing
- As it is discussed before, part period balancing heuristic chooses a lot size that equals the demand of some K periods in future, where K>0.
- Holding and ordering costs are computed for each K=1, 2, 3, etc. starting from K=1 and increasing K by 1 until the holding cost exceeds the ordering cost. The best K is the one that minimizes the (absolute) difference between the holding and ordering costs.
- Note the similarity of this method with the Silver-Meal heuristic and least unit cost heuristic. Part period balancing heuristic chooses K on the basis of the (absolute) difference between the holding and ordering costs.

- Part Period Balancing

Lot-Sizing

- Some important notes
- Inventory costs are computed on the ending inventory.
- L4L minimizes carrying cost
- Silver-Meal Heuristic, LUC and Part Period Balancing are similar
- Silver-Meal Heuristic and LUC perform best if the costs change over time
- Part Period Balancing perform best if the costs do not change over time
- The problem extended to all items is difficult to solve

Lot-Sizing

Example 2: The MRP gross requirements for Item A are shown here for the next 10 weeks. Lead time for A is three weeks and setup cost is $10. There is a carrying cost of $0.01 per unit per week. Beginning inventory is 90 units.

Week Gross requirements Week Gross requirements

1 30 6 80

2 50 7 20

3 10 8 60

4 20 9 200

5 70 10 50

Determine the lot sizes.

Lot-Sizing: Lot-for-Lot

Use the above table to compute ending inventory of various periods.

Lot-Sizing: Lot-for-Lot

20

Week 4 net requirement = 20 > 0. So, an order is required.

Lot-Sizing: Lot-for-Lot

20

20

20

A delivery of 20 units is planned for the 4th period..

Lot-Sizing: Lot-for-Lot

Exercise

0

70

20

20

20

0

The net requirement of the 5th period is 70 periods.

Lot-Sizing: EOQ

- First, compute EOQ
- Annual demand is not given. Annual demand is estimated from the known demand of 10 weeks.
- Compute annual holding cost per unit

Lot-Sizing: EOQ

- First, compute EOQ
- Therefore, whenever it will be necessary to place an order, the order size will be 344 units. This will now be shown in more detail.

Lot-Sizing: EOQ

Use the above table to compute ending inventory of various periods.

Lot-Sizing: EOQ

20

Week 4 net requirement = 20 > 0. So, an order is required.

Lot-Sizing: EOQ

20

344

344

Order size = EOQ = 344, whenever it is required to place an order.

Lot-Sizing: EOQ

Exercise

324

20

344

344

324

Week 5 b. inv=344-20=324>70= gross req. So, no order is required.

Lot-Sizing: Silver-Meal-Heuristic

The order is placed for K periods, for some K>0. Use the above table to find K.

Lot-Sizing: Silver-Meal-Heuristic

20 0.00 10 10.0

If K=1, order is placed for 1 week and the order size = 20. Then, the ending inventory = inventory holding cost =0. The order cost = $10. Average cost per period = (0+10)/1=$10.

Lot-Sizing: Silver-Meal-Heuristic

Exercise

20 0.00 10 10.0

90 70 0.70 10 5.35

If K=2, order is placed for 2 weeks and the order size = 20+70=90.

Then, inventory at the end of week 4 = 90-20=70 and holding cost =70 0.01. = 0.70. Average cost per period = (0.70+10)/2=$5.35.

Lot-Sizing: Silver-Meal-Heuristic

Use the above table to compute ending inventory of various periods.

Lot-Sizing: Silver-Meal-Heuristic

Exercise

20

Week 4 net requirement = 20 > 0. So, an order is required.

Lot-Sizing: Least Unit Cost

The order is placed for K periods, for some K>0. Use the above table to find K.

Lot-Sizing: Least Unit Cost

20 0.00 10 .500

If K=1, order is placed for 1 week and the order size = 20. Then, the ending inventory = inventory holding cost =0. The order cost = $10. Average cost per unit = (0+10)/20=$0.50

Lot-Sizing: Least Unit Cost

Exercise

20 0.00 10 .500

90 70 0.70 10 .119

If K=2, order is placed for 2 weeks and the order size = 20+70=90.

Then, inventory at the end of week 4 = 90-20=70 and holding cost =70 0.01. = 0.70. Average cost per unit = (0.70+10)/90=$0.119.

Lot-Sizing: Least Unit Cost

Use the above table to compute ending inventory of various periods.

Lot-Sizing: Least Unit Cost

Exercise

20

Week 4 net requirement = 20 > 0. So, an order is required.

Lot-Sizing: Part Period Balancing

The order is placed for K periods, for some K>0. Use the above table to find K.

Lot-Sizing: Part Period Balancing

20 0.00 10 10.0

90 70 0.70 10 9.30

170 150 80 2.30 10 7.70

190 170 100 20 2.90 10 7.10

250 230 160 80 60 5.30 10 4.70

450 430 360 280 260 200 15.30 10 5.30

NOT COMPUTED

1 week, week 9

200 0.00 10 10.0

250 50 0.50 10 9.50

2 weeks, weeks 9 to 10

The above computation is similar to that of the Silver-Meal heuristic. The primary difference is that the (absolute) difference between holding and ordering cost is shown in the last column.

Lot-Sizing: Part Period Balancing

Use the above table to compute ending inventory of various periods.

Lot-Sizing: Part Period Balancing

230 160 80 60 0 50

200

20 200

250 250

250 250

230 160 80 60 0 50 0

The computation is similar to that of the Silver-Meal heuristic.

Cost Comparison

- Lot-for-Lot
- See the last slide entitled “lot-sizing: lot-for-lot”
- Number of orders: 7
- Ordering cost = 7 $10/order = $70
- Holding cost = (60+10) $0.01/unit/week = $0.70
- Total cost = 70+0.70 =$70.70

- EOQ
- See the last slide entitled “lot-sizing: EOQ”
- Number of orders: 2
- Ordering cost = 2 $10/order = $20
- Holding cost = (60 +10 +324 +254 +174 +154 +94 +237 +187) $0.01/unit/week = $14.94
- Total cost = 20+14.94 =$34.94

Cost Comparison

- Silver-Meal Heuristic
- See the last slide entitled “lot-sizing: Silver-Meal heuristic”
- Number of orders: 2
- Ordering cost = 2 $10/order = $20
- Holding cost = (60 +10 +230 +160 +80 +60 +50) $0.01/unit/week = $6.50
- Total cost = 20+6.50 =$26.50

- Least Unit Cost
- See the last slide entitled “lot-sizing: least unit cost”
- Number of orders: 2
- Ordering cost = 2 $10/order = $20
- Holding cost = (60 +10 +430 +360 +280 +260 +200) $0.01/unit/week = $16.00
- Total cost = 20+16.00 =$36.00

Cost Comparison

- Part-Period Balancing
- See the last slide entitled “lot-sizing: part-period balancing”
- Number of orders: 2
- Ordering cost = 2 $10/order = $20
- Holding cost = (60 +10 +230 +160 +80 +60 +50) $0.01/unit/week = $6.50
- Total cost = 20+6.50 =$26.50

- Conclusion: In this particular case, Silver-Meal heuristic and part period balancing yield the least total holding and ordering cost of $26.50 over the planning period of 10 weeks.

READING AND EXERCISES

Lesson 22

Reading:

Section 7.2-7.3 pp. 366-375 (4th Ed.), pp. 358-366 (5th Ed.)

Exercise:

17 and 25 pp. 371-373, 375 (4th Ed.), pp. 363, 366