Question 4 of 8: Lot-Sizing a Requirements Schedule — EOQ, Periodic Reorder and Part-Period Balancing

Question 4: Lot-Sizing a Requirements Schedule — EOQ, Periodic Reorder and Part-Period Balancing (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A twelve-week discrete requirements schedule for a dependent-demand component, with a set-up (ordering) charge per lot and a carrying charge per unit held at the end of each week.

Find. The week-by-week inventory record (lot receipts and ending balances) and the total incremental cost, meaning set-up cost plus carrying cost over the twelve weeks, under each of the three lot-size policies, and hence which policy is cheapest for this schedule.

Approach. Compute the fixed economic lot size and the economic order interval from the classical square-root formulas, run each policy week by week against the tabulated requirements to obtain its inventory record, and cost each record as (number of lots) times the set-up charge plus (sum of the weekly ending balances) times the carrying charge.

1. Part (a) — compute the economic lot size. Balancing the set-up cost against the carrying cost of the average cycle stock gives the classical square-root lot size $$Q_{o}=\sqrt{\frac{2\,\bar{R}\,c_{p}}{c_{H}}}=\sqrt{\frac{2(116.7)(400)}{4}}=\sqrt{23{,}340}=152.8\;\Rightarrow\;\boxed{Q_{o}=153\ \text{units per lot}}$$ The lot is rounded to a whole unit; the total-cost curve is very flat near its minimum, so nothing is lost by the rounding.
2. Run the fixed lot against the schedule. The rule is mechanical: enter each week with the balance carried forward, and if it cannot cover that week's requirement, receive as many whole lots of 153 units as are needed. Week 1 opens empty, so one lot arrives and 128 units are left over; weeks 2 and 3 are met from stock; week 6 needs 325 against an opening balance of 4, so three lots (459 units) arrive; week 7 needs two more.

   Inventory record, policy (a): fixed lot of 153 units

   Week	1	2	3	4	5	6	7	8	9	10	11	12	Total
   Requirement	25	30	75	125	200	325	400	100	0	100	0	100	1480
   Lot received	153	—	—	153	153	459	306	153	—	153	—	—	1530
   Ending balance	128	98	23	51	4	138	44	97	97	150	150	50	1030
   Carrying cost	$512	$392	$92	$204	$16	$552	$176	$388	$388	$600	$600	$200	$4,120

3. Cost the fixed-lot record. Ten lots are released, and the ending balances sum to 1,030 unit-weeks: $$TC_{a}=10(400)+1{,}030(4)=4{,}000+4{,}120=\boxed{\$8{,}120}$$ The fixed lot is a poor fit here. It was derived from an average requirement, but the actual schedule is lumpy, so the policy leaves 128 units sitting through weeks in which almost nothing is used, and still needs three lots in a single week when demand spikes.
4. Part (b) — convert the lot size into an order interval. The economic periodic reorder model orders at a fixed interval rather than in a fixed quantity, and the economic order interval is the lot size expressed in weeks of average demand: $$EOI=\frac{Q_{o}}{\bar{R}}=\sqrt{\frac{2\,c_{p}}{c_{H}\,\bar{R}}}=\sqrt{\frac{2(400)}{4(116.7)}}=1.31\ \text{weeks}\;\Rightarrow\;\boxed{T=1\ \text{week}}$$ Since orders can only be placed at the weekly buckets of the schedule, the interval is rounded to the nearest whole week.
5. Run the periodic policy. With a one-week interval the rule degenerates into lot-for-lot ordering: each week receives exactly its own requirement and nothing is carried, except that the two weeks with no requirement need no order at all.

   Inventory record, policy (b): reorder every week, order the week's requirement

   Week	1	2	3	4	5	6	7	8	9	10	11	12	Total
   Requirement	25	30	75	125	200	325	400	100	0	100	0	100	1480
   Lot received	25	30	75	125	200	325	400	100	—	100	—	100	1480
   Ending balance	0	0	0	0	0	0	0	0	0	0	0	0	0
   Carrying cost	—	—	—	—	—	—	—	—	—	—	—	—	$0

   Ten lots are released and nothing is ever held, so $$TC_{b}=10(400)+0(4)=\boxed{\$4{,}000}$$ For comparison, rounding the interval up to two weeks instead would give six lots and 780 unit-weeks of stock, $2,400 plus $3,120, a total of $5,520 — so the one-week rounding is the correct one.
6. Part (c) — find the economic part-period. Part-period total cost balancing extends each lot forward in time until the cost of carrying the added requirement is about to exceed the set-up cost that would be saved. The balance point is the economic part-period $$EPP=\frac{c_{p}}{c_{H}}=\frac{400}{4}=\boxed{100\ \text{part-periods}}$$ meaning that carrying 100 units for one week, or one unit for 100 weeks, costs exactly as much as one additional set-up.
7. Accumulate part-periods lot by lot. Starting in week 1, adding week 2 to the lot costs $30\times1=30$ part-periods; adding week 3 as well would cost $30+75\times2=180$, which overshoots 100 by more than 30 undershoots it, so the first lot covers weeks 1 and 2 only, 55 units. The same test applied from week 3 gives $125\times1=125$ part-periods, nearer to 100 than 0 is, so the second lot covers weeks 3 and 4, 200 units. Continuing in this way places lots in weeks 1, 3, 5, 6, 7, 10 and 12.

   Inventory record, policy (c): part-period balancing, EPP = 100

   Week	1	2	3	4	5	6	7	8	9	10	11	12	Total
   Requirement	25	30	75	125	200	325	400	100	0	100	0	100	1480
   Lot received	55	—	200	—	200	325	500	—	—	100	—	100	1480
   Ending balance	30	0	125	0	0	0	100	0	0	0	0	0	255
   Carrying cost	$120	—	$500	—	—	—	$400	—	—	—	—	—	$1,020

8. Cost the part-period record and compare the three policies. Seven lots are released and the ending balances sum to 255 unit-weeks: $$TC_{c}=7(400)+255(4)=2{,}800+1{,}020=\boxed{\$3{,}820}$$ which is the cheapest of the three. The week-10 decision is a genuine tie on the part-period test — extending that lot to cover week 12 as well would accumulate 200 part-periods, exactly as far above 100 as 0 is below it — and the tie is broken by costing both: covering weeks 10 to 12 in one lot saves one set-up ($400) but adds 200 unit-weeks of stock ($800), a net penalty of $400. Placing the separate week-12 lot is therefore correct.