Question 4 of 7: Seasonal Lawn-Game Production — Dynamic Lot Sizing

Question 4: Seasonal Lawn-Game Production — Dynamic Lot Sizing (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. March–August demand and cost data, opening inventory (end of February) $=0$:

Find. (a) The cost-minimizing production quantity for every month, March–August; (b) whether a finer (weekly/daily) or coarser (quarterly) planning period would change the result.

Approach. This is a dynamic (multi-period) lot-sizing problem with time-varying setup, unit-production and holding costs — the appropriate algorithm is the Wagner–Whitin dynamic program, but because the unit production cost itself changes part-way through the horizon (unlike the textbook's constant-unit-cost version), the production-cost term cannot be dropped from the recursion: it must be evaluated for every candidate lot. With only 6 periods, every one of the $2^5=32$ feasible "produce / don't produce" patterns (March must produce, since opening inventory is zero) can be evaluated exactly by computer, which is both simpler and more reliable here than working the Wagner–Whitin recursion by hand.

1. Zero-inventory-ordering property. As in standard lot-sizing, the optimal plan never produces in a period while carrying inventory into it from an earlier lot (any such plan is weakly dominated by delaying that production) — so the search reduces to choosing which subset of months are "production months," each production month's lot exactly covering its own demand plus the demand of every following month up to (but not including) the next production month.
2. Cost of one lot covering months $i$ to $j$. With unit cost $c_i$ and setup $S_i$ charged at the producing month's own rate, and holding cost $h_t$ charged on whatever is still in stock at the end of each month $t$ in the lot: $$\text{Cost}(i\!\to\!j)=S_i+c_i\!\sum_{t=i}^{j}d_t+\sum_{t=i}^{j-1}h_t\!\left(\sum_{k=t+1}^{j}d_k\right).$$
3. Exhaustive evaluation (32 patterns). Evaluating every one of the 32 feasible production-month subsets gives a minimum total cost of $$\boxed{\text{Total cost}=\$421{,}100},$$ achieved by producing in March (covering March–May, quantity $3{,}000+1{,}500+500=5{,}000$ units) and again in June (covering June–August, quantity $300+800+2{,}000=3{,}100$ units): $$\text{March lot: }S+c\!\cdot\!5{,}000+h_{\text{Mar}}(1{,}500\!+\!500)+h_{\text{Apr}}(500)=2{,}000+250{,}000+2{,}000+500=\$254{,}500,$$ $$\text{June lot: }S+c\!\cdot\!3{,}100+h_{\text{Jun}}(800\!+\!2{,}000)+h_{\text{Jul}}(2{,}000)=2{,}000+155{,}000+5{,}600+4{,}000=\$166{,}600.$$ Sum $=\$254{,}500+\$166{,}600=\boxed{\$421{,}100}$, matching the exhaustive search.
4. A genuine tie exists. Splitting the March lot into two separate lots (produce March alone, $3{,}000$ units; then April, covering April–May, $2{,}000$ units) costs exactly the same, $\$421{,}100$: the extra $\$2{,}000$ setup is offset exactly by $\$2{,}000$ less holding cost, since April's demand is no longer carried an extra month from March. Both plans are optimal; the two-lot (March/June) plan is reported as the primary answer for simplicity (fewer setups, same cost).

(b) Effect of a Different Planning-Period Length

A coarser (quarterly) period would force the algorithm to lump each quarter's demand into one bucket, which happens to align reasonably well here (March–May is Q1, June–August is Q2, matching the optimal lot boundaries found above) — but this is a coincidence of this particular cost break, not a general guarantee. In general, a quarterly bucket removes the algorithm's ability to place a lot's boundary anywhere except a quarter edge, so if the true cost-minimizing switch point fell in the middle of a quarter, the coarser model would be forced into a strictly worse (or at best equally good) plan; a quarterly period can never improve on the monthly answer, only match or worsen it.

A finer (weekly or daily) period would not change the qualitative structure of the answer but could improve it: the feasible region of the finer-grained lot-sizing problem strictly contains every plan expressible at the monthly level (any monthly plan can be reproduced by lots that start/end exactly on month boundaries), so the finer model's optimum is guaranteed to cost the same or less, never more — assuming setup and holding costs scale down proportionally with the shorter period. In practice, a weekly or daily model could reveal a genuine improvement if demand is not actually uniform within a month (e.g., if June's 300 units are concentrated in the first two weeks rather than spread evenly), letting a lot's end boundary land closer to the true demand pattern and shave a little more holding cost than the monthly aggregation can see.