Question 2 of 7: MacBig Wrapper Inventory — EOQ, Decentralization, and Shrinkage

Question 2: MacBig Wrapper Inventory — EOQ, Decentralization, and Shrinkage (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. Head-office-held inventory, shipped to every store as needed; wrapper demand is constant and continuous (stores never close, no holidays).

Find. (a) An inventory-control policy for the centralized (head-office) system; (b) the annual cost of a decentralized (store-level) alternative and a recommendation; (c) how a known per-store demand split would change the policy; (d) whether differential shrinkage rates reverse the recommendation.

Approach. Treat wrappers as a classic deterministic, constant-rate EOQ item: compute the system-wide annual demand and per-unit holding/ordering costs, size a continuous-review $(Q,R)$ order policy centrally, then repeat the EOQ calculation as if each store ordered independently and compare total relevant costs to make the centralize-vs-decentralize decision.

1. Annual system-wide demand. With 500 stores at 12,000 wrappers/day, operating 7 days/week for the full 52-week year (no holidays): $$D=500\times12{,}000\times7\times52=\boxed{2{,}184{,}000{,}000\ \text{wrappers/yr}}.$$
2. Unit cost and EOQ cost parameters. A box of 10,000 costs $100, so the item cost is $c=\$100/10{,}000=\$0.01$/wrapper; holding cost $H=10\%\times c=\$0.001$/wrapper/yr; ordering cost $S=\$100$/order.
3. Centralized EOQ (part a). Applying the EOQ formula to the full system demand: $$Q^*=\sqrt{\frac{2DS}{H}}=\sqrt{\frac{2(2{,}184{,}000{,}000)(100)}{0.001}}\approx 20{,}899{,}761\ \text{wrappers} = 2089.98\ \text{boxes}.$$ Rounding to a whole number of boxes (minimum order = 1 box) gives $\boxed{Q^*\approx2{,}090\ \text{boxes} = 20{,}900{,}000\ \text{wrappers}}$, i.e. about $D/Q^*\approx104.5$ orders/yr, one every $\approx3.5$ days. The control system is therefore a continuous-review order policy at head office: monitor the aggregate wrapper stock daily, and place a fixed order of 2,090 boxes whenever the position drops to the reorder point $R=(\text{daily system demand})\times(\text{lead time})$ (lead time not stated, so $R$ is left symbolic; with the ~3.5-day cycle, even a short lead time keeps very little safety stock on hand). Annual relevant cost (ordering + holding, excluding purchase cost, which is fixed regardless of policy): $$TC_{HQ}=\frac{D}{Q^*}S+\frac{Q^*}{2}H\approx\boxed{\$20{,}900/\text{yr}}.$$
4. Decentralized alternative (part b). If each of the 500 stores orders independently at its own demand $d=12{,}000\times7\times52=4{,}368{,}000$ wrappers/yr, each still faces $S=\$100$ and $H=\$0.001$/unit/yr, so each store's own EOQ is $$Q_{store}^*=\sqrt{\frac{2dS}{H}}\approx934{,}666\ \text{wrappers}\approx93.5\ \text{boxes}\ \Rightarrow\ 93\ \text{boxes}=930{,}000\ \text{wrappers},$$ with annual relevant cost per store $TC_{store}\approx\$934.68$/yr, and across all 500 stores $$TC_{decentralized}=500\times TC_{store}\approx\boxed{\$467{,}339/\text{yr}}.$$ This is about 22$\times$ the centralized cost ($\sqrt{500}\approx22.4$, the classic EOQ scale factor: pooling $n$ independent streams into one order stream cuts relevant cost by $\sqrt{n}$). Recommendation: keep inventory centralized at head office — the saving is roughly $\$446{,}000$/yr, with no offsetting benefit from decentralizing (both policies ship the same total volume; decentralizing only fragments the ordering).

(c) Knowing each store's individual demand. The problem as posed assumes every store demands the same 12,000 wrappers/day; in reality demand almost certainly varies by store (city size, drive-thru traffic, seasonality). Knowing the true per-store split, I would keep the ordering centralized (the pooling argument above does not depend on demand being equal across stores — it only requires aggregation), but I would stop shipping every store the same allocation. Instead I would allocate the centrally ordered stock to stores in proportion to each store's actual demand, hold safety stock centrally rather than duplicating it at every store (risk pooling reduces the total safety stock needed to hit a given service level, because a demand spike at one store is partly offset by slack at another), and apply an ABC-style classification so the highest-volume stores are replenished more frequently/tightly monitored while low-volume stores are batched onto a lower-frequency route. This keeps the $\sqrt{n}$ ordering-cost advantage of part (b) while removing the inefficiency of a uniform-allocation policy that over-stocks slow stores and risks stock-outs at fast ones.

(d) Differential shrinkage. Shrinkage acts as an additional, non-recoverable holding cost on top of the stated 10% financial rate. A crude but useful way to fold it in is to annualize the monthly rate linearly and add it to $H$: at head office, $H_{HQ}\approx(10\%+12\times5\%)\times c=70\%\times\$0.01=\$0.007$/unit/yr; at the stores, $H_{store}\approx(10\%+12\times1\%)\times c=22\%\times\$0.01=\$0.0022$/unit/yr. Recomputing the two total costs with these shrinkage-adjusted holding rates: $$TC_{HQ}'=\sqrt{2DSH_{HQ}}\approx\$55{,}296/\text{yr},\qquad TC_{decentralized}'=500\sqrt{2dSH_{store}}\approx\$693{,}167/\text{yr}.$$ The centralized option's cost roughly triples (its dramatically shorter order cycle, $\approx3.5$ days versus the stores' $\approx77.5$-day cycle, means HQ stock is exposed to the 5%/month shrinkage rate for far less time per unit than a store shelf is exposed to 1%/month — fast turnover is itself a mitigant against a high loss rate), but it is still over 12$\times$ cheaper than decentralizing. The recommendation does not change — centralization remains the right call — but the shrinkage figures are a strong argument for also tightening loss-prevention controls at the head-office warehouse specifically (the 5%/month rate is itself the real problem to fix, independent of where inventory is held).