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

Question 2: MacBig Wrapper Inventory — EOQ, Decentralization, and Annual 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 the stated 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 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 3,000 wrappers/day, operating 7 days/week for the full 52-week year (no holidays): $$D=500\times3{,}000\times7\times52=\boxed{546{,}000{,}000\ \text{wrappers/yr}}.$$
2. Unit cost and EOQ cost parameters. A box of 10,000 costs $\$10,$ so the item cost is $c=\$10/10{,}000=\$0.001$/wrapper; holding cost $H=15\%\times c=\$0.00015$/wrapper/yr; ordering cost $S=\$100$/order.
3. Centralized EOQ (part a). Applying the EOQ formula to the full system demand: $$Q^*=\sqrt{\frac{2DS}{H}}=\sqrt{\frac{2(546{,}000{,}000)(100)}{0.00015}}\approx 26{,}981{,}475\ \text{wrappers} = 2{,}698.1\ \text{boxes}.$$ Rounding to a whole number of boxes (minimum order = 1 box) gives $\boxed{Q^*\approx2{,}698\ \text{boxes} = 26{,}980{,}000\ \text{wrappers}}$, i.e. about $D/Q^*\approx20.2$ orders/yr, one every $\approx18.0$ 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 2,698 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 ~18-day cycle, even a modest lead time keeps very little safety stock on hand). Annual relevant cost (ordering + holding, excluding purchase cost, which is fixed regardless of policy): $$TC_{HQ}=\frac{D}{Q^*}S+\frac{Q^*}{2}H\approx\boxed{\$4{,}047/\text{yr}}.$$
4. Decentralized alternative (part b). If each of the 500 stores orders independently at its own demand $d=3{,}000\times7\times52=1{,}092{,}000$ wrappers/yr, each still faces $S=\$100$ and $H=\$0.00015$/unit/yr, so each store's own EOQ is $$Q_{store}^*=\sqrt{\frac{2dS}{H}}\approx1{,}206{,}648\ \text{wrappers}\approx120.7\ \text{boxes}\ \Rightarrow\ 121\ \text{boxes}=1{,}210{,}000\ \text{wrappers},$$ i.e. only $d/Q_{store}^*\approx0.90$ orders/yr — a single store's order cycle stretches past 400 days, so each store would place barely one giant order every 13–14 months. Annual relevant cost per store $TC_{store}\approx\$181$/yr, and across all 500 stores $$TC_{decentralized}=500\times TC_{store}\approx\boxed{\$90{,}499/\text{yr}}.$$ 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 $\$86{,}500$/yr, with no offsetting benefit from decentralizing (both policies ship the same total volume; decentralizing only fragments the ordering, and drags each store's own order cycle out to over a year, which is itself an operational red flag for a perishable-adjacent, custom-printed item).

(c) Knowing each store's individual demand. The problem as posed assumes every store demands the same 3,000 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) Annual shrinkage. Both rates are stated directly per year, so they add straight onto the 15% financial holding rate without any annualizing conversion: at head office, $H_{HQ}'=(15\%+18\%)\times c=33\%\times\$0.001=\$0.00033$/unit/yr; at the stores, $H_{store}'=(15\%+1\%)\times c=16\%\times\$0.001=\$0.00016$/unit/yr. Recomputing the two total costs with these shrinkage-adjusted holding rates: $$TC_{HQ}'=\sqrt{2DSH_{HQ}'}\approx\$6{,}003/\text{yr},\qquad TC_{decentralized}'=500\sqrt{2dSH_{store}'}\approx\$93{,}467/\text{yr}.$$ The centralized option's cost rises by about 48% (from $\$4{,}047$ to $\$6{,}003$), while the decentralized option barely moves (from $\$90{,}499$ to $\$93{,}467$) — the store-level 1%/yr rate is much closer to the financial 15%/yr baseline than head office's steep 18%/yr rate is, so it changes store economics comparatively little. The recommendation does not change — centralization remains over 15$\times$ cheaper — but an 18%/year warehouse shrinkage rate is itself alarmingly high (more than the 15% financial holding cost on its own) and is a strong argument for tightening loss-prevention controls at the head-office warehouse specifically, independent of where inventory is held.