Quantitative - Choosing a parcel carrier: fixed fee versus volume growth
A Riyadh online store ships 960 parcels a month and expects that figure to hold for the first three months of a twelve-month planning period. From month four, a marketing campaign is expected to lift volume to 1,200 parcels a month for the remaining nine months. Carrier A charges 21 SAR per parcel, has no monthly fee, and absorbs the cost of returned parcels itself. Carrier B charges a fixed 3,000 SAR a month plus 18 SAR per parcel, and bills an extra 4 SAR for each returned parcel; returns run at 3% of parcels. Moving to Carrier B also requires a one-time 5,000 SAR integration fee. The owner will sign with a single carrier for all twelve months and wants a recommendation supported by numbers, not impressions.
1. At what monthly parcel volume do the two carriers cost the same, once return charges are included? State the equation and any assumption your answer depends on.
Read the worked answer
Step 1 - convert Carrier B into a cost per parcel. B charges 18 SAR per parcel plus 4 SAR on the 3% that come back: 18 + (0.03 × 4) = 18 + 0.12 = 18.12 SAR per parcel, plus the 3,000 SAR fixed fee.
Step 2 - set the bills equal. 21P = 3,000 + 18.12P (21 − 18.12)P = 3,000 2.88P = 3,000 → P = 1,041.7, i.e. about 1,042 parcels a month.
Below 1,042 parcels A is cheaper; above it B is cheaper.
Assumption that must be checked: the calculation bills B's 18 SAR on every parcel, including returned ones. If B refunds the 18 SAR on a returned parcel and only charges the 4 SAR, B's cost drops to (18 × 0.97) + 0.12 = 17.58 SAR, and the crossover jumps to 3,000 ÷ 3.42 ≈ 877 parcels. The contract wording, not arithmetic, decides the answer.
2. Compare the two carriers over the full twelve months, including the integration fee, and give the owner a recommendation with a number attached.
Read the worked answer
Months 1–3 (960 parcels):
- A: 960 × 21 = 20,160 SAR per month → 60,480 SAR
- B: 3,000 + 960 × 18.12 = 3,000 + 17,395.20 = 20,395.20 SAR per month → 61,185.60 SAR A wins these three months by 235.20 SAR each.
Months 4–12 (1,200 parcels):
- A: 1,200 × 21 = 25,200 SAR per month → 226,800 SAR
- B: 3,000 + 1,200 × 18.12 = 3,000 + 21,744 = 24,744 SAR per month → 222,696 SAR B wins these nine months by 456 SAR each.
Totals: A = 60,480 + 226,800 = 287,280 SAR. B = 61,185.60 + 222,696 + 5,000 = 288,881.60 SAR.
Recommendation: stay with Carrier A, but only narrowly - A is cheaper by about 1,600 SAR, roughly 0.6% of spend. The decision is fragile, so the owner should ask B to waive or reduce the integration fee; at zero fee B wins by about 3,400 SAR and becomes the better contract.
3. The accountant argues the 5,000 SAR integration fee should be ignored because it is one-time. Evaluate that reasoning and identify the volume condition under which B recovers the fee.
Read the worked answer
The reasoning is wrong in a way that is easy to miss: a cost does not stop counting because it is paid once. What matters is whether the recurring saving is large enough, over the life of the contract, to repay the one-time cost.
B's saving at 1,200 parcels is 25,200 − 24,744 = 456 SAR per month. Recovery time = 5,000 ÷ 456 ≈ 11 months of high-volume shipping.
The plan gives only nine such months, so B recovers about 4,104 SAR of the 5,000 and ends the year behind. The fee would be recovered only if the 1,200-parcel level continued for roughly two more months - that is, if the campaign's effect persists into a second term or volume keeps growing.
Useful sensitivity numbers: at 1,300 parcels from month four, B saves 744 SAR a month and clears the fee within the nine months; if volume stays at 960 all year, A wins comfortably. The recommendation therefore depends on a forecast, and the honest answer names that dependency rather than hiding it.