Newsvendor Model — Weekly Sausage Batch Size: 23-Ind-A1 Operations Research, December 2018, Q8

Question 8: Newsvendor Model — Weekly Sausage Batch Size (15 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. Demand $D\sim \text{Uniform}(1000,\,2000)$ kg/week; gross profit on fresh sale $=\$0.50$/kg; gross loss on excess sold to pet food $=\$0.10$/kg.

Find. The batch size $Q^*$ that maximizes expected net profit (the newsvendor optimum).

Approach. Single-period (newsvendor) inventory model: the underage cost is the margin lost on any demand not covered by the batch, the overage cost is the loss taken on any batch in excess of demand; the optimum sets the batch at the critical-fractile quantile of demand.

Identify underage and overage costs. Underage cost (profit foregone per kg of unmet demand) $C_u=\$0.50$/kg. Overage cost (loss per kg sold to the pet food buyer instead) $C_o=\$0.10$/kg.
Compute the critical ratio. $$CR=\frac{C_u}{C_u+C_o}=\frac{0.50}{0.50+0.10}=\boxed{0.8333}$$
Apply the uniform-demand newsvendor formula $Q^*=a+CR(b-a)$ for $D\sim\text{Uniform}(a,b)$: $$Q^*=1000+0.8333(2000-1000)=1000+833.33=\boxed{1833.33\text{ kg}}$$
Expected net profit at the optimum (a check that the fractile really pays). For uniform demand the expected leftover is $E[(Q-D)^+]=\dfrac{(Q-a)^2}{2(b-a)}=\dfrac{833.33^2}{2000}=347.22$ kg, so expected fresh sales are $1833.33-347.22=1486.11$ kg and $$E[\text{net profit}]=0.50(1486.11)-0.10(347.22)=\boxed{\$708.33/\text{week}}$$ By comparison, baking only the mean demand (1500 kg) earns $0.50(1375)-0.10(125)=\$675.00$, so the fractile batch is worth about $33 a week more. The exam text says "minimizes the expected total net profit"; since profit is the quantity to be made as large as possible, this is read as a slip for "maximizes".

Final results — Question 8
Item	Value
Underage cost $C_u$	$0.50/kg
Overage cost $C_o$	$0.10/kg
Critical ratio	0.8333 (5/6)
Optimal batch size $Q^*$	1833.33 kg
Expected net profit at $Q^*$	$708.33/week

Topic: Newsvendor (single-period inventory) model with a uniform demand distribution.

Key relations: Critical ratio $CR=C_u/(C_u+C_o)$; for $D\sim\text{Uniform}(a,b)$, $Q^*=a+CR(b-a)$ (the $CR$-quantile of a uniform distribution is a linear interpolation between its bounds).

Why this works: Expected profit is concave in the batch size; the marginal expected benefit of one more kg is $C_u\cdot P(D\ge Q)$ and the marginal expected cost is $C_o\cdot P(D\lt Q)$ — setting them equal gives $P(D\le Q^*)=CR$.

Common pitfall: Confusing which margin is underage vs. overage (the fresh-sale profit is the cost of running SHORT, not the cost of running long), or using the demand mean (1500 kg) instead of the critical-fractile quantile.

Source: Nahmias, Production and Operations Analysis — single-period (newsvendor) inventory model.

Question 8 of 10: Newsvendor Model — Weekly Sausage Batch Size

Question 8: Newsvendor Model — Weekly Sausage Batch Size (15 marks)