When Is the Optimal Time to Sell Pigs? A Profit Maximization Model

Problem

A pig farm invests 4 yuan each day for feed, labor, and equipment, which is estimated to increase the weight of an 80 kg pig by 2 kg. The market price is currently 8 yuan per kilogram but is expected to drop by 0.1 yuan each day. The question is when the pig should be sold to maximize profit, and how estimation errors affect the result.

Analysis

Because daily investment raises the pig’s weight while the selling price declines over time, there exists an optimal selling time that yields the highest profit.

Modeling and Solution

Let t be the number of days until sale. The pig’s weight after t days is W(t)=80+ r·t , where r is the daily weight‑gain rate (≈2 kg per 4 yuan). The selling price after t days is P(t)=8 - g·t , where g is the daily price‑drop rate (0.1 yuan). Daily cost is 4 yuan, so total cost is C(t)=4·t . Revenue is R(t)=W(t)·P(t) . Profit is π(t)=R(t)-C(t) . Maximizing π(t) with respect to t gives the optimal selling day; the solution indicates selling after a certain number of days (e.g., 7 days) can increase profit by about 20 yuan.

Sensitivity Analysis

The model’s response to parameter changes is examined:

If the daily weight‑gain rate r increases by 1 %, the optimal selling time is delayed by roughly 3 %.

If the daily price‑drop rate g increases by 1 %, the optimal selling time advances by about 3 %.

These percentages represent the relative sensitivities of the optimal selling time to the two parameters.

Robustness Analysis

When parameters are not constant, the recommendation is to keep the pig until the incremental profit equals the daily cost, then sell. It is suggested to re‑estimate the parameters after one week (t = 7) and recompute the optimal selling day.

Source: Mathematical Modeling (5th Edition) by Jiang Qiyuan, Xie Jinxing, Ye Jun.