How to Model the Optimal Factory Price for Toothpaste Packs

Problem

Determine the reasonable factory price for a specific toothpaste pack size based on given factory prices of other pack sizes.

Modeling Process

(1) Model Preparation

In everyday life, larger packaging often offers better value due to factory pricing. Suppose a factory produces toothpaste packs of size A with a factory price of X yuan per pack, and packs of size B with a price of 25 yuan per pack. The factory now needs to produce packs of size C and we must determine a reasonable factory price for this size.

(2) Model Assumptions

(i) The factory price of toothpaste depends only on production cost and packaging cost. (ii) Production cost is proportional to the mass of the toothpaste (excluding the tube). (iii) Packaging cost is proportional to the surface area of the tube. (iv) The tube is fully filled.

(3) Model Formulation

Let k1 be the proportionality coefficient between production cost and toothpaste mass, and k2 be the coefficient between packaging cost and tube surface area. Then the factory price P can be expressed as:

P = k1 * mass + k2 * surface_area

For the known pack sizes we obtain equations that relate k1 and k2 to the given prices, allowing us to solve for the price of the target pack.

(4) Model Solution

Using the known price of the A‑size pack (X yuan) and the B‑size pack (25 yuan), we solve the system of equations to find k1 and k2, and then compute the price of the C‑size pack:

P_C = ... (computed value) yuan

(5) Model Analysis

(i) The actual factory price also includes other costs such as outer box packaging, which are not considered in this simplified model. (iii) Additional factors like transportation cost and retailer profit can be incorporated to refine the model and estimate a market selling price.

(6) Model Validation

The computed price is compared with the real factory price of the C‑size pack. If the discrepancy is within an acceptable error range, the model is validated; otherwise, reasons for deviation are analyzed and the model is improved.

Shen Wenxuan, Yang Qingtiao, “Mathematical Modeling Attempts”