Profit Optimization at Soda Factory

by Panith Thiruvenkatasamy

Submitted : Spring 2018

A Local soda company is trying to maximize its profit per phone manufactured. The soda bottles are sold for $5 per unit. After analyzing the cost of direct labor, indirect labor, materials, indirect materials, utilities, repair, maintenance, taxes, and insurance, the company’s analysts come up with the formula x³-5.2x²+8x to calculate the total manufacturing cost of 1000x units. This equation is to be used in accordance with the Law of Diminishing Marginal Returns. The objective is to find the number of units to produce to maximize the profit.

The presented problem is a profit margin optimization problem. The function for the revenue produced when 1,000x bottles are sold was found by multiplying the price of each bottle by the multiple of 1,000 sodas sold. It was kept in mind that the function represented thousands of dollars, since 1x represents 1000 bottles. Then, the function for the cost of manufacturing was subtracted from the revenue function to obtain the profit per 1,000 bottles produced and sold function. This function was used to optimize the profit margin. The critical points of the profit function were found by first deriving the function, and then finding the zero values of the first derivative. Then, the critical points were tested to be either maxima or minima by inputting them to the second derivative of the profit function. The x value of the maximum point is the multiple of 1,000 bottles that need to be produced to maximize profit. The maximum point, or the point were the second derivative function outputted a negative value, was then inputted to the original profit function to find the maximum profit margin obtainable.

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