Question

Tom's, Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom's, Inc., makes two salsa products: Western Foods Salsa and Mexico City Salsa. Essentially, the two products fave different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20% tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs 10 ounces. For the current production period Tom's, Inc., can purchase up to 280 pounds of whole tomatoes, 130 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound for these ingredients is $0.96,$0.64, and $0.56, respectively. The cost of the spices and the other ingredients is approximately $0.10 per jar. Tom's, Inc., buys empty glass jars for $0.02 each, and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Tom's contract with Western Foods results in sales revenue of $1.64 for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa.

a. Develop a linear programming model that will enable Tom's to determine the mix of salsa products that will maximize the total profit contribution.
b. Find the optimal solution.

Answer 1

A linear programming model for Tom's, Inc. involves setting up variables for the number of salsa jars produced, constraints based on ingredient availability and recipes, and an objective to maximize profit by determining the optimal production quantities.

Step-by-step explanation:

To develop a linear programming model for maximizing profit for Tom's, Inc., we need to determine the variables, constraints, and objective function. Let X be the number of Western Foods Salsa jars and Y be the number of Mexico City Salsa jars produced.

The constraints are based on the availability of whole tomatoes, tomato sauce, and tomato paste, along with the salsa recipes. We also account for the cost of the jars, labels, and additional ingredients per jar.

The objective is to maximize the profit function, P = $1.64X + $1.93Y - total cost. Total cost includes the cost of tomatoes, tomato sauce, paste, spices, jars, labeling, and filling for X and Y jars.

By setting up inequalities for the constraints and a profit function, we can solve the linear programming problem using methods such as the Simplex algorithm, graphical methods, or optimization software to find the optimal solution of jars to produce for maximum profit.