Application of the TORA Software for Linear Programming Using the Big M Method to Maximize Profits from Scent, Aroma, and Fresh Perfume Products

Abstract

Linear Programming (LP) is a mathematical method used to find the optimal solution in resource allocation problems to achieve specific objectives, such as maximizing profit or minimizing cost. This method is widely applied in various fields such as business, manufacturing, and economics. In this study, linear programming is applied to a perfume company that produces three types of perfumes: Scent, Aroma, and Fresh. Each perfume requires limited raw materials, namely rose, jasmine, and kenanga flower extracts. The available resources today are only 1.7 kg of rose extract, 1 kg of jasmine extract, and 0.3 kg of kenanga extract. The company aims to maximize profit, with a profit of Rp30,000 per bottle for Scent, Rp17,000 for Aroma, and Rp19,000 for Fresh. Given the constraints on raw materials, linear programming is used to determine the optimal number of perfume bottles to produce in order to maximize the company's profit. The problem is solved using the Big-M method, which handles inequality constraints in the form of ≥ by adding artificial variables. In this approach, a very large penalty is assigned to artificial variables to ensure that these variables are not part of the optimal solution. The TORA software is used to perform the calculations and analysis in solving this problem. The results show that the company can achieve a maximum profit of Rp7,390,000.00 by producing the optimal number of perfume bottles. This method proves effective in solving resource allocation problems to maximize profit in a manufacturing context.