Linear Programming Model for Maximizing Weekly Profit in Toy Manufacturing, Study notes of Mathematics

Download Linear Programming Model for Maximizing Weekly Profit in Toy Manufacturing and more Study notes Mathematics in PDF only on Docsity! MT235 Introduction prepared by Professor Jenny Baglivo, September 2002 In this brief introduction, I will discuss the modeling process, formulate a two-variable linear programming model, and solve the problem graphically. References: Sections 1.1-1.4 and 2.1-2.2. In making decisions, an analyst needs to • Evaluate the alternatives and • Choose the best course of action. Mathematics can help. A mathematical model uses • Symbols to represent decision variables and • Functions to describe a real system or decision problem. There are benefits to using models. In particular, • Models are less costly to analyze than real systems. • Models can be analyzed more rapidly. • Models facilitate "what if" analyses. • Models provide insight about the real system under study. 1 ‡ § 1. Modeling Process The modeling process includes the following steps: (1) Identify the problem. (2) Formulate the model. (3) Analyze the model. (4) Test the results. (5) Implement the solution. We will work with steps (2) through (4): From an identified problem, we will formulate and analyze the model, and interpret the results. We will ask "what if" certain aspects of the problem change, analyze the new model, and interpret the results. The models we will use are often referred to as optimization models. 2 Decision variables and mathematical model: x1 = number of dozen Zappers produced in one week x2 = number of dozen Space Rays produced in one week MAXIMIZE: Profit = 5 x1 + 8 x2 SUBJECT TO THE FOLLOWING CONSTAINTS: Plastic : x1 + 2 x2 <= 1200 Production Time : 4 x1 + 3 x2 <= 2400 Production Total : x1 + x2 <= 800 Production Mix : –x1 + x2 <= 450 Nonnegativity : x1 >= 0 x2 >= 0 Solution highlights 200 400 600 800 1000 1200 x1 200 400 600 800 x2 5 (1) The lines in the display are graphs of the linear equations x1 + 2 x2 = 1200 4 x1 + 3 x2 = 2400 x1 + x2 = 800 –x1 + x2 = 450 x1 = 0 x2 = 0 (2) The shaded area is the solution to the system of linear inequalities x1 + 2 x2 <= 1200 4 x1 + 3 x2 <= 2400 x1 + x2 <= 800 –x1 + x2 <= 450 x1 >= 0 x2 >= 0 (3) The form of the problem (maximizing a linear function over a region defined by a system of linear equations and inequalities) means that the method of linear programming can be used to find the best combination of Zappers and Space Rays. The solution is Produce • x1 = 240 dozen Zappers and • x2 = 480 dozen Space Rays each week. The weekly profit will be $5040. This production schedule is an improvement over the current schedule. 6