49928 - Design Optimisation for Manufacturing Project -1 Problem - 9 Student Name Student Id Number Sri Prashanth K. Sukumar 12467256 Swaroop M. Kunyil 11852074 Karthikeayan Somasundaram 12767315 Problem 9 A company assembles four products (1, 2, 3, 4) from delivered components. The profit per unit for each product (1, 2, 3, 4) is £10, £15, £22 and £17 respectively. The maximum demand in the next week for each product (1, 2, 3, 4) is 50, 60, 85 and 70 units respectively. There are three stages (A, B, C) in the manual assembly of each product and the man- hours needed for each stage per unit of product are shown below: The nominal time available in the next week for assembly at each stage (A, B, C) is 160, 200 and 80 man-hours respectively. It is possible to vary the man-hours spent on assembly at each stage such that workers previously employed on stage B assembly could spend up to 20% of their time on stage A assembly and workers previously employed on stage C assembly could spend up to 30% of their time on stage A assembly. Production constraints also require that the ratio (product 1 units assembled)/(product 4 units assembled) must lie between 0.9 and 1.15. Formulate the problem of deciding how much to produce next week as a linear program and solve it. Solution for Problem 9: Mathematical Model Decision Variables: A company assembles four products from delivery component and their profit are described as X1 is profit of product 1 X2 is profit of product 2 X3 is profit of product 3 X4 is profit of product 4 Non-Decision Variables: Tba is maximum man-hour that may be spend by B for A, where Tba<= 0.2*200 Tca is maximum man-hour that may be spend by C for A, where, where Tca= 0.3*80 Let us assume man hour as Time(hrs) and Profit (£) Objective Function: Maximize 10x1 + 15x2 + 22x3 + 17x4 Even though our objective function is based on the decision variables,due to the time constraint variable relate the function we have to include them also in the objective function when we solve it by MATLAB or Excel Solver. Therefore, the Objective function becomes, 10x1 + 15x2 + 22x3 + 17x4+ 0Tba + 0 Tca Subject to the following constraints, Process Constraints: Time Constraint, Production Ratio Constraint and Maximum produce are the three constraints that we are going to deal. 1)Time Constraints: 2x1 + 2x2 + 3x3 + 4x4 <= 160 + Tba + Tca 2x1 + 2x2 + 4x3 + 2x4 <= 200 – Tba x1 + 3x2 + 3x3 +x4 <= 80 – Tca 2)Production Ratio Constraint : -x1 + 0.9x4 <= 0 x1 - 1.15x4 <= 0 3)Maximum to produce next week: x1=50 x2=60 x3=85 x4=70 Solution: MATLAB Toolbox Matrix form Objective function f = [10 15 22 17 0 0] Inequality constrain a = [■(2&2&3&4&-1&-1@2&2&4&2&1&0@1&3&3&1&0&1@-1&0&0&0.9&0&0@1&0&0&-1.15&0&0)] b = [160 200 80 0 0] No equality constrains aequ = [ ] bequ = [ ] Lower Boundary L = [0 0 0 0 0 0] Upper Boundary U = [50 60 85 70 40 24] MATLAB CODE: Using the following data in MATLAB optimization tool box, EXCEL SOLVER: SENSITIVITY ANALYSIS: RESULTS: Since we solved the problem in linear program, we have ignored that the values of x1,x2,x3,x4 to be an integer. Maximum products to be produced next week to obtain a maximum profit of 957.94 pounds are X1 = Product 1 = 27.692. X2 = Product 2 = 0. X3 = Product 3 = 7.179. X4 = Product 4 = 30.769. The results obtained are within the maximum value of the products that are permitted to be produced for the next week. The production ratio constraints of x1/x4=0.89999 which is close to the constraints said value. Tba= 40 hours. Tca = 0 hour. The time constraint is also within the limit of process B workers 20% and process C workers 30%.