Statistics and Operation Research

Question: Describe about the report of optimum production unit of the company for both machine is estimated and discussed to produce that machine much, which will provide more profit and operating cost will be least? Answer: Executive summary In this report of optimum production unit of the company for both machine is estimated and discussed to produce that machine much, which will provide more profit and operating cost will be least. Further, the report shows the profit margin of the company may be enhanced due to usage of linear programming in industrial application as it provides the accurate solution for the operational problem and further, it will be able to use it in different conditions of physical changes of the activities. Introduction In this report of production of machines for Body fitness 100 and 200, emphasize is given on the profit margin of the machine and operational hour of the machine. The numerical programming method is used for solving the constraints present in production with maintaining the profitability of the products. Further, it is also reported that two machines are to be produced in such a manner that Body Plus 200 must be of 25% of total production of the two machines. The cost and profit associated with the machines are estimated in the report and further, report consists of optimum profit margin for the company at optimum production level. Discussion Better Fitness is a company that is engaged in producing and selling of fitness machines in Freeport, Long Island. It is observed that in a trade show, the two new designed machines, Body Plus 100 and 200 have been appraised by the customers and distributors. Company got many orders exceeding the total capacity of the plant in the current period. For making the fitness machines, company needs raw materials costing total 1606 for body plus 100 and 2292 for body plus 200. The selling price of the two machines may not be more than the 2062.5 and 2812.5 respectively for Body Plus 100 and 200 respectively. Further, it is also estimated that company may not get time for the three activities equally as the management has decided to produce machine of Body plus 200 more than 25% of total production. The distributors of the company may not be able to buy the machines more than 25% of the marked price of two machines except in some different conditions. Therefore, companys management has d ecided to give more emphasize on Body Plus 200, although, there is no estimation from their side. Activities for Body Plus 100 There are four stages for making one Body Plus 100 frame unit, press station and pec-dec station. These four are the basic unit to be assembled for making the machine where the main activities are machining and welding, painting and packaging and testing and assembling. For ease of reporting purpose, the activities are termed as A1, A2 and A3 as said in the above order. For A1, A2 and A3, company needs total of 14 hours to complete one Body Plus 100. It is also observed that cost for each stage is different due to different requirements of raw materials. The labour charge per hour is also different for every activity as estimated by the management. Stages Raw material cost (in ) Activities Total cost Frame unit 550 Machining and welding Press station 400 Painting Pre-dec station 350 Testing and assembling Packaging 25 1375 Table 1: stages and raw material cost per unit (Source: created by author) Activities Frame unit Press station Pre-dec station Total time Total labour cost Machining and welding 4 1 1 6 120 Painting 2 1 2 5 75 Testing and assembling 3 36 14 231 Table 2: activity cost per unit of Body Plus 100 (Source: created by author) It is observed that to manufacture 1 Body plus 100, company needs to work 14 hour and it is also noted that it needs 1375 for raw material and labour charges of 231of making one machine. Activities for Body Plus 200 For Body Plus 200, the stages were increased one more from the previous machine there is leg-press station for making the machine. The details of the stages are as follows: Stages Raw material cost (in ) Activities Total cost Frame unit 750 Machining and welding Press station 500 Painting Pre-dec station 350 Testing and assembling Leg-press station 300 Packaging 25 1975 Table 3: stages and raw material cost per unit (Source: created by author) Activities Frame unit Press station Pre-dec station Leg-press station Total time Total labour cost Machining and welding 4 2 2 3 11 220 Painting 3 3 2 1 9 135 Testing and assembling 1 12 21 367 Table 4: activity cost per unit of Body Plus 200 (Source: created by author) Total cost for raw material for making 1 Body Plus 200 is 1975 and total labour charges associated with it is 367 as estimated in the table above. The labour cost is given in per hour as estimated and that is specific for machining and welding activity as 20 per hour and for painting 15 per hour. For testing and packaging, the labour cost is 12 per hour. The management has faced the problem of making decision on production units to be produced for the two machines separately. The volume must be in a number so that production may provide optimum position of the profitability and cost function of the company. The marked price for the two is considered as 2750 and 3750 for Body Plus 100 and 200 respectively. Distributors can afford to pay only 75% of the marked price of the machines. Considering the cost of machine production is C1 and C2, the constraints related to labour hour on every activity is estimated on basis of cost function of per unit. The constraints for the management are as follows: C1 = 0 C2 = 0 C1 + C2 = 0 C2 = (C1 + C2) / 4 C1 = 3 (C1 + C2) / 4 12000 = C1 + C2 = 340 6750 = C1 + C2 = 210 1680 = C1 + C2 = 48 The above equations provide the management to get the key constraints for solving the problems regarding the required units for producing within the period so that the cost of production becomes optimum. In the mean time, company needs to maintain the decision of producing the Body Plus 200 at least 25 % of total production. For solving this problem, the help of linear programming is considered as it is a branch of mathematics; specifically of operation research, which helps to evaluate the optimum solution of cost of production so that the optimum numbers of producing units may be estimated (Yang, 2008). According to Luenberger and Ye (2010), linear programming is a method of solving the linear equations where the problems arise due to obtain the best value of the constraints. For solving this problem the traditional method is to use the graphical method of calculation and another method is linear programming that helps to estimating in every activity of operation at an optimum cost (Bazaraa et al. 2010). The graphical method cannot solve more than three variables at a time while linear programming may solve problems with unknown numbers of variables. As opined by Derhy (2010), it is a process of optimization used in resource allocation efficiently. For using the LP in this model, the constraints are changed slightly in terms of cost of production activities for getting the results of unit to be produced in total and separately. Further, it is observed that cost of unit production activities are as follows BP 100 BP 200 total machining 120 220 340 paint 75 135 210 testing 36 12 48 raw material cost 1375 1925 3300 Table 5: per unit cost for both production (Source: created by author) Resources available for the production are of 20430 for labour cost for minimum production of one unit of each for Body Plus 100 and 200. Total cost for available resources are - 12000 for machining, 6750 for painting and 1680 for testing. Raw material cost may not be produced, as there is no information of maximum production capacity of the machines. Constraint function: 340 = resource available for machining = 12000 210 = painting = 6750 48 = testing = 1680 The analysis from the Appendix, shows that optimum production may be for the period will be 15354508083 and BP 100 must be produced of 13421773. The other machine must be produced of 53687091. The value provided here shows that company may produce BP 100 much more than BP 200. Recommendations It is recommended to the management of the company to produce the Body Plus 100 more than the Body Plus 200. However, the manager has decided to make a market of Body Plus 200 in case, for making a brand value of the product in future. However, it is observed that company has better profit in Body Plus 100 than the 200. Therefore, it is optimum to produce the BP 200 just 25% of total production. Further, it is also recommended that company must consider to sell body Plus 200 at higher price as it would provide much profit for the company. Conclusion From the above estimation, it is shown that company must evaluate the problems related to optimum operation using the linear programming, as this technique is good for getting better production capacity within the limited capacity and the constraints of production. Further, it is also advised that company may produce Body Plus 100 more than Body Plus 200 as that will be the optimum solution for the company and further, producing 25% Body Plus 200 of total production will provide company a chance to make its brand popular in future. Reference lists Bazaraa, M., Jarvis, J. and Sherali, H. (2010). Linear programming and network flows. Hoboken, N.J.: John Wiley Sons. Derhy, M. (2010). Linear programming, sensitivity analysis and related topics. New York: Prentice Hall. Luenberger, D. and Ye, Y. (2010). Linear and nonlinear programming. New York: Springer. Yang, X. (2008). Introduction to mathematical optimization. Cambridge, UK: Cambridge International Science Pub. Appendix BP 100 BP 200 unit profit 1144 1408 resource used resource available mchinig 120 220 340 0 12000 paint 75 135 210 0 6750 testing 36 12 48 0 1680 raw material cost 1375 1925 3300 BP 100 BP 200 total production production unit 13421773 53687091 15354508083