Manufacturing Plants Across the Country

Question: Case studies are used to enable you to apply new concepts, use the tools you have mastered, and improve your technical skills you have attained. Through the individual case studies you will discover for yourself the usefulness of quantitative problem solving methods, how to apply them in practice, and their benefit to organizational decision-makers. In this case study, you will act as a consultant for a manufacturing company looking to maximize net profit generated by a production facility subject to a number of production constraints. You will develop a linear programming model and is solve it using Excels Solver tool. Further, you will interpret the generated Answer and Sensitivity Reports to develop recommendations for optimal product mix and future profitability of the company. Both a written report and an Excel spreadsheet model are required to be submitted. Answer: Introduction ABCD Ltd. is a sports equipment manufacturing company. The company owns and operates number of manufacturing plants across the country. The company has a plant which manufactures basketball and football. The company has a limit on the maximum and minimum number of basketball and football it can manufacture. Also the production capacity of the plant is limited due to the maximum and minimum number of machine hours available. The company wants to optimize the production of basketball and football such that the profit generated by the company is maximized. The purpose of the report is to contain the detailed analysis of the operational activity of the company for this plant and the number of footballs and basketballs the company needs to produce to maximize profits. The report also contains information regarding the constraints which has limited the profits of the company and how and by what amount the company can increase the profits by changing the limiting resources. Description of the Problem The problem faced by the company is profit maximization by the sales of basketball and football. The profit of the company is dependent on the number of basketball and football the company can produce. The manufacturing of each basketball and football requires certain number of machine hours that is to be consumed. The company is subjected to capacity constraints due to the minimum and maximum machine hours available. Also the company has a demand constraint and hence has to produce between the range of minimum and maximum demand. Thus even if the company can earn more profits, it wont be able to produce number of units because of the above constraints. Methodology The objective of the company is to maximize profit. Let the number of basketballs to be produced be X and the number of footballs to be produced be Y. Given, The time taken to manufacture a Basketball = 0.5 hours The time taken to manufacture a Football = 0.3 hours Thus the total machine hours used = 0.5* Y + 0.3* X The cost of material for a football = $1.25 The cost of material for a basketball = $2.00 Thus the Total material cost = 1.25* Y + 2*X The cost of labor per machine hour used = $6 Total labor cost = 6* (0.5* Y + 0.3* X) Thus total cost of the company = (1.25* Y + 2*X) + 6* (0.5* Y + 0.3* X) The selling price of a football = $11 The selling price of a basketball = $14 Total revenue = 14*X + 11* Y Thus, Profit = Total revenue Total cost Profit after taxes = Profit taxes (= 28% of the profit) The following are the constraints of the company The number of basketball that can be produced Min 30000 and Max 60000 The number of football that can be produced Min 20000 and Max 40000 Total machine hours available Min 39000 and Max 40000 Thus the following parameters were used in Excel Solver and solved for the number of basketballs and the number of footballs and the results were generated. Results Thus using the Excel Solver, the following results were generated The number of basketballs to be produced = 56000 units and The number of footballs to be produced = 40000 units The total machine hours used = 40000 hours Thus the total revenue earned = $1224000, and the total cost = $402000 Profit = 1224000 402000 = 822000 Thus Profit after taxes = $230160 From the Answer report it can be seen that the machine hour and the number of footballs to be produced are the binding constraints i.e. the maximum profit can be increased if the company can increase the maximum demand of the football or it can increase the machine hours available.(Bertsimas, 1997) The maximum demand of the basketball has no effect on the maximum profit produced as it is not binding and the slack is 4000 i.e. even if the demand of the basketball decreases by 4000 units, there will be no change in the maximum profit obtained from the analysis. From the Sensitivity report it can be seen that the objective coefficient of the number of basketball and football is 7.95 and 9 respectively i.e. the profit earned by selling one unit of basketball is 7.95 and one unit of football is 9. The reduced cost is the amount by which the co efficients of the objective function can be changed without affecting the output. In this case, the reduced cost is 0 for both the basketball and football. Thus the profit of the company will change with the change in the co efficients of the objective function. The shadow price is the amount by the optimal solution will increase or decrease with unit change in the limiting constraint i.e. the profit of the company will increase by the amount in the shadow price if the limiting constraint is increased or decreased by a unit. (Bertsimas, 1997). In this case, the shadow price of the machine hour is 18. Thus if the machine hours available is increased by 1 unit, the profit will increase by $18. The shadow price of the number of football is 2.55. Thus if the maximum demand of football is increased by 1 unit, the profit will increase by 2.55 units. The allowable increase and decrease provides the range till which the calculated shadow price is valid i.e. if the constraint limit changes by more than the allowable limits then the shadow price will change. In this case, the allowable increase and decrease for the machine hours available is 2000 and 1000. Thus, the shadow price will remain same if the machine hour is between 39000 and 42000 hours. Similarly, the allowable increase and decrease for the number of football produced is 43333.33 and 6666.67 units. Thus, the shadow price will remain same if the number of football produced is between 33333.33 units and 83333.33 units. Conclusion Thus the number of basketballs and footballs to be produced by the company is 56000 and 40000 units respectively. The limiting constraints for the company are the maximum demand of football and the total machine hours available. Also it was found that if the company can increase the total machine hours available by 1 unit the profits will increase by $18 and if the company can increase the maximum demand of football by 1 unit the profits will increase by $2.55. Recommendations The following recommendations will help the company ABCD Ltd in smooth functioning of the plant As seen from the analysis, the profit of the company can be greatly improved if the machine hours are improved. The company should find methods to improve the efficiency of the labors so that the utilization of the machine is high and time consumed to produce each unit of basketball and football be reduced and thus the profits can be improved. For each hour saved, the company will be able to generate $18 profit. The company should also find methods to increase the maximum demand of football with the help of advertisement or finding new markets it can serve so that the profitability of the company can be increased. For each unit increase in demand, the company will be able to generate $2.55 profit. The company should use the depreciation of the machine used to help in reducing the taxable income and hence increase the profit of the company. The cost of the material purchased by the company affects the profitability of the company. The company should find ways to reduce the cost of the material to reduce the material cost and increase profits. This can be done by bulk purchasing and find new vendors who can provide the same quality at a lower price. References Bertsimas,D. (1997). Introduction to linear optimization. Publication: Athena Scientific Dynamic Ideas. Massachusetts. Sensitivity Analysis. (n.d.). Retrieved on August 6, 2016 from https://www.excel-easy.com/examples/sensitivity-analysis.html