Production Plan for Brass Limited - Essay Example

1. Executive Summary Brass Ltd. requires an optimal production plan for the two types of products it manufactures. This report aims to find an optimal solution for the same keeping in mind the constraints of available capacity and government regulations. In order to do this, a linear programming model has been developed. The model has been solved using a manual graphical approach to find the optimal values of quantities of two products and the maximum profit. A sensitivity analysis has been done to help Brass Ltd. take decision on increasing available capacity. Finally, a few recommendations are provided to help Brass Ltd. manage its operations well. 2. Problem Brass Ltd. manufactures two types of products – Masso and Russo which incur machining and assembly costs in their production. There are constraints on the availability of machining and assembly hours which are 700 hours and 1000 hours respectively. The capacity is fixed and the cost on the same is incurred irrespective of the usage. The maximum allowed production of each product type is 400 units. The selling price and costs incurred for the two products are available. The time spent on each machine for each product is also available. The company needs to know the optimal production of each product type to maximize profit. It also needs to gauge the impact of increasing production marginally on the profitability. 3. Model Formulation A linear programming model can be formulated to find a solution to the given problem. All the constraints here are maximization constraints. The model can hence be developed as follows: Constraints: M + 2R < 700 2.5M + 2R < 1000 M < 400 R < 400 M > 0 R > 0 Objective Function: Z = 40M + 50 R = Maximum Where, M = Number of Masso R = Number of Russo 4. Key Assumptions There are certain inherent assumptions involved while formulating the above Linear Programming Model. The first assumption is that the constraints and the objective function can be represented through linear equations. In other words, the constraints on machining hours, assembly hours and the maximum production are directly proportional to both the number of Masso and the number of Russo. The second assumption is that the production of Masso is independent of the production of Russo and hence the impact of their production on the constraints as well as the objective function is perfectly additive. The third assumption is that the immediate objective of the firm is to recover the variable costs on the manufacturing of the two products and therefore fixed costs of machining and assembly are not considered for model formulation. Ideally, these fixed costs are distributed over the number of products manufactured. 5. Optimal Solution For obtaining a solution to the above model, the 6 inequalities shown above are plotted on a graph as shown in Figure 4.1 (www.maths.unp.ac.za). The two constraints regarding machining and assembly hours are plotted as straight lines. The two constraints on the number of each product type are given by two lines parallel to the axes while the non-negative constraints are given by the two axes themselves (Bazaraa et. al., 2009). The area contained within all the lines shows the feasible region. The 4 end-points of this region are given as (0,0), (0,400),(200,250) and (350,0). Figure 4.1: Optimal Solution through Graphical Approach It is found that the objective function increase in the direction of the end-point (200,250). Hence, this is the point that gives the optimal solution in the feasible range (Taha, 2009). The optimal solution can therefore be obtained as: Number of Masso = 200 Number of Russo = 250 The value of objective function or profit for optimal solution is given by: Z (optimal) = 200*40 + 250*50 = $20500 6. Sensitivity Analysis Sensitivity Analysis can be performed in order to gauge the impact of independent marginal increase in the available capacity for machining and assembly. The increase in one department is done keeping the other constant. The output maxima are also kept constant. The increase in profit or the objective function noticed as a result of this marginal increase in a constraint is also called as the shadow price of that constraint. Shadow price can be manually calculated as the instantaneous increase in the profit with a unit increase in either constraint (Dantzig & Thapa, 2003). Using a similar approach as above, if the number of machining hours available are increased from 700 hours to 701 hours, keeping the assembly hours constant as 1000 hours, the number of Masso and Russo can be calculated. These values come out as 199.333 and 250.833 respectively. The objective function in this case is received as: Z = 40*199.333 + 50 * 250.833 =$ 20515 Hence, change in objective function = shadow price = $(20515 – 20500) = $ 15 Similarly, if the number of assembly hours available are increased from 1000 hours to 1001 hours, keeping the machining hours constant as 700 hours, the number of Masso and Russo can be calculated. These come out as 200.667 and 249.667 respectively. The objective function in this case is received as: Z = 40*200.667 + 50 * 249.667 =$ 20510 Hence, change in objective function = shadow price = $(20510 – 20500) = $ 10 On the basis of the sensitivity analysis, it can be commented that the marginal independent increase in machining hours increase profit by $15 while the marginal independent increase in assembly hours increases the profit by $10. A similar analysis can be done alternatively by noticing the decrease in profit or objective function if the constraints are marginally decreased (Vaserstein, 2003). 7. Recommendations On the basis of above analysis, the company is recommended to manufacture products at the optimal level with the given capacity in order to yield maximum profits. Thus, Brass Ltd. should aim to produce 200 units of Masso and 250 units of Russo per month. At the optimal level, a profit of $20500 is realized. If profit maximization is the only objective of the firm, this is the best it can do with given constraints. However, if the company has other objectives as well such as revenue growth, capacity expansion and international expansion, it may decide to produce more at lower margins. This especially holds true given the fact that the demand for the products is highly unfulfilled. It can be easily seen that the constraints of maximum units to be produced by the government (400 units each) are yet to be exhausted. Therefore, the major barrier to produce more units is the available capacity of machining and assembly hours. The company can ponder over increasing this capacity to increase both profits and revenues in the long term. This would involve huge capital expenditure and other government clearances. Also, it is clear from the sensitivity analysis that increasing machining hours should be given a greater priority over increasing assembly hours since the shadow price of the former is greater. Also, since the selling prices are fixed by the government, there is not much scope to take steps on the pricing side unless some negotiations can be made with the government. However, steps may be taken to reduce variable costs by adopting Total Quality management practices. 8. References Bazaraa, M. S., Jarvis, J. J. and Sherali, H. D. 2009, Linear Programming and Network Flows, Wiley Dantzig, G. B. and Thapa, M. N. 2003, Linear Programming 2: Theory and Extensions, Springer Taha, Hamdy A. 2009. Operations research: An Introduction. Pearson Education, Eighth Edition “Solving Linear Programming problems using the graphical method”, Available online April 11, 2012 from Vaserstein, L. N. 2003, Introduction to Linear Programming, Prentice Hall Read More