Contribution
The contribution is the difference between sales value and the marginal cost of sales.
The contribution is very important in marginal costing. The term contribution is frequently used to refer to activities aimed at covering fixed overheads and making a profit. In the case below, the contribution per unit for each computer peripherals is computed as below.
Let the computer peripherals be x1,x2,x3,x4,x5, and x6 where
The internal modem is x1, External modem is x2, Circuit board is x3, CD Drive is x4, Hard disk drive is x5, and Memory board is x6. Computation of variable labor costs for each of the computer peripherals is done by multiplying the time required for each device by the cost per unit for the test device as follows. First, we convert the cost per hour to the cost per unit by dividing each variable cost per unit by 60minutes.
- Internal modem, x1
- Variable cost = 2×25/60 + 1×15/60 + 5×18/60 + 3×12/60 = $3.18
- External modem, x2
- Variable cost = 3×25/60 + 5×15/60 + 2×18/60 + 4×12/60 = $3.90
- Circuit board, x3
- Variable cost = 5×25/60 + 3×15/60 + 3×18/60 + 2×12/60 = $4.13
- CD drive, x4
- Variable cost = 6×25/60 + 2×15/60 + 3×18/60 + 1×12/60 = $4.10
- Hard disk drive, x5
- Variable cost = 4×25/60 + 5×15/60 + 3×18/60 + 2×12/60 = $4.22
- Memory board, x6
- Variable cost = 8×25/60 + 5×15/60 + 2×18/60 + 3×12/60 = $5.78
Linear Programming
Linear programming is a technique of decision-making used by managers to allocate limited resources, such as machinery, raw materials, and labor to minimize costs or maximize production. During the formulation of the linear programming function, the following steps are very important, one being identifying variables ( product x and product y). Another step is identifying the objective, which is to maximize contribution or to minimize cost and writing down its mathematical presentation in terms of variables.
Furthermore, identifying the constraints that are, the limited resources shared among the variables, and writing down its mathematical representation in terms of variables is another step. Writing down the objectives and the constraints in terms of the variables are the last step. These steps apply to the computation of variables regardless of the number of variables. It is important to note that if only two variables are involved, a graphical solution can be used otherwise for multivariable problems, a simple method is applied to find the solution.
Formulating a linear programming model
Identifying variables
The variables here are the number of units of computer peripherals produced by Northern Hi-Tech E Ltd per day. We can represent them as:
- X1= a unit of Internal modem.
- X2= a unit of External modem.
- X3= a unit of Circuit board.
- X4= a unit of CD drive.
- X5= a unit of Hard disk drive.
- X6= a unit of Memory board.
Identify the objective
An objective is the desired result that is, optimization of a function that is dependent on the decision variable and is subject to some constraints. The objective of Northern Hi-Tech E Ltd is to maximize its contribution. The objective function is the formula that will give us the total contribution in a day for an internal modem, External modem, Circuit board, CD drive, Hard disk drive, and Memory board.
The information above can thus be represented in a tabular form as:
Objective function = 156.82X1 + 156.10X2+250.87X3 + 167.90X4+290.78X5 +274.22X6
The objective is to maximize156.82X1+156.10X2+250.87X3+167.90X4+290.78X5 +274.22X6
Identifying constraints (constraints formulation)
Therefore production must be such that the number of machine-hours required is less than or equal to the available machine hours per week as follows.
- For Test device 1 is 2x1+3x2+5x3+6x4+4x5+8x6 ≤ 150
- Test device 2 is x1+5x2+3x3+2x4+5x5+5x6 ≤ 130
- Test device 3 is 5x1+2x2+3x3+3x4+3x5+3x6 ≤ 110
- Test device 4 is 3x1+4x2+2x3+x4+2x5+3x6 ≤ 102
Non-Negativity: It is logical to assume that the company cannot manufacture negative amounts of a product. Thus, it can only manufacture either zero products or more. Therefore, we have:
X1, X2, X3, X4, X5, X6≥0
Thus the complete linear programming model is;
Maximize 156.82X1+156.10X2+250.87X3+167.90X4+290.78X5 +274.22X6
Subject to the constraints;
- 2x1+3x2+5x3+6x4+4x5+8x6 ≤ 150.
- x1+5x2+3x3+2x4+5x5+5x6 ≤ 130.
- 5x1+2x2+3x3+3x4+3x5+3x6 ≤ 110.
- 3x1+4x2+2x3+x4+2x5+3x6 ≤ 102.
- X1, X2, X3, X4, X5, X6≥0.
Solving linear programming problems
The question requires us to optimize (in our case, maximize) the objective (the contribution function), or in simple terms, we are required to solve the linear programming model. Solving the linear programming model entails finding the values of variables that satisfy all inequalities simultaneously and optimize the objective. To convert this problem to a system of linear equation, we introduce slack variables to each constraint.
- Z = 156.82X1 + 156.10X2+250.87X3 + 167.90X4+290.78X5 +274.22X6
Subject to 2x1+3x2+5x3+6x4+4x5+8x6 +S1 = 150.
X1+5x2+3x3+2x4+5x5+5x6 +S2 = 130.
5x1+2x2+3x3+3x4+3x5+3x6 + S3 = 110.
3x1+4x2+2x3+x4+2x5+3x6 +S4 = 102.
Where the structural variables S1, S2, S3, and S4 are slack variables. We then place this information in a tabular form known as a tableau
Initial tableau
Computer-generated values
It is optimum to manufacture 2 internal modems, 16 circuit boards, and 15 hard disk drives. We observe that all the time except when test device 4 is binding. This slack has 29.60 minutes. The reduction of the profit margin of the internal modem reduces the total contribution by $28.00 per unit. Additional time per week will not help the company much since it has already reached its optimal solution with the already available resources.
Decision variables are the amounts of each product to be made in a given time. Linear programming assumes that the variable has a linear relationship. Linear programming is used in several departments, including the production department to whereby it is used to decide on the number of pots to be produced, which is subject to limited resources (constraints) such as labor, power, machine hours, and raw materials.
The marketing department would use it to allocate sales representatives to different sales regions, which is also subject to their expected performance. Finally, the human resource department uses it to schedule the personnel’s work hours and job description to either maximize production or minimize cost. In this case, we consider the manufacturing of Internal modem, External modem, Circuit board, CD Drive is, Hard disk drive is, and Memory board.
Constraints are circumstances that govern the achievement of an objective (Kline 21). Limitations must be quantified mathematically and they must be linear. For Northern Hi-Tech E Ltd, we have limited machine-hours, which must be shared among the production of the internal modem, External modem, Circuit board, CD drive, Hard disk drive, and Memory board. In this case, the optimum solution is where the company manufactures 2 internal modems, 16 circuit boards, and 15 hard disk drives. This is the optimum solution, which maximizes profits at $9,081. However, additional time will lead to increased contribution.
Analysis of the computer-generated figures or values shows that the company is maximizing its profits by producing 2 internal modems, 16 circuit boards, and 15 hard disk drives and these are the decision variables that the objective function is based on. This is the best combination the company can achieve with the available resources. This combination maximizes profit for the company at $9,089.53. The time available for test device 1, test device 2, and test device 3 is in abundance and has not been fully exploited. This means that increasing their resources do not help the company much. The time constraint available in test device 4 is limited.
This means that by adding a single unit of this constraint, the company would be improving its position in terms of profit maximization since it is binding to the optimal solution. The slack value in this solution is 29.60. This means that to solve part of the problem, the company should utilize this available resource but this cannot be possible without affecting the others that have already reached the optimum stage.
The variable labor costs have a one on one impact on the contribution per unit that the company makes. If the company decides to increase the variable labor cost per unit then this is going to have a downward effect on the contribution per unit. An increase in the variable labor cost would mean that the company is producing more computer peripherals, which can cover these unit costs. It, therefore, follows that the company is at optimum profit-making by manufacturing 2 internal modems, 16 circuit boards, and 15 hard disk drives and there cannot be any other better combination with the already available limited resources (Sadler 87).
Works Cited
Desvaux, Martin. “A Synopsis Of Clive Ponting‘s: A Green History Of The World.” Optimum Population Trust Journal, 5.2 (2010): 45-67. Print.
Freeman, Edward. Strategic Management: A Stakeholder Approach. New York: Cambridge University Press, 2010. Print.
Kline, John. Ethics for International Business: Decision-Making in a Global Political Economy. New York: Routledge, 2010. Print.
Sadler, Philip. Strategic Management. London: Kogan Page Publishers, 2003. Print.