Abstract
This paper analyzes the case of TeflaGong, a company producing two types of electrical products: ACs (air conditioners) and L.F (large fans). The board wants a hired consultant to determine the maximization and minimization points regarding the company’s production as well as profits. The case study is presented and an analysis provided along with the related linear programming solution. For reasons of technicalities beyond the scope of this paper, the linear programming variables have taken the non-negative value; that is, values ate equal to zero or greater than zero. This is because, in several cases where, for instance the variables may indicate the quantities of a selection of activities or quantities of certain resources use, the need for non-negativity is reasonable – and even necessary ( Schrijver, 1998) .
Introduction
Linear Programming is one of the most powerful systems for describing and findings solutions for optimization issues. It enables an individual to specify a given set of solution variables, as well as a linear goal and a set of constraints on the identified or listed variables ( Schrijver, 1998) . To offer a widely used and simple linear programming instance, consider the issue of reducing the cost of a choice of medications that satisfies all the allowed daily intake guidelines. The linear programming model would be characterized by a set of solution variables capturing the quantity of each pharmaceutical drug intake, a linear goal or objective minimizing the overall cost of buying the chosen drug, as a linear constraint for every drug, requiring that the selected medicine together contains an adequate amount of that wellness or treatment ingredient.
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The Case
TeflaGong Corporation produces two electrical merchandises: large fans and air conditioners. The process of assembly for each product is similar in that they both need a certain amount of drilling and wiring. Each conditioner requires 2hrs of drilling and 3 wiring-hours. Each fan has to undergo 1hr of drilling and 2 wiring hours. During the next period of production, about 140 drilling time may be experienced and 240 wiring time are available. Each sold air conditioner produces $25 profit. Each assembled fan can be sold for a profit of $15. The management board of TeflaGong Corporation has hired you as a consultant to formulate as well as solve the above Linear Programming production mix case to identify the best combination of fans and air conditioners that will produce the maximum or highest profits. The board asks for a comprehensive analysis of linear programming models, including the application of corner point graphical technique to understand the maximization and minimization points regarding profits as well as production.
TeflaGong’s management realizes that it failed to include two vital constraints. Specifically, administration decides that there needs to be a minimum quantity of air conditioners manufactured so as to satisfy a contract. Further, as a result of an oversupply of fans in the past business year, a boundary needs to be placed on the overall quantity of manufactured fans. The consultant considers the following suggestions from the concerns placed by the board: a) If TeflaGong takes to producing at least 20 air conditioners, what would be the optimal/maximum result? How much surplus or slack is available for each of the four variables? b) If TeflaGong takes to producing at least 30 air conditioners but fans should not exceed a maximum of 50 units, what would be the optimal result? How much surplus or slack is available for each of the 4 variables at the optimal result?
Analysis and Solution
In TeflaGong Corporation, the merchandise produced include large fans and air conditioners. A summary of the manufacturing needs are as follows: a) hours required at each phase/unit of product; b) overall available time per week; and c) profits from each merchandise sold. From the information given in the case, a table can be constructed as follows:
| Machine Type | L.F (large fan) | A.C (air conditioner) | Available time per week (hours) |
| Wiring Machine – X 1 | 2 hrs | 3 hrs | 240 |
| Drilling Machine – X 2 | 1 hr | 2 hrs | 140 |
| Profits per unit sales | $15 | $25 |
The blend of these two merchandises can be achieved through the following step. However, prior to proceeding to identifying the best combination, it is noteworthy that the machines listed in the table above are given expressive values of X 1 and X 2 for simpler problem solving as well as for the best of choice variables. In a linear program, variables refer to a collection of amounts that needs to be addressed in order to find solutions to the problem; that is, the issue is solved only when the best value concerning the variables are identified (Forest Resource Management, 2000). Sometimes, variables are referred to as decision variables because the issue to identify the solution as to what value or amount each variable needs to take. Primarily, variables determine the quantity of a resource to put into production or the degree of a certain activity (Forest Resource Management, 2000). For instance, a variable might indicate the amount of acres to cut from a specific section of the forest in a given period. Often, explaining the variables of an issue is one of the most crucial and hardest phases in formulating linear programming problem. The following procedures explains the ways of finding the best maximum solution to the case. To maximize the profits, the applicable objective function includes; Maximize = 25X 1 +15X 2
In Linear Programming, the Objective Function, or the objective of the problem is to minimize or maximize a particular numerical value (Forest Resource Management, 2000). The value can be expected property value, net present value, or even a project cost. Also, it could refer to the quantity of produced wood or the anticipated quantity of a park’s visitor days. The objective function explains how each variable adds to the optimization of the value in finding solution to the issue.
Other constraints determining the objective functions available for profit maximization are presented as follows:
3X 1 +2X2≤240….(1) [indicated by blue line in graph]
2X 1 +X 2 ≤140……(2) [indicated by green line]
X 1 ≥ 0…………….(3) [x-axis]
X 2 ≥ 0…………….(4) [y-axis]
Through the application of Equation 1 and Equation 2, it is possible to ascertain the maximum objective value of 1900 with the vertex indicated as (40,60). Application of equation 1 and equation 3 gets the objective value of 1800 having 0,120 as the vertex. Also, applying equation 2 and 4 produces objective value of 1750 having 70,0 as the vertex. Finally, combining equation 3 and 4 produces an outcome of the inner region for a graph with vertex 0,0.
A gridline variation of 10 is used for all equations. Therefore, the drilling machine should have a maximum of 60 and the wiring machine having 40 so as to produce an optimal profit of 1900. The graph is provided below:
The following section deals with the second problem expressed in the case study concerning TeflaGong Corporation. Using the calculations above, it is possible to identify the non-negative values expressed as X 1 and X 2 .
The maximum solution for fitting the overall number of ACs to be manufactured to the 20-level minimum is as follows. This part also involves determining the quantity of surplus or slack.
X 1 ≥ 20…..(5) [the graph is presented below]
The maximum solution is to manufacture more air conditioners exceeding 20 and not any quantities below. Therefore, the feasible area in the graph is greatly lessened, thereby, reducing the additional corner point as well as the surplus.
Reduction of the corner points affects the satisfaction of corner solution, a special case of an variable’s maximizing issue in which the amount of one of the constraints in the optimizing function is zero (Forest Resource Management, 2000). Often, a corner solution come along with the case where the “best” choice (that’s, maximizing utility or profit) is attained with the basis on brute-force limit conditions, but not on the related quantities’ market-efficient maximization. Such decisions lacks mathematical approval, and several instances are characterized by externally compelled situations (for example, “variable X,Y cannot be negative”), placing the actual local extrema away from the allowed values.
Slack or Surplus values are reported for every constraint. Slack applies to equal or ‘less than’ constraints, while Surplus is applied to equal or “greater than” constraints (Forest Resource Management, 2000). Supposing that a constraint is constraint is compelling or binding, then the equivalent surplus or slack value equals zero. In the occasion when an equal-or-less-than constraint is not compelling or binding , then there is a likely occurrence of certain extra or slack resource. The value of slack refers to the quantity of the variable, as described by the equal-or-less-than constraint, that is unused. In the instance where an equal-or-greater-than constraint is not forceful, then the extra/surplus denotes the surplus quantity over the variable that is being utilized or manufactured.
On a different not, if the company takes to manufacturing air conditioners in a range of a maximum of 50 large fans and a minimum of 30 air conditioners, the maximum result is subjected to limited feasibility region through the impact of limiting variables of 30 ≥ X1≤50. This results to a surplus of 20 units of the ACs.
Conclusion
Linear Programming refers to a mathematical framework for determining optimal decisions to issues that can be illustrated through linear inequalities and equalities. Supposing that a real-world case can be relayed accurately through mathematical equations available in linear program, the technique will determine the best decision to the issue. Indeed, limited sophisticated real-world issues can be relayed accurately with regards to a list of linear functions. Nonetheless, linear programming models can offer reasonably actual representations of several real-world issues – particularly if an extra creativity is applied to the mathematical rendition of the issue. The case analysis finds that in order to maximize profits to 1900, the wiring machine needs a maximum of 40 and the drilling machine will require a maximum of 60.
References
Forest Resource Management. (2000). Chapter II: Basic linear programming concepts. Lecture Presentations, University of Washington.
Schrijver, A. (1998). Theory of linear and integer programming . John Wiley & Sons.
