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Fuel management problem as a linear program
The fuel management problem can be solved using the linear programming model. The issues of consideration in this exercise are consumption, weight and fuel cost. Linear programming is one of management science techniques that can be used to solve a problem. Generally, a management science model allows a number of steps to be carried out in finding the best solution to a problem.
These steps include the following:
- Formulating the problem where its definition is derived, the objectives are specified and the area of study located.
- Observation of the system where parameters affecting the problem are determined. Alternatively, data representing the estimate values are collected at this stage.
- Formulating a mathematically representative model of the problem under consideration.
- This representation is then verified and the resulting model is used to make predictions.
- Constituting a number of alternative to include in the model from which the best solution can be chosen.
The models used are either descriptive or prescriptive. Prescriptive models seek to optimize finding the best values of given decision variables (Balakrishnan, Render and Stair 2006). The linear programming approach is used to solve optimization problems
Typically, a linear programming model includes problems that seek to maximize or minimize some quantity usually cost or profit. This is the objective function of the prescriptive model. For this fuel management assignment, the objective is to minimize the cost of fuel consumed during the flight trips between Birmingham and Athens.
The linear programming model also has a number of restrictions or constraints which provide boundaries on the objective function. For this problem the two most important constraints relate to the fuel weights at take off and landing. These determine how much should be added at each of the stops and this can be calculated in terms of the cost of fuel at each of the stops based on the rates charged at each of the stops.
Based on various outcomes from the linear programming model, different courses of action can be recommended and implemented. The objective function and constraints in a linear programming model can then be expressed in terms of linear equations as a final stage. Basing on the fuel management problem, the decision variables are based on the maximum takeoff weight (TW), and maximum landing weight (LW).
- α = fuel consumption per trip
- β = fuel consumption per take off
- TW= Maximum takeoff weight
- LW= Maximum landing weight
- Rj = safety reserve
- Cj = cost of fuel per lb
- Maximum fuel capacity = 46300
The objective function:
(α + (β X TW)) * Cj >= ((46300 – (α + (β X TW)) + Rj) * Cj
Opening up the brackets to rearrange the equation we have the following:
- α Cj + β Cj TW >= 46300 Cj – α Cj + β Cj TW + Rj Cj
- 2 α Cj – Rj Cj >= 46300 Cj
Dividing through by Cj
We have
Cj (2 α – Rj)/ Cj >= 46300 Cj / Cj
The Cj cancels out giving
2 α – Rj >= 46300
Cj, α, β and Rj are variables.
Subject to the following constraints
- TW =< 154000
- LW =< 129000 TW, LW > 0 (non negative constraint)
- α, β >0 (non negative constraint)
The optimal solution
Based on the results by the Solver the most optimal cost of fuel per lb is 0.136. This is a necessary condition to satisfy all the conditions optimally in the fuel management problem. It was assumed for this exercise that the objective is to calculate the optimal cost of fuel and hence determine where among the three fuel stops the plane can optimally refuel. Based on this a decision can be made on where to maximally refuel when the plane does its round trip from Birmingham through Amsterdam to Athens and back. The decision is necessary in order to readjust the maximum take off weight and consumptions at each of these stopovers.
How the solution changes when the fuel cost at Schiphol CAMS varies between £0.050 per lb and £0.300 per lb
The objective function is
2 α – Rj >= 46300
Let Cj = cost of fuel at Schiphol. We can factor the cost of fuel Cj into the objective function above by multiplying this variable to both sides of the equation.
Therefore, 2 α Cj – Rj Cj >= 46300 Cj
The table here below is calculated based on the variation of fuel cost at Schiphol. The variation is from 0.03 to 0.05.
We have increased the standard cost of fuel of 0.160 at Schiphol by 0.03, 0.04 and 0.05 respectively as shown in the table below. The results however indicate negligible change in α which is the fuel consumed during the trip to Amsterdam (Schiphol).
The advantages and limitations of this specific model
Management science assists in decision making by designing and operating a system that has scarce resources. Descriptive models which are part of the tools used in management science are used to focus on behaviour of a system which may involve a number of factors. Prescriptive models on the other hand assign behaviour to an organization that will enable it meet its goals. Linear programming as a management science model has several advantages to its credit. However, there are various limitations. Some of these typical limitations are highlighted here below:
- Not all problems encountered in real life are linear and as such linear programming may not be practically suited for all problems.
- Linear programming also assumes that the variables are real valued, however is some cases these fractional values may not be suitable to use. This implies that it may not be practical to treat integer problems using linear programming while rounding off the values as integers which may result in an infeasible solution
- Linear programming also assumes that the data being used is perfectly accurate which may not be the case practically.
- The decisions in a linear programming environment are made under uncertainty and hence it may not be suitable to apply to all real life scenarios
- Redefining or formulating real world scenario into a set of linear equations is usually a very difficult step to undertake.
The following are the advantages of using linear programming as a modeling tool in management science.
- Linear programming offers a simplified approach to deriving prescriptive model for certain problems
- Linear programming supports the use of software to generate suitable solutions.
- Linear programming supports a large variety of simplex based algorithms which can be used to solve different problems
- Linear programming also supports the algorithms such as Khachian and Kamankar.
Reference List
Balakrishnan, N., Render, B. and Stair, R., 2006. Managerial decision modelling with spread sheets. 2nd ed. New Jersey: Prentice Hall.
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