#212

Dear Iraq T. Abbas, Manar Naji Ghayyib

I read your paper

Using Sensitivity Analysis in Linear Programming with Practical Physical Applications

My comments:

1- Abstract You say “It is also the simplest and easiest type of model that can be created to address industrial, commercial, military and other dilemmas”

You are absolutely right. LP is the easiest way in modelling MCDM problems. There no assumptions, no weights, no pairwise comparisons. The user just writes the initial decision matrix with the data, including the targets for each criterion, and assigns the corresponding mathematical symbols ‘≥’, ‘≤’ or ‘=’. That is all, and the DM can follow at will the computer processing step-by-step or asking for a final result.

I very gladly noticed that that you include the shadow prices concepts that are also furnished by the LP methods automatically. However, sensitivity analysis (SA) does not use the shadow prices but that targets values. It can, but in that case, it determines how much increases an objective function due to a unit increase in a certain criterion, while SA finds out how much a basic criterion can increase or decrease when I unit of the target is added or subtracted.

2- “Any change in the values of the model constants or what is known as the inputs of the model will change the problem of linear programming and will affect the optimal solution”

Sorry, this statement is incorrect; variations of criterion values may or may not change the best solution; it depends on how much a basic criterion can vary. A basic criterion is which affects the solution found, and as that, responsible for its selection, while the other criteria are irrelevant.

3- You start mentioning many aspects that most probably are not familiar to readers like shadow prices, targets, right hand side, the Simplex algorithm etc. In my opinion, you should start by explaining the meaning of each of these terms, otherwise, the reader may be in the dark.

You can first define in a few words what LP is, that solves two problems at the same time, the primal, which determines the optimal choice, and its dual, that delivers the importance of the basic criteria, or its marginal value or marginal utility.

These are the shadow prices, and that indicated how much increases an objective by each unit increase of a basic criterion

You should say that LP uses inequations not equations, and then work with spaces. Each criterion has two parts, the left one called LHS or Left-Hand Side, and the right one, the Right-Hand side (RHS). Both sides are separated by any of the three symbols mentioned. The RHS is the target of each inequation. The LHS gives the data, while the RHS established the restrictions or targets, which normally are resources, to which the data must respect.

4- Page 3 “Whether it seeks to maximize or reduce a particular function which is related to the idea of the binary model that supports the original model”

In MCDM a binary value means that it is the one or two values, i.e. 0 or 1, not both. It appears that you interpret that it refers to the two LP results, which in reality can be considered as the two sides of a coin. Binary values can form criteria rows, as well as complete matrices, and in reality, are very useful. Binary criteria may be intercalated in the initial matrix, to nominal performance values without any difficulty, although the corresponding targets may have different meanings. You can say for instance that 1 + 1 = 1, when only one alternative must be selected out of two. As for instance, you cannot select two restaurants at the same time, when you only need one to dine, it is restaurant xxxx or yyyy

In the case that both alternatives can be considered as the same time, then 1 + 1 = 2. For instance, a railway track can be built at the same time than a parallel road, because the rail is normally used for grain transportation, oil, steel products, cars etc., while the road is built to buses, cars and trucks, and both linking the same two areas A and B.

5- “Since the primary objective of the original linear model was to maximize marginal profits, this marginal profit is the return on variable costs or the difference between the selling price”

The primary objective of LP is to optimize a function, not marginal profits, which means optimizing the use of resources or targets. Putting it simple. If you do not have enough resources to produce something, you cannot reach the goal you had. As an example, if a manufacturer needs to improve sales and profits, he must have enough raw material, low costs, enough workers, etc. to manufacture and improving sales, and LP is the unique tool that can balance all resources to produce something. If the manufacturer does not have enough fun ds, – a resource - the problem is unfeasible, and LP tells you so.

6- “The binary model is built from the standard formula for the inequalities of the linear programming model, as it is shown in the next table”

This is not called ‘binary model’, it is simple the primal and the dual aspects of the same problem, and both deliver the same result.

7- “The standard approach to resolving the linear programming model is the shadow prices and opportunity cost, which are defined as the lost profits for the best alternative that comes after the chosen alternative or as an expected theoretical value for the alternatives abandoned as a result of choosing a specific alternative, these are two examples of information that the simplex method provides which the graphic method does not”

I disagree. You do not need the shadow prices to solve a LP problem, and they are not the same as opportunity costs. Normally, solving the primal is more than enough. In solving the primal you certainly use the cot of opportunity principle, that selects the alternative that enters in the solution. The graphic method also provides this, since you only have one tangent vertex with the Z or objective line.

8- “The internal opportunity cost is the rate of return the institution that can get in exchange for helping to pay its fixed costs and earn a profit”

This is the first time that I read this definition. The IRR is simple the point of the Net Present Value line is zero

9- An external opportunity cost is that cost that should be added like externalities, for instanced to the GDP, the externality produced by oil extraction, because a country is losing a part of its natural capital.

10- Page 7 “The accepted solution does not change when a new variable is added to the model since it becomes a non-essential variable and its value is zero at the lower bounds”

This is incorrect. The accepted solution may or may not change if a new variable is added, and this is not related to SA. In SA you are increasing a criterion value, n ot adding a new alternative. I do not understand why you are mixing SA with Rank Reversal. RR may occur due to the change of the object to a new one want you add or delete an alternative., and of course, you get a new decision matrix. That is, by adding a new alternative you are changing the area and the form of the space tghat contains all feasible solutions

11- Why do you introduce he the fuzzy logic? It has no relation to SA

These are my comments. I hope they can be useful

Nolberto Munier