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Abstract: The operation research tools vary significantly. In this paper, several operation research tools that can handle uncertainty are investigated. They include sensitivity analysis and parametric analysis, mean-variance analysis, stochastic linear programming, fuzzy linear programming, and value at risk (VaR). In addition, these tools are compare and contrast based on their applicability, time and technical requirements.

Keywords: Risk management, sensitivity analysis, parametric analysis, mean-variance analysis, stochastic linear programming, fuzzy linear programming, value at risk (VaR.)

I. INTRODUCTION

The operation research tools have reached a stage of maturity and sophistication. They vary significantly, with some handling nonlinear models while some not, and with some handling uncertainty in the models while some not. [1] presents several operation research tools, including the linear programming, nonlinear programming, mixed integer programming, and stochastic programming to optimize the financial models. However, except for the stochastic programming, none is capable of handling uncertainty in the financial models. Uncertainty is the condition in which the possibility of error exists, because we have less than the total information about our environment. These conditions occur all the time in all industries. Risk arises when the models are bounded to uncertainty. To measure and control the risks, operation research tools capable of handling uncertainty are needed.

This research is initially intended to explore different operation research tools to handle the uncertainty in serving customer demand in the power industry [2]. The operation research tools investigated are intended for linear constraints and linear objective model. They include sensitivity analysis and parametric analysis, mean-variance analysis, stochastic linear programming, fuzzy linear programming, and value at risk. However, as these tools can be used as well in the financial industry, we wish to share our perspectives on these operation research tools.

This paper first describes the linear programming models in section III. Then, several operation research tools are explored in section IV through section VIII. Finally, the different tools are commented on their applicability, technical requirement, and time requirement.

II. NOMENCLATURE

A: The constraint matrix.

: The right-hand-side vector.

: The cost vector.

: Right-hand-side vector in the recourse function that is uncertain (stochastic linear programming.)

: Transition matrix corresponding to x variables that is uncertain (stochastic linear programming.)

:Objective vector corresponding to y variables that is uncertain (stochastic linear programming.)

: Technology matrix corresponding to y variables that is uncertain (stochastic linear programming.)

x: Deterministic decision variable.

y: Uncertain decision variable.

z: Aspiration level (fuzzy linear programming.)

III. LINEAR PROGRAMMING

The canonical form of a classical maximization linear programming problem can be stated as in (1).

subject to:

(1)

is the vector of cost coefficients. Since (1) maximizes the objective, represents the profit per unit of x. is the constraint matrix. is the right-hand-side vector, representing the minimal requirements to be satisfied. All coefficients of , , and are known deterministically, the inequality sign, “”, is not to be violated, and that the objective is a strict imperative [3]. Linear programming is the most commonly used approach nowadays.

IV. SENSITIVITY ANALYSIS AND PARAMETRIC ANALYSIS

Sensitivity analysis and parametric analysis are the two most readily available approaches to assess and manage the risk if the solution methodology used to optimize the model is the linear programming. They are post-optimality analysis available from the optimal solution reached by the linear programming model.

Sensitivity analysis examines the effect of relaxing some of the constraints on the value of the optimal objective without having to resolve the problem. The analysis [3, 4] includes change in the cost vector, , change in the right-hand-side vector, b, change in the constraint matrix, A, addition of a new constraint, and addition of a new variable. Sensitivity analysis is used only when there are not many changes to be analyzed. Otherwise, sensitivity analysis is no different than solving a new linear programming problem. Furthermore, to conduct sensitivity analysis, the changes must be added one by one. For example, to analyze the effect on the changes in two cost coefficients, the technique requires first analyzing the effect of the first coefficients. Then, the risk manager may choose to analyze the effect of the second coefficient based on the original optimal solution or the optimal solution reached after including the effect of the first coefficient.

Parametric analysis analyzes how the optimal solution changes as several parameters change simultaneously over some range. The analysis includes simultaneous changes in the cost vector, c, or the right-hand-side vector, b. Parametric analysis is not suitable when the risk manager intends to change the constraint matrix, A, either by changing the parametric values, or by adding new constraints and activities.

V. MEAN-VARIANCE ANALYSIS

The mean-variance analysis assumes that risk management is based on the expected profit and the associated variance that represents the risk of decision. There are several ways to model the approach [1, 5, 6]. Equation (2) shows one of them.

subject to:

(2)

E and V are the expected value and covariance matrix of the objective function, . They are shown in (3) and (4) respectively.

(3)

(4)

Equation (2) is a quadratic linear programming problem. When V is a convex function (,) (6) can be solved using the modified simplex method to search for the optimal solution for the given value. The modified simplex method is described in [4]. On the other hand, when V is not a convex function (,) the solution reached using the modified simplex method may not be globally optimal. There are various techniques that have been proposed to solve the quadratic programming models. Notably, Hazell developed the Minimization of the Total Absolute Deviation (MOTAD) technique for the quadratic programming model. He uses the variance estimates based on the sample Mean Absolute Deviation (MAD.) Even though the sample MAD is a less efficient estimator of the population variance than the sample variance, MAD is sufficient in ranking the alternative activities, which is the initial intention of the quadratic programming model. Thus, the technique is found to be as good as solving the quadratic programming model [6].

VI. STOCHASTIC LINEAR PROGRAMMING

The stochastic linear programming approach has been researched since 1950s [7]. The approach borrows its technique from statistics. A stochastic linear programming model is shown in (5).

subject to:

(5)

The matrix, A, and vectors, and b, are known with certainty. The matrices, transition matrix and technology matrix , objective vector and right-hand-side vector , are uncertain parameters. To solve (5), the Bender Decomposition technique and the Monte Carlo simulation are employed. Equation (5) is solved using a two-stage linear program with recourse. The approach is an iterative procedure that alternatively solving the master problem and the sub-problems. The master problem solves the deterministic variables and Benders cuts while the sub-problems solve the uncertain variables that are realized as deterministic variables through Monte Carlo simulation [7].

VII. FUZZY LINEAR PROGRAMMING

In describing the fuzzy linear programming, the concept of satisfaction of criteria is important. The criteria can either be the constraints or the objectives. In general, the objective is to find that would satisfy the set of criteria or equations in (6).

(6)

The value of represents the aspiration level of the objective function. The goal of fuzzy linear programming is to determine a solution that would satisfy the criteria mostly when uncertainty or fuzziness exists in (6).

There are two major streams in solving (6) using the fuzzy logic extension theory. The first [8] represents the objective function and the constraints using fuzzy sets that will be aggregated to derive at a maximizing decision. Equivalently, the inequality signs, , are replaced by the fuzzified version, . [8] presents several ways to solve the problem. The second [9] describes the coefficients (A, b, z, and ) as fuzzy functions and solves the fuzzy functions using fuzzy logic extension theory. The first work has an advantage where the formulated problem can be solved using linear programming model. However, as the technique stresses on fuzziness in the inequality signs, the individual effect of the coefficients (A, b, z, and ) may not be properly accounted for. Since the second work describes the coefficients as fuzzy functions, the individual effects of the coefficients are taken into account during the decision making. However, the resulted formulation is nonlinear in the constraint matrix. To solve the problem, an iterative procedure is needed to search for the optimal solution [9].

VIII. VALUE AT RISK

Value at risk (VaR) is a risk assessment tools that has been used by the financial institutions to evaluate the risk of holding a portfolio of assets. The VaR approach differs from all previously discussed approaches. The four approaches presented in section IV through section VII deciding the set of actions to be taken (x and y,) while including the uncertain factors (A, b, , etc.) in the decision making process. VaR, however, assumes that certain actions have been decided (x and y are determined.) The issue is to determine the monetary risk of taking such actions. Figure 2 shows the graphical representation of VaR.

There are in general three techniques that can be used to evaluate VaR. The first technique is the historical simulation. Historical simulation applies the historical data to evaluate the VaR. The second technique is the covariance technique. To apply the covariance technique, the correlation matrix of the uncertain factors is assumed available. The third technique is the Monte Carlo simulation. Monte Carlo simulation involves artificially generating a very large set of events, correlated changes, from which VaR is derived [10].

The VaR analysis differs from one situation to another. For instance, the VaR in serving customer demand in the power industry is different than holding the portfolio of assets in the financial industry. The difference is further explored in [11].

Figure 2. Graphical representation of VaR.

IX. REMARKS

Section IV through section VIII present the various approaches that can be used to manage and assess the uncertainty. Each of these approaches has its own strengths and weaknesses. In this section, the various approaches are discussed in the following aspects: (1) applicability, (2) technical requirement, (3) time requirement.

Applicability refers to how easy the uncertainty in the models may be addressed by the various tools. The mean-variance, sensitivity analysis, and parametric analysis allow studies on the correlated changes in either the cost coefficient, c, or the right-hand-side vector, b, only. Even though the sensitivity analysis evaluates the perturbation in c, A, and b, it evaluates the perturbation one at a time. It would be easier to resolve the linear programming problem if too many perturbations are needed in the sensitivity analysis. The stochastic linear programming approach studies the uncertainties in c, A, and b simultaneously. Part of the model (A , c and b, in section VI) can be deterministic while the rest stochastic (, , , and ). A distinct advantage of the stochastic linear programming is that some of the parameters in a constraining equation can be stochastic while the rest remains deterministic. The fuzzy linear programming approach utilizes a broad range of techniques. The approach, in general, studies the changes in c, A, and b simultaneously. However, to use the fuzzy linear programming, either all parameters within a constraining equation be considered fuzzy (i.e., being perturbed) or none at all. The VaR approach, to my belief, ranks the best of all in applicability. The reason is that all risk factors that are hard to be included in the formulation can be included in the VaR. By distinguishing the VaR to three separate components, the approach allows the decision makers to evaluate the financial risk independently.

Technical requirement refers to the degree of knowledge needed in analyzing the uncertainty using the different tools. Mean-variance, sensitivity analysis and parametric analysis, and fuzzy linear programming approaches are based on the linear programming technique. Understanding the concept of linear programming is sufficient to model the financial operation using these tools. Fuzzy linear programming approach requires additional technicality as the approach borrows its concept from the fuzzy set extension theory. The tool, nevertheless, is similar to the sensitivity analysis and parametric analysis. The only exception is the ability of the fuzzy linear programming approach in explaining the why and the how perturbations may affect the risk of the financial operation. To use the stochastic linear programming approach requires knowledge in the Bender decomposition and the Monte Carlo simulation techniques. As a result, the approach is more math-oriented than the rest. Fortunately, there are software packages available, reducing the requirements to understand the techniques at great length. Understanding the Monte Carlo simulation technique is sufficient in the VaR approach. However, as the steps involved in the VaR are dependent on the operational conditions, it is the hardest among all. For instance, Best describes the VaR [10] from a financial analyst’s perspective, however, the approach he described is not sufficient to handle the operational condition in serving customer demand in the power industry [11]. In addition, to evaluate the VaR, a decision choice must be chosen. This means that some other approaches must be used in addition to the VaR approach. For instance, to evaluate the VaR of holding a portfolio of stocks, some other techniques must be used to determine the number and amount of stocks that the portfolio manager should consider [5].

Time requirement refers to the time needed to analyze using the different approaches. The mean-variance, sensitivity analysis and parametric analysis, and fuzzy linear programming approaches are based on the linear programming technique. Sensitivity analysis and parametric analysis is the easiest approach since solving the linear programming model has already provided information needed to analyze using the approach. Mean-variance and fuzzy linear programming approaches require remodeling the models presented in section III. However, since the linear programming technique can still be used to solve the model derived from the two approaches, the time requirement is considered moderate. The stochastic linear programming and VaR approaches require the Monte Carlo simulation to obtain the random changes. The time requirements for the two approaches are considerably higher. Since the stochastic linear programming approach requires solving the model iteratively, time requirement for the approach is the highest

X. REFERENCES

[1] S. A. Zenios, Financial Optimization, New York: Cambridge University Press, 1993.

[2] K. H. Ng, and G. B. Sheblé, “Risk Management Tools Risk for an ESCO Operation,” submitted to the 6^th International Conference on PMAPS, March 2000.

[3] M. S. Bazaraa, J. J. Jarvis, and H. D. Sherali, Linear Programming and Network Flows, New York: John Wiley & Sons, Inc., 1990.

[4] F. S. Hillier, G. J. Lieberman, Introduction to Operations Research, New Yrok: McGraw-Hill, Inc., 1995

[5] G. L. Thompson and S. Thore, Computational Economics: Economic Modeling with Optimization Software, San Francisco, California: The Scientific Press, 1992.

[6] P. B. R. Hazell, and R. D. Norton, Mathematical Programming for Economic Analysis in Agriculture, New York: Macmillan Publishing Company, 1986.

[7] G. Infanger, Planning under Uncertainty: Solving Large-Scale Stochastic Linear Programs, Danvers, Massachusetts: Boyd & Fraser Publishing Company, 1994.

[8] H. J. Zimmermann, Fuzzy Set Theory – and Its Applications, Boston: Kluwer-Nijhoff Publishing, 1985.

[9] H. Tanaka and K. Asai, “Fuzzy Linear Programming Problems with Fuzzy Numbers,” Fuzzy Sets and Systems, 1984.

[10] P. Best, Implementing Value at Risk, Chichester: New York, 1998.

[11] K. H. Ng, and G. B. Sheblé, “Value at Risk for an ESCO Operation,” submitted to the 6^th International Conference on PMAPS, March 2000.

XI. BIOGRAPHIES

Kah-Hoe Ng received his BSEE with distinction (May, 1995) and MSEE (May, 1997) from Iowa State University (ISU). He is currently pursuing Ph.D. in Electrical Engineering and MS in Economics at ISU. His research interests include power system economics and optimization.

Gerald B. Sheblé (M 71, SM 85) is a Professor of Electrical Engineering, ISU, Ames, Iowa. Dr. Sheblé received his B.S. and M.S. degrees in Electrical Engineering from Purdue University and his Ph.D. in Electrical Engineering from Virginia Tech. His industrial experience includes over fifteen years with a public utility, a research and development firm, a computer vendor, and a consulting firm. His research interests include power system economics and optimization, especially scheduling and control.