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Abstract
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Introduction
Branch and Bound Method
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Computer Science
Artificial Intelligence

Operational Research Approach to Decision Making

Profile image of oleg pokrovskyoleg pokrovsky
https://doi.org/10.1007/978-1-4020-9253-4_12
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24 pages

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Abstract

The decision making (DM) problem is of great practical value in many areas of human activities. Most widely used DM methods are based on probabilistic approaches. The well-known Bayesian theorem for a conditional probability density function (PDF) is a background for such techniques. It is needed due to some uncertainty in many parameters entered in any model which describes the functioning of many real systems or objects. Uncertainty in our knowledge might be expressed in an alternative form. We offer to employ appropriate confidence intervals for model parameters instead of a relevant PDF. Thus one can formulate a prior uncertainty in model parameters by means of a set of linear constraints. The related cost or goal function should be defined at a corresponding set of parameters. That leads us to stating the problem in terms of operational research or mathematical linear programming. It is more convenient to formulate such optimization problems for discreet or Boolean variables. A review of relevant problem statements and numerical techniques are presented as well as many examples.

FAQs

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AI

What explains the sequential steps in the decision-making process?add

The paper identifies five sequential steps in decision-making: problem identification, information gathering, solution generation, solution evaluation, and strategy selection, emphasizing the goal-directed nature of decisions.

How does Bayesian inference impact hypothesis evaluation?add

Bayesian inference updates the probability of a hypothesis based on new evidence, enabling discrimination between conflicting hypotheses; however, initial beliefs may bias the results.

What distinguishes info-gap theory from traditional decision-making models?add

Info-gap theory focuses on local robustness under severe uncertainty instead of probabilistic assessments, contrasting sharply with Wald's maximin paradigm that employs global worst-case analysis.

What are the practical applications of linear programming in decision-making?add

Linear programming serves as a decision support tool, optimizing managerial decisions by rationalizing production and resource allocation, exemplified by maximizing profits in a production scenario.

What obstacles does Integer Programming face in yielding optimal solutions?add

Integer Programming may yield suboptimal solutions due to rounding of Linear Programming outputs and difficulty in formulating constraints for discrete decisions.

Figures (8)
Fig. 1 Half-planes: the feasible region of a linear inequality.
Fig. 1 Half-planes: the feasible region of a linear inequality.
Fig. 3. Graphical solution of the example LP (6). Operational Research Approach to Decision Making
Fig. 3. Graphical solution of the example LP (6). Operational Research Approach to Decision Making
There are two fail options related to LP problem solution. First is absence of any solution, when the feasible region is empty. Consider again the original example (6), modified by the additional requirements (imposed by the company’s marketing department) that the daily production of product X; must be at least 30 units, and that of product Xz should exceed 20 units. These requirements introduce two new constraints into the problem formulation, i.e., X; > 30,X2 > 20 . Attempting to plot the feasible region for this new problem, we get Figure 4, which indicates that there are no points in the (X1, X2)-plane that satisfy all constraints, and therefore our problem is infeasible (over-constrained).
There are two fail options related to LP problem solution. First is absence of any solution, when the feasible region is empty. Consider again the original example (6), modified by the additional requirements (imposed by the company’s marketing department) that the daily production of product X; must be at least 30 units, and that of product Xz should exceed 20 units. These requirements introduce two new constraints into the problem formulation, i.e., X; > 30,X2 > 20 . Attempting to plot the feasible region for this new problem, we get Figure 4, which indicates that there are no points in the (X1, X2)-plane that satisfy all constraints, and therefore our problem is infeasible (over-constrained).
fore, this is an example of a (two-var) LP whose objective function can take arbi- trarily large values. Such an LP is characterized as unbounded. Notice, however, that even though an unbounded feasible region is a necessary condition for an LP to be unbounded, it is not sufficient; to convince yourself, try to graphically iden- tify the optimal solution for the above LP in the case that the objective function is changed to: maxi2dX, Sx J/<- —V>. Operational Research Approach to Decision Making
fore, this is an example of a (two-var) LP whose objective function can take arbi- trarily large values. Such an LP is characterized as unbounded. Notice, however, that even though an unbounded feasible region is a necessary condition for an LP to be unbounded, it is not sufficient; to convince yourself, try to graphically iden- tify the optimal solution for the above LP in the case that the objective function is changed to: maxi2dX, Sx J/<- —V>. Operational Research Approach to Decision Making
Fig. 8 Third branching. Our best integer solution now has objective value 21. The subproblem that gen- erates that is not active due to integrality of the solution. The other subproblem generated is not active due to infeasibility. There is still a subproblem that is active. It is the subproblem with solution value 21.65. By our “round-down”’ result, there is no better solution for this subproblem than 21. But we already have a solution with
Fig. 8 Third branching. Our best integer solution now has objective value 21. The subproblem that gen- erates that is not active due to integrality of the solution. The other subproblem generated is not active due to infeasibility. There is still a subproblem that is active. It is the subproblem with solution value 21.65. By our “round-down”’ result, there is no better solution for this subproblem than 21. But we already have a solution with
where e,, is an m—dimensional column vector of “1”s. The set covering problem (P2) is (P1) with less than or equal to constraints replaced by greater than or equal to constraints and the objective is to minimize rather than maximize. The set par- titioning problem (P3) is (P1) with the constraints written as equalities. The set partitioning problem can be converted to a set packing problem or a set covering problem (see [9]) using standard transformations. If the right hand side vector e,, is replaced by a non-negative integer vector b, (P1) is referred to as the generalized The three problems (P1), (P2) and (P3) can be formulated as integer linear pro- grams as follows: Let A denotes the m*n matrix where
where e,, is an m—dimensional column vector of “1”s. The set covering problem (P2) is (P1) with less than or equal to constraints replaced by greater than or equal to constraints and the objective is to minimize rather than maximize. The set par- titioning problem (P3) is (P1) with the constraints written as equalities. The set partitioning problem can be converted to a set packing problem or a set covering problem (see [9]) using standard transformations. If the right hand side vector e,, is replaced by a non-negative integer vector b, (P1) is referred to as the generalized The three problems (P1), (P2) and (P3) can be formulated as integer linear pro- grams as follows: Let A denotes the m*n matrix where
Fig. 9 Ensemble Prediction System forecast relative values (usefullness) responded to multi-user and multi-event case [15]. Let us consider multi-user decision making for many meteorological events. We used the European Center for Medium Range Weather Forecasting (ECMWF) Ensemble Prediction System (EPS) forecast for the T850 (air temperature field at the standard level of 850 mb) anomaly, Europe, Jan—Feb, 1998 (Figure 9) (see details in [15]) with n = 3 (number of users), m = 4 (number of meteorological events). The matrix of EPS forecast relative economic values are presented in Table 1, the minimal efficiency for each user in Table 2. In the case of equal importance of users we came to the optimal solution xo,; for (14) constrained by (15). This solution
Fig. 9 Ensemble Prediction System forecast relative values (usefullness) responded to multi-user and multi-event case [15]. Let us consider multi-user decision making for many meteorological events. We used the European Center for Medium Range Weather Forecasting (ECMWF) Ensemble Prediction System (EPS) forecast for the T850 (air temperature field at the standard level of 850 mb) anomaly, Europe, Jan—Feb, 1998 (Figure 9) (see details in [15]) with n = 3 (number of users), m = 4 (number of meteorological events). The matrix of EPS forecast relative economic values are presented in Table 1, the minimal efficiency for each user in Table 2. In the case of equal importance of users we came to the optimal solution xo,; for (14) constrained by (15). This solution

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