Measurability of evaluative criteria used by optimization methods

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1 Proceedings of the th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November -3, Measurability of evaluative criteria used by optimization methods JAN PANUS, SANISLAVA SIMONOVA Institute of System Engineering and Informatics Faculty of Economics and Administration University of Pardubice Studentska 95, Pardubice CZECH REPUBLIC Abstract: - he fundamental requirements are to optimize certain suggested solution in directive and decision-making activities. Used methods are derived on mathematical basis and their typical implementation is searching of optimal traffic connection between the two junctions. It means we look for the shortest distance among starting and final points. Evaluative indicator can be another criterion than distance, however. If weighted standpoint is price of realization of certain projects or number of workers that are necessary for proposed business then it is possible to utilize optimization methods for supporting of decision-making operation in common manager profession. Key-Words: - Simulated Annealing Method, Optimization Global Optimization, Measurable Criteria Introduction Optimization methods evaluate optimal variation of solution on their defined basis. Reviewing criterion is able to be measurable but also hardly measurable or nonmeasurable sometimes. Measurable criterions could be financial costs, time, number of workers etc. Hardly measurable values could be social influences, ethical standpoint etc. Sometimes it is necessary to make provision for scalable or non-measurable standpoints for findings of optimal solution. hese standpoints could be e.g. numbers of interested workers or reputation ( good name of company ). All indexes are necessary to assign to unified form of worth representation for using of optimization methods. In general, it is not an easy task to formulate mathematical models that are suitable for such complex problems. If the model is found then it may be described with the use of an algebraic modeling language and hence, solved by numerical methods implemented by solvers. Having the optimal solution, it may be interesting to change model parameters and focus on the sensitivity analysis results. his may be realized with current modeling languages and their theoretical background without significant problems. Efficient techniques for restarting the solution process are known for a broad class of optimization models. 2 Problem Formulation Measurable criteria can be valuable expressed like results of measurement or calculation. Difficult measurable or non-measurable criteria are possible to express by expert qualified estimation. Criterion that is expressing of manager opinion about credibility of company is not possible to numerically quantify. But it is good to determine unit process of assessment by transferring into non-dimensional numbers for using in worth formulation of criteria. It means, we choose unit process of worth expression for measurable and nonmeasurable criteria in some interval (e.g. from zero to number one). Every criterion is able to gather two extreme values criterion is totally replete: criterion gather the value of number one, it is the most positive classification, criterion is totally unfulfilled: criterion gather the value of number zero, it is the most negative classification, however it is unpractical to work with the value of zero therefore the value of criteria at this point is set to the number that is close to zero but it is greater than zero. of criterion middle Fig. Assessment of Criterion criterion measurable non-measurable

2 Proceedings of the th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November -3, It is evident that criterion is able to gather many values when it is replete just in part way. Fig. shows transmission of measurable and non-measurable criteria into unit process of formulation. Measurable criterion (e.g. number of concerned workers on the project) is expressed exactly defined values (, 25, 5, 75 and see fig. ). Criteria that are hardly measurable (credibility of company) record uncertain expressed valuables (since low credibility that is expressed by symbol as far as high credibility that is expressed by symbol + + +). assessment of criterion,75,5, middle criterion measurable non-measur. Fig. 2: Assessment of criterion non-dimensional number he criterion is assign to non-dimensional number in interval of zero to one. It is possible to start from all scale of interval zero to one or it is possible to take in the way of expression on predefine number of particular values within the interval of zero to number one see fig. 2. One of the possible ways how to set the interval is to set the criteria into five possible values. here are two boundary values (zero and one) and three values among this boundary values (,25,,5,,75). his way is better to use for hardly measurable criteria whereas the way keeps necessary diversification of resultant valuation. Adjustment of the range of valuation schedule of criterion is optional of course and it depends on what size of diversification every manager has to make provision for utilization of optimization method. 2. Optimization methods he goal of an optimization problem can be formulated as follows: find the combination of parameters (independent variables) which optimize a given quantity, possibly subject to some restrictions on the allowed parameter ranges. he quantity to be optimized (maximized or minimized) is termed the objective function; the parameters which may be changed in the quest for the optimum are called control or decision variables; the restrictions on allowed parameter values are known as constraints. A general constrained nonlinear programming problem (NLP) takes the following form minimize f(x) subject to h(x) = x = (x,..., xn) () g(x) where f(x) is an objective function that we want to minimize. h(x) = [h (x),...., h m (x)] is a set of m equality constraints, and g(x) = [g (x),..., g k (x)] is a set of k inequality constraints. All f(x), h(x), and g(x) are either linear or nonlinear, convex or nonconvex, continuous or discontinuous, analytic (i.e., in closedform) or procedural (i.e., evaluated by some procedure or simulation). Variable space X is composed of all possible combinations of variables x i, i =, 2,...., n. In contrast to many existing NLP theory and methods, our formulation has no requirements on convexity, differentiability, and continuity of the objective and constraint functions. With respect to minimization problem (), we make the following assumptions. Objective function f(x) is lower bounded, but constraints h(x) and g(x) can be either bounded or unbounded. All variables x i (i =, 2,...., n) are bounded. All functions f(x), h(x), and g(x) can be either linear or nonlinear, convex or nonconvex, continuous or discontinuous, differentiable or non-differentiable. Active research in the past four decades has produced a variety of methods for solving general constrained nonlinear programming problems. hey fall into one of two general formulations, direct solution or transformation-based. he former aims to directly solve constrained NLP () by searching its feasible regions, while the latter first transforms () into another form before solving it. ransformation-based formulations can be further divided into penalty-based and Lagrangian-based. For each formulation, strategies that can be applied are classified as local search, global search, and global optimization. 2.. Local search Local search methods use local information, such as gradients and Hessian matrices, to generate iterative points and attempt to locate constrained local minima (CLM) quickly. Local search methods may not guarantee to find CLM, and their solution quality is heavily dependent on starting points. hese CLM are constrained global minima (CGM) Nocedal and Wright (3) only if () is convex, namely, the objective function f(x) is convex, every inequality constraint g i (x) is convex, and every equality constraint h i (x) is linear.

3 Proceedings of the th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November -3, Global search Global search methods employ local search methods to find CLM and, as they get stuck at local minima, utilize some mechanisms, such as multistart, to escape from these local minima. Hence, one can seek as many local minima as possible and pick the best one as the result. hese mechanisms can be either deterministic or probabilistic and do not guarantee to find CGM Global optimization Global optimization methods are methods that are able to find CGM of constrained NLPs. hey can either hit a CGM during their search or converge to a CGM when they stop. In this paper, we survey one of the existing methods for solving each class of discrete, continuous, and mixed-integer constrained NLPs. It is simulated annealing (SA), that was developed for solving unconstrained NLPs. SA searches in variable space X and it (SA) does probalistic descents in variable x space with acceptance probalistic governed by a temperature. 3 he problem of Monitored Criterion In general, it is not an easy task to formulate mathematical models that are suitable for such complex problems using in regional desicions. Optimalization methods mostly deal with one basic criterion; it means one characteristic is changing for finding the best solution. But the solution of regional questions often means wide complex of input reflected characteristic. It is dificult question how to take into account all input regional characteristic and create one non-dimensional number. his number is our criterion. Regional data sources involve huge number of data, it means there is no problem in quantity of data. But a big problem is in the area of strategic question Simonova and Capek (6); it means it is difficult to find optimal solution for particular regional strategic processes. More mathematical methods make possible to search optimum in problem solution. One of them is method of Simulated Annealing that finds global optimum. 3. Method of Simulated Annealing Annealing is the metallurgical process of heating up a solid and then cooling slowly until it crystallizes. he atoms of this material have high energies at very high temperatures. his gives the atoms a great deal of freedom in their ability to restructure themselves. As the temperature is reduced the energy of these atoms decreases. If this cooling process is carried out too quickly many irregularities and defects will be seen in the crystal structure. he process of too rapid of cooling is known as rapid quenching. Ideally the temperature should be deceased at a slower rate. A slower fall to the lower energy rates will allow a more consistent crystal structure to form. his more stable crystal form will allow the metal to be much more durable. Simulated annealing seeks to emulate this process. Simulated annealing begins at a very high temperature where the input values are allowed to assume a great range of random values. As the training progresses the temperature is allowed to fall. his restricts the degree to which the inputs are allowed to vary. his often leads the simulated annealing algorithm to a better solution, just as a metal achieves a better crystal structure through the actual annealing process. We briefly overview SA and its theory Berthsekas () for solving discrete unconstrained NLPs or combinatorial optimization problems. A general unconstrained NLP is defined as minimize i f(i) for i S. procedure SA 2. set starting point i = i ; 3. set starting temperature = and cooling rate < α < ; 4. set N (number of trials per temperature); 5. while stopping condition is not satisfied do 6. for k to N do 7. generate trial point i` from S i using q(i; i`); 8. accept i` with probability A (i; i`) 9. end for. reduce temperature by α x ;. end while 2. end procedure Fig. 3: Simulated annealing (SA) algorithm. where f(i) is an objective function to be minimized, and S is the solution space denoting the finite set of all possible solutions. A solution i opt is called a global minimum if it satisfies f(i opt ) f(i), for all i S. Let S opt be the set of all the global minima and f opt = f(i opt ) be their objective value. Neighborhood S i of solution i is the set of discrete points j satisfying j Si, i Sj. Fig. shows the procedure of SA for solving unconstrained problem (2). q(i; i`), the generation probability, is defined as q(i; i`) = / Si for all i` Si, and A (i; i`), the acceptance probability of accepting solution point i, is defined by: A (i; i) = exp ( f ( i`) f ( i) ), where a + = a if a >, and a+ = otherwise. + (2) (3)

4 Proceedings of the th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November -3, Accordingly, SA works as follows. Given current solution i, SA first generates trial point i`. If f(i`) < f(i), i` is accepted as a starting point for the next iteration; otherwise, solution i` is accepted with probability exp ( i ) f ( i) f ; N k log k + e ( ),. (4) he worse the i` is, the smaller is the probability that i` is accepted for the next iteration. he above procedure is repeated N times until temperature is reduced. heoretically, if is reduced sufficiently slowly in logarithmic scale, then SA will converge asymptotically to an optimal solution i opt S opt Bertsekas (). In practice, a geometric cooling schedule, α, is generally utilized to have SA settle down at some solution i* in a finite amount of time. SA can be modeled by an inhomogeneous Markov chain that consists of a sequence of homogeneous Markov chains of finite length, each at a specific temperature in a given temperature schedule. According to generation probability q(i; i`) and acceptance probability A (i; i`), the one-step transition probability of the Markov chain is: ( i,i`) ) A ( i,i`) P ( i, j) q( if i` Si j Si,j i if i`= i otherwise P (i; i`) = (5) and the corresponding transition matrix is P = [P (i; i`)]. It is assumed that, by choosing neighborhood S i properly, the Markov chain is irreducible, meaning that for each pair of solutions i and j, there is a positive probability of reaching j from i in a finite number of steps. Consider the sequence of temperatures { k ; k =,, 2,...}, where k > k+ and lim k k =, and choose N to be the maximum of the minimum number of steps required to reach an i opt from every j S. Since the Markov is irreducible and search space S is finite, such N always exists. he asymptotic convergence theorem of SA is stated as follows. 3.2 heorem he Markov chain modeling SA converges asymptotically to a global minimum of S opt if the sequence of temperatures satisfies: where Δ = max i,j S {f(j) - f(i) j S i }. (6) he proof of this theorem is based on so-called local balance equation Bertsekas(), meaning that: π ( i )P ( i, i`) = π ( i`)p ( i`, i ), (7) where π (i) is the stationary probability of state i at temperature. Although SA works for solving unconstrained NLPs, it cannot be used directly to solve constrained NLPs that have a set of constraints to be satisfied, in addition to minimizing the objective. he widely used strategy is to transform constrained NLP () into an unconstrained NLP using penalty formulations Luenberger (2). For static penalty formulation Luenberger(2), it is very difficult to choose suitable penalty γ: if the penalty is too large, SA tends to find feasible solutions than optimal solutions. For dynamic penalty formulation Humpries and Hawkins (4), unconstrained problem (5) at every stage of λ(k) has to be solved optimally (, 2) in order to have asymptotic convergence. However, this requirement is difficult to achieve in practice, given only a finite amount of time in each stage. If the result in one stage is not a global minimum, then the process cannot be guaranteed to find constrained global minima. herefore, applying SA to a dynamic-penalty formulation does not always lead to asymptotic convergence. Besides, SA cannot be used to search in a Lagrangian space, because minimizing Lagrangian function. For special case of (), the generalized discrete augmented Lagrangian function is defined as L d (x, λ) = f(x) + λ H(h(x)) +/2 h(x) 2 (8) where λ = { λ,λ 2,... λ m } is a set of Lagrange multipliers, H is a continuous transformation function that satisfies H(x) =, x =, and h(x) 2 = ( ) m 2 i = hi x 4 Pre-Computation he basic principle of the method of Simulated Annealing deals with the optimum searching by help of criteria changing. his principle seems to be suitable for strategic regional decisions. he management of local authorities needs optimum solution for particular problems in the context of various regional indicators. Example of these strategic regional questions is using of investment grants in the region. One example could be construction of new highway. One large project is divided into smaller projects (or stages see fig. 4). here are many companies (Com, Com2,...) in the project. Every company is possible to work on specific stage of (9)

5 Proceedings of the th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November -3, whole project. Company Com is able to complete all project (with every stages) but company Com2 is able to complete just stage S, S2 and S4. Others companies is able to complete just a few stages or that stages for that they have sufficient devices (vehicles, material, knowledge etc.) Region (R) x Services (W) x Extra (A) x Reliability () x Com Com2 Com3 Com4 Com5 Com6 Com7 S S S2 S3 S4 A2 A3 A4 A5 A B B2 B4 C C3 C4 D2 D3 E2 E5 F F4 G3 F5 G Fig. 4: Separate stages of the project S5 5 Conclusion Using of optimization methods is necessary for efficient managing of special areas. One of these areas is managing of huge projects for developing of region. hat problem can contain huge number of data which hides large potential for answering of strategic inquiries. Management of institution that manages the projects often needs to find optimum with solving concrete problem if the optimum is contingent by various criteria. Funds of criteria are available in line time, structural rows and at others rows. Method of SA offers possibility of very effective algorithm to solving combinatorial exercise and gained solutions are either identical or very near to optimum solution. Regional authority can make some conditions and set criterions for every stage of whole project. It is no problem to evaluate table where we can find whether concrete company is able to realize defined conditions. able shows one stage of whole project with possibility of every company to realize this stage. Company Com A is transnational company that is able to realize the project for a good price. But the problem is that the company is not able to employ workers who are from region where is the project realized. he valuating of criterion R is very low. On the opposite way is reliability () or Extra services (A) of the company. It is because the company is very strong with established resources or knowledge. here are many different companies in the table with classification of their possibilities to realize the stage. he classification depends on manager decision and on the priority of every local authority. It is no problem to make another classification depends on this set criterions. hat classification can use applicable matrix for simulated annealing as optimizing method. It is just about determining of utilizable criterions. It means we have to determine symbols using in the table to specific numbers (matrix) and then use them in the optimization method. able : Setting of Criterions for one stage of the project Company A B C D E F 6 References. Bertsekas, D. P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, Luenberger, D. G.: Linear and Nonlinear Programming. Addison-Wesley Publishing Company, Reading, MA, Nocedal, J., Wright, S. J.: Numerical Optimization. Springer Verlag, New York Rardin, R. L.: Optimization in Operations Research. Upper Saddle River, New Jersey: Prentice Hall, Humphries, M., Hawkins, M, W.: Data warehousing Architecture and implementation (Czech translation) Computer Press Prague Simonova, S., Capek, J. Fuzzy Approach in Enquiry to Regional Data Sources for Municipalities, WSEAS ransactions on Information Science and Applications, Vol.3, No.2, 24, pp Ait-Aoudia, S., Mana I.: Numerical Solving of Geometric Constraints by bisection A Distributed Approach, WSEAS ransactions on Information Science and Applications, Vol., No.5, 24, pp Popela, P.: Computer Aided Optimization. In Proceedings of 3 rd Conference, p. 3-. Ostrava, Czech republic, 998. Price (F) 4 x 6 4