Iranian Journal Of Operations Research
Volume:3 Issue: 1, Winter and Spring 2012

We consider a family of damped quasi-Newton methods for solving unconstrained optimization problems. This family resembles that of Broyden with line searches, except that the change in gradients is replaced by a certain hybrid vector before updating the current Hessian approximation. This damped technique modifies the Hessian approximations so that they are maintained sufficiently positive definite. Hence, the objective function is reduced sufficiently on each iteration. The recent result that the damped technique maintains the global and superlinear convergence properties of a restricted class of quasi-Newton methods for convex functions is tested on a set of standard unconstrained optimization problems. The behavior of the methods is studied on the basis of the numerical results required to solve these test problems. It is shown that the damped technique improves the performance of quasi-Newton methods substantially in some robust cases (as the BFGS method) and significantly in certain inefficient cases (as the DFP method).

We present a new algorithm obtained by changing the search directions in the algorithm given in [8]. This algorithm is based on a new technique for finding the search direction and the strategy of the central path. At each iteration, we use only the full Nesterov-Todd (NT)step. Moreover, we obtain the currently best known iteration bound for the infeasible interior-point algorithms with full NT steps, namely, which is as good as the linear analogue.

We present an application of ABS algorithms for multiple sequence alignment (MSA). The Markov decision process (MDP) based model leads to a linear programming problem (LPP), whose solution is linked to a suggested alignment. The important features of our work include the facility of alignment of multiple sequences simultaneously and no limit for the length of the sequences. Our goal here is to avoid the excessive computing time, needed by dynamic programming based algorithms for alignment of a large number of sequences. In an attempt to demonstrate the integration of the ABS approach with complex mathematical frameworks, we apply the ABS implicit LX algorithm to elucidate the LPP, constructed with the assistance of MDP. The MDP applied for MSA is a pragmatic approach and entails a scope for future work. Programming is done in the MATLAB environment.

We consider a mathematical model in the form of a system of ordinary differential equations (ODE) for optimally administrating cancer treatments. The ODE system dynamics characterized by locating equilibrium points and stability properties are determined by linearization and using appropriate Lyapunov functions. By applying optimal control theory, we seek to minimize the cost function associated with the vaccine therapy looking for minimization of the tumor cells. Global existence of a solution is shown for this model and existence of an optimal control is proven. The optimality conditions and characterization of the control are discussed.

We consider an open shop scheduling problem. At first, a bi-objective possibilistic mixed-integer programming formulation is developed. The inherent uncertainty in processing times and due dates as fuzzy parameters, machine-dependent setup times and removal times are the special features of this model. The considered bi-objectives are to minimize the weighted mean tardiness and weighted mean completion times. After converting the original formulation into a single-objective crisp one by using an interactive approach and obtaining the Pareto-optimal solutions for small-sized instances, an efficient multi-objective particle swarm optimization (MOPSO) is proposed in order to achieve a good approximate Pareto-optimal set for medium and large-sized examples. This algorithm exploits new selection regimes of the literature for the global best and personal best. Furthermore, a modified decoding scheme is designed to reduce the search area in the solution space, and a local search algorithm is proposed to generate initial particle positions. Finally, the efficiency of the proposed MOPSO (PMOPSO) is shown by comparing with the common MOPSO (CMOPSO) by the use of the design of experiments (DOE) based on three comparison metrics.

In recent years, enriching traditional revenue management models by considering the customer choice behavior has been a main challenge for researchers. The terminology for the airline application is used as representative of the problem. A popular and an efficient model considering these behaviors is choice-based deterministic linear programming (CDLP). This model assumes that each customer belongs to a segment, which is characterized by a consideration set, which is a subset of the products provided by the firm that a customer views as options. Initial models consider a market segmentation, in which each customer belongs to one specific segment. In this case, the segments are defined by disjoint consideration sets of products. Recent models consider the extension of the CDLP to the general case of overlapping segments. The main difficulty, from a computational standpoint, in this approach is solving the CDLP efficiently by column generation. Indeed, it turns out that the column generation subproblem is difficult on its own. It has been shown that for the case of nonoverlapping segments, this can be done in polynomial time. For the more general case of overlapping segments, the column generation sub-problem is NP-hard for which greedy heuristics are proposed for computing approximate solutions. Here, we present a new approach to solve this problem by using a genetic algorithm and compare it with the column generation method. We comparatively investigate the effect of using the new approach for firm’s revenue.

We develop an optimization model based on Markovian approach to determine the optimum value of thresholds in a proposed acceptance sampling design. Consider an acceptance sampling plan where items are inspected and when the number of conforming items between successive defective items falls below a lower control threshold value, then the batch is rejected, and if it falls above a control threshold value, then the batch is accepted and if it falls within the thresholds, the process of inspecting the items continues. A decision is made to accept or reject the batch. We begin with developing a Markov model for determining performance measures of sampling designs, resulting in an acceptance sampling plan optimized based on the minimum angle method. Then, the performance measures of the acceptance sampling plan are determined and the optimum values of thresholds are selected in order to optimize the objective functions. In order to demonstrate the application of the proposed methodology, numerical examples are illustrated.