Oper Res Research Articles

Branch-and-bound algorithms for scheduling in an m-machine no-wait flowshop

In this paper, we develop branch-and-bound algorithms for objectives such as sum of weighted flowtime, weighted tardiness and weighted earliness of jobs, for an $$m-$$ machine no-wait (continuous) flowshop. We believe that there has been no prior work on exact algorithms for this problem setup with a variety of objective functions. For the interest of space, we confine our discussion to a subset of certain combination of these objectives and the extension to other objective combinations is quite straight-forward. We explore the active nodes of a branch-and-bound tree by deriving an assignment-matrix based lower bound, that ensures one-to-one correspondence of a job with its due date and weight. This idea is based on our earlier paper on general $$m-$$ machine permutation flowshop (Madhushini et al. in J Oper Res Soc 60(7):991–1004, 2009) and here we exploit the intricate features of a no-wait flowshop to develop efficient lower bounds. Finally, we conclude our paper with the numerical evaluation of our branch-and-bound algorithms.

A functional equation of tail-balance for continuous signals in the Condorcet Jury Theorem

Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability p, then the probability of a correct verdict tends to one as the jury size tends to infinity (Marquis de Condorcet in Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix, Imprim. Royale, Paris, 1785). Recently, Alpern and Chen (Eur J Oper Res 258:1072–1081, 2017, Theory Decis 83:259–282, 2017) developed a model where jurors sequentially receive independent signals from an interval according to a distribution which depends on the state of Nature and on the juror’s “ability”, and vote sequentially. This paper shows that, to mimic Condorcet’s binary signal, such a distribution must satisfy a functional equation related to tail-balance, that is, to the ratio alpha (t) of the probability that a mean-zero random variable satisfies X>t given that |X|>t. In particular, we show that under natural symmetry assumptions the tail-balances alpha (t) uniquely determine the signal distribution and so the distributions assumed in Alpern and Chen (Eur J Oper Res 258:1072–1081, 2017, Theory Decis 83:259–282, 2017) are uniquely determined for alpha (t) linear.

The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds

ABSTRACT In this paper, we study the Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function on Hadamard manifolds. The gH-directional differentiability for interval-valued function is defined by using the generalized Hukuhara difference. The concepts of interval-valued convexity and pseudoconvexity are introduced on Hadamard manifolds, and several properties involving such functions are also given. Under these settings, we derive the KKT optimality conditions and give a numerical example to show that the results obtained in this paper are more general than the corresponding conclusions of Wu [The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res. 2007;176:46–59] in solving the optimization problem with interval-valued objective function.

A survey on the cutting and packing problems

This paper presents a unified approach, based on a geometrical method (see Faina in Eur J Oper Res 114:542–556, 1999; Eur J Oper Res 126:340–354, 2000), which reduces the general two and three dimensional cutting and packing type problems to a finite enumeration scheme.

Decentralization and mutual liability rules

This paper builds on the recent work of Groote Schaarsberg et al. (Math Methods Oper Res 87(3):383–409, 2018) on mutual liability problems. In essence, a mutual liability problem comprises a financial network in which agents may have both monetary individual assets and mutual liabilities. Here, mutual liabilities reflect rightful monetary obligations from past bilateral transactions. To settle these liabilities by reallocating the individual assets, mutual liability rules are analyzed that are based on centralized bilateral transfer schemes which use a certain bankruptcy rule as its leading allocation mechanism. In this paper we derive a new characterization of mutual liability rules by taking a decentralized approach instead, which is based on a recursive individual settlement procedure. We show that for bankruptcy rules that satisfy composition, this decentralized procedure always leads to the same allocation as the one prescribed by the corresponding mutual liability rule based on centralized bilateral transfer schemes. Finally, we introduce a new reduction method for mutual liability problems and prove that any bankruptcy-rule-based mutual liability rule is invariant with respect to such a reduction.

Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets

Choo (Oper Res 32:216–220, 1984) has proved that any efficient solution of a linear fractional vector optimization problem with a bounded constraint set is properly efficient in the sense of Geoffrion. This paper studies Geoffrion’s properness of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. By examples, we show that there exist linear fractional vector optimization problems with the efficient solution set being a proper subset of the unbounded constraint set, which have improperly efficient solutions. Then, we establish verifiable sufficient conditions for an efficient solution of a linear fractional vector optimization to be a Geoffrion properly efficient solution by using the recession cone of the constraint set. For bicriteria problems, it is enough to employ a system of two regularity conditions. If the number of criteria exceeds two, a third regularity condition must be added to the system. The obtained results complement the above-mentioned remarkable theorem of Choo and are analyzed through several interesting examples, including those given by Hoa et al. (J Ind Manag Optim 1:477–486, 2005).

Efficient nearest neighbors methods for support vector machines in high dimensional feature spaces

In the context of support vector machines, identifying the support vectors is a key issue when dealing with large data sets. In Camelo et al. (Ann Oper Res 235:85–101, 2015), the authors present a promising approach to finding or approximating most of the support vectors through a procedure based on sub-sampling and enriching the support vector sets by nearest neighbors. This method has been shown to improve the computational efficiency of support vector machines on large data sets with low or intermediate feature space dimension. In the present article we discuss ways of adapting the nearest neighbor enriching methodology to the context of very high dimensional data, such as text data or other high dimensional data types, for which nearest neighbor queries involve, in principle, a high computational cost. Our approach incorporates the proximity preserving order search algorithm of Chavez et al. (MICAI 2005: advances in artificial intelligence, Springer, Berlin, pp 405–414, 2005), into the nearest neighbor enriching method of Camelo et al. (2015), in order to adapt this procedure to the high dimension setting. For the required set of pivots, both random pivots and the base prototype pivot set of Mico et al. (Pattern Recogn Lett 15:9–17, 2015), are considered. The methodology proposed is evaluated on real data sets.

On Approximate Stationary Points of the Regularized Mathematical Program with Complementarity Constraints

We discuss the convergence of regularization methods for mathematical programs with complementarity constraints with approximate sequence of stationary points. It is now well accepted in the literature that, under some tailored constraint qualification, the genuine necessary optimality condition for this problem is the M-stationarity condition. It has been pointed out, (Kanzow and Schwartz in Math Oper Res 40(2):253–275. 2015), that relaxation methods with approximate stationary points fail to ensure convergence to M-stationary points. We define a new strong approximate stationarity concept, and we prove that a sequence of strong approximate stationary points always converges to an M-stationary solution. We also prove under weak assumptions the existence of strong approximate stationary points in the neighborhood of an M-stationary solution.

A regularization method for constrained nonlinear least squares

We propose a regularization method for nonlinear least-squares problems with equality constraints. Our approach is modeled after those of Arreckx and Orban (SIAM J Optim 28(2):1613–1639, 2018. https://doi.org/10.1137/16M1088570) and Dehghani et al. (INFOR Inf Syst Oper Res, 2019. https://doi.org/10.1080/03155986.2018.1559428) and applies a selective regularization scheme that may be viewed as a reformulation of an augmented Lagrangian. Our formulation avoids the occurrence of the operator $$A(x)^T A(x)$$, where A is the Jacobian of the nonlinear residual, which typically contributes to the density and ill conditioning of subproblems. Under boundedness of the derivatives, we establish global convergence to a KKT point or a stationary point of an infeasibility measure. If second derivatives are Lipschitz continuous and a second-order sufficient condition is satisfied, we establish superlinear convergence without requiring a constraint qualification to hold. The convergence rate is determined by a Dennis–More-type condition. We describe our implementation in the Julia language, which supports multiple floating-point systems. We illustrate a simple progressive scheme to obtain solutions in quadruple precision. Because our approach is similar to applying an SQP method with an exact merit function on a related problem, we show that our implementation compares favorably to IPOPT in IEEE double precision.

Optimal bi-criterion planning of rescue and evacuation operations for marine accidents using an iterative scheduling algorithm

We consider the problem of real-life evacuation of people at sea. The primary disaster response goal is to minimize the time to save all the people during the evacuation operation, taking into account different groups at risk (children, women, seniors etc.) and the evacuation processing time (including the routing time), subject to a budget constraint. There are different evacuation tools (e.g., lifeboats, salvage ships, sea robots, helicopters etc.) for rescuing groups at risk to some safe points (e.g., hospitals, other ships, police offices etc.). The evacuation processing time of a group at risk depends on the group and the evacuation tool used. The secondary goal is to minimize the cost among all the alternative optimal solutions for the primary goal. We present a new mathematical rescue-evacuation model and design a fast solution method for real-time emergency response for different population groups and different evacuation tools, based on iterative utilization of a modification of the scheduling algorithm introduced by Leung and Ng (Eur J Oper Res 260:507–513, 2017).

Generalisation of A-equitable preference in multiobjective optimisation problems

The concept of A-equitable efficiency in solving the multiobjective optimisation problems have recently been introduced by Mut and Wiecek [Generalised equitable preference in multiobjective programming. Euro J Oper Res. 2011;212:535–551], where A is an arbitrary matrix with non-negative entries. The preference relation of this concept solution, , does not satisfy the strict monotonicity and strict principle of transfers axioms in general, therefore the set of A-equitably efficient solutions is not contained within the set of equitably efficient solutions and the set of Pareto-optimal solutions for the same problem. In this paper, we extend the work done by Mut and Wiecek and state the new conditions on the matrix A that guarantee to satisfy these axioms by the preference relation . The other contribution of this paper is the use of the preference relation to solve the multio objective optimisation problems instead of . This has the advantage that the decision-maker has more freedom to choose the preferences matrix A. The relation has become an equitable rational preference relation by imposing the weaker conditions on entries of A, in comparison to the relation .

A polyhedral study of event-based models for the resource-constrained project scheduling problem

In this paper, we study event-based mixed-integer programming (MIP) formulations for the resource-constrained project scheduling problem (RCPSP) that represent an alternative to the more common time-indexed model (DDT) (Pritsker et al. in Manag Sci 16(1):93–108, 1969; Christofides et al. in Eur J Oper Res 29(3):262–273, 1987) for the case when the scheduling horizon is large. In contrast to time-indexed models, the size of event-based models does not depend on the time horizon. For two event-based models OOE and SEE introduced by Koné et al. (Comput Oper Res 38(1):3–13, 2011), we first present new valid inequalities that strengthen the original formulation. Furthermore, we state a new event-based model, the Interval Event-Based Model (IEE), and deduce natural linear transformations between all three models. Those transformations yield the strict domination order IEE succ SEE succ OOE for their respective linear programming relaxations, meaning that the new IEE model has the strongest linear relaxation among all known event-based models. In addition, we show that DDT can be retrieved from IEE by subsequent expansion and projection of the underlying solution space. This yields a unified polyhedral view on a complete branch of MIP models for the RCPSP. Finally, we also compare the computational performance between all models on common test instances of the PSPLIB (Kolisch and Sprecher in Eur J Oper Res 96(1):205–216, 1997).

A pricing problem with unknown arrival rate and price sensitivity

We study a pricing problem with finite inventory and semi-parametric demand uncertainty. Demand is a price-dependent Poisson process whose mean is the product of buyers’ arrival rate, which is a constant lambda , and buyers’ purchase probability q(p), where p is the price. The seller observes arrivals and sales, and knows neither lambda nor q. Based on a non-parametric maximum-likelihood estimator of (lambda ,q), we construct an estimator of mean demand and show that as the system size and number of prices grow, it is asymptotically more efficient than the maximum likelihood estimator based only on sale data. Based on this estimator, we develop a pricing algorithm paralleling (Besbes and Zeevi in Oper Res 57:1407–1420, 2009) and study its performance in an asymptotic regime similar to theirs: the initial inventory and the arrival rate grow proportionally to a scale parameter n. If q and its inverse function are Lipschitz continuous, then the worst-case regret is shown to be O((log n / n)^{1/4}). A second model considered is the one in Besbes and Zeevi (2009, Section 4.2), where no arrivals are involved; we modify their algorithm and improve the worst-case regret to O((log n / n)^{1/4}). In each setting, the regret order is the best known, and is obtained by refining their proof methods. We also prove an Omega (n^{-1/2}) lower bound on the regret. Numerical comparisons to state-of-the-art alternatives indicate the effectiveness of our arrivals-based approach.

A minimax regret portfolio model based on the investor’s utility loss

The out-of-sample performance of relative robust portfolio optimization methodologies is still little explored in portfolio literature. In this paper, a new minimax regret portfolio optimization model is presented, where regret is defined as the utility loss for the investor. The real benefits of the proposed methodology were analyzed by comparing in-sample and out-of-sample performances of robust and non-robust portfolios. The results suggest that the proposed relative robust model has more value for risk-taking investors. Furthermore, the proposed relative-robust portfolio outperforms the non-robust portfolios, in many of the time windows under analysis, with the exception of the global minimum variance portfolio. Comparatively to the minimax regret model presented by Xidonas et al. (Eur J Oper Res 262:299–305,2017) and the absolute robust approach developed by Kim et al. (Econ Lett 122:154–158, 2014a), the proposed methodology presents itself as a more consistent approach since it generates portfolios that reveal greater stability concerning in-sample and out-of-sample performances. The developed relative robustness approach stands out as a valuable contribution for the assertion of robust optimization, in particular of relative robust methodologies, within the field of portfolio selection under uncertainty.

A Novel MILP Model for N-vehicle Exploration Problem

The N-vehicle exploration problem (NVEP) is a nonlinear discrete scheduling problem, and the complexity status remains open. To our knowledge, there is no literature until now employing mixed integer linear programming (MILP) technology to solve this problem except for Wang (J Oper Res Soc China 3(4):489–498, 2015). However, they did not give numerical experiments since their model existed strictly inequalities and the number of constraints was up to $$O(n^3)$$, which was inefficient to solve practical problems. This paper establishes a more concise MILP model, where the number of constraints is just $$O(n^2)$$. Therefore, the existing efficient MILP algorithms can be used to solve NVEP. Secondly, we provide some properties of N-vehicle problem and give three methods for cutting plane construction, which can increase the solving speed significantly. Finally, a numerical experiment is provided to verify the effectiveness and robustness for different instances and scales of acceleration techniques.

Stronger MIP formulations for the Steiner forest problem

The Steiner forest problem asks for a minimum weight forest that spans a given number of terminal sets. We propose new cut- and flow-based integer linear programming formulations for the problem which yield stronger linear programming bounds than the two previous strongest formulations: The directed cut formulation (Balakrishnan et al. in Oper Res 37(5):716–740, 1989; Chopra and Rao in Math Prog 64(1):209–229, 1994) and the advanced flow formulation by Magnanti and Raghavan (Networks 45:61–79, 2005). We further introduce strengthening constraints and provide an example where the integrality gap of our models is 1.5. In an experimental evaluation, we show that the linear programming bounds of the new formulations are indeed strong on practical instances and that the related branch-and-cut algorithm outperforms algorithms based on the previous formulations.

Robust Solutions to the Life-Cycle Consumption Problem

This paper demonstrates how the well-known discrete life-cycle consumption problem (LCP) can be solved using the Robust Counterpart (RC) formulation technique, as defined in Ben-Tal and Nemirovski (Math Oper Res 23(4):769–805, 1998). To do this, we propose a methodology that involves applying a change of variables over the original consumption before deriving the RC. These transformations allow deriving a closed solution to the inner problem, and thus to solve the LCP without facing the curse of dimensionality and without needing to specify the prior distribution for the investment opportunity set. We generalize the methodology and illustrate how it can be used to solve other type of problems. The results show that our methodology enables solving long-term instances of the LCP (30 years). We also show it provides an alternative consumption pattern as to the one provided by a benchmark that uses a dynamic programming approach. Rather than finding a consumption that maximizes the expected lifetime utility, our solution delivers higher utilities for worst-case scenarios of future returns.

On fractional continuous weighted OWA (FCWOWA) operator with applications

In order statistics, CWOWA operator in real line based on Choquet integral was recently introduced by Narukawa et al. (Ann Oper Res 244(2):571–581, 2016). Recently, the relation of Choquet integral with fractional integral on a continuous support was recently discussed by Sugeno (IEEE Trans Fuzzy Syst 23:1439–1457, 2015). As an open problem, Sugeno emphasized that “it is necessary to consider fractional Choquet integral with respect to any monotone measures”. This reason permit us to consider this problem and introduce the fractional Choquet integral with respect to any monotone measures. We also introduce the fractional CWOWA operator (FCWOWA) which includes CWOWA and WOWA operators as special cases. As an application, we also present some important inequalities for FCWOWA operator.

A revisit to the markup practice of irreversible dynamic pricing

We consider an irreversible dynamic pricing situation in which a firm uses real-time inventory information to decide the most opportune time to raise its sales prices. Feng and Xiao (Oper Res 48:332–343, 2000b) has studied this problem along with the opposite markdown case. In quite symmetric fashions, they established the optimality of threshold policies for both cases. Though the earlier work has made dramatic advances in dynamic pricing and at the same time pioneered with many relevant techniques, we believe its treatment of the markup case warrants some revision. In particular, we find it is in possession of an erstwhile-unknown complementarity property between price flexibility and inventory, whose counterpart is not true for the markdown case. This property is needed in the derivation leading to the optimality of a threshold policy. Our development also allows demand to be time-dependent in a product form, and naturally leads to an efficient policy-computing algorithm.

A note and new extensions on “interval efficiency measures in data envelopment analysis with imprecise data”

This paper deals with imprecise data in data envelopment analysis (DEA). We construct a new pair of mathematical programming models by using the concepts of ‘inf’ and ‘sup’ to calculate the exact values of the lower- and upper-bound efficiency scores in the presence of interval and ordinal data. The method proposed in this study is motivated by the approach introduced by Kao (Eur J Oper Res 174(2):1087–1099, 2006) where a pair of two-level mathematical DEA models are converted into linear programming (LP) models to calculate the lower- and upper-bound efficiency scores in the presence of pure ordinal data. We show that the LP model proposed by Kao (2006) for finding the lower-bound efficiency score yields the upper-bound efficiency score. We propose an improved model that overcomes this drawback and successfully calculates the lower- and upper-bound efficiency scores. We demonstrate the applicability of our models with a numerical example and exhibit its efficacy through comparison with Kao’s (2006) approach.