Max-min Approach Research Articles

RATING AND CONTROL OF COMPLEX OBJECTS IN UNCERTAINTY CASES

Statement of the rating and control of complex objects problem under uncertainty is formulated. Seven methods are presented for assessing the state of individual properties of a complex object, designed for various conditions of uncertainty. The source of uncertainty can serve as objective tools of control, that is, estimates of particular properties of an object can be either exact values ​​on a set of real numbers or interval estimates, in addition to this, the state of individual properties can have a probability distribution about the possible state of an object. The qualitatively described properties of an object are subject to uncertainty, the source of which is the subjectivity of the judgments of experts involved in their evaluation. It is important that experts can also evaluate quantitative indicators, for example, in cases where the tools of objective control are not available, that is, subjective input methods are applicable not only to qualitative indicators, but also quantitative ones. In order to aggregate information about private properties, it is proposed to use the matrix mechanisms of complex assessment that are known and obtained by the author. The defined matrix mechanisms of complex evaluation are as follows: the tree fuzzy rating mechanism with an additive-multiplicative approach and two max-min approaches to set theoretic operations, as well as two continuous complex evaluation mechanisms. Six model examples demonstrate procedures of aggregating two criteria with accurate real values, as well as with different uncertainty such as interval values, fuzzy values and F-Fuzzy values.

Assortment optimisation problem: A distribution-free approach

Assortment optimisation is a critical decision that is regularly made by retailers. The decision involves a trade-off between offering a larger assortment of products but smaller inventories of each product and offering a smaller number of varieties with more inventory of each product. We propose a robust, distribution-free formulation of the assortment optimisation problem such that the assortment and inventory levels can be jointly optimised without making specific assumptions on the demand distributions of each product. We take a max-min approach to the problem that provides a guaranteed lower bound to the expected profit when only the mean and variance of the demand distribution are known. We propose and test three heuristic algorithms that provide solutions in O(nlog (n)) time and identify two cases where one of the heuristics is guaranteed to return optimal policies. Through numerical studies, we demonstrate that one of the heuristics performs extremely well, with an average optimality gap of 0.07% when simulated under varying conditions. We perform a sensitivity analysis of product and store demand attributes on the performance of the heuristic. Finally, we extend the problem by including maximum cardinality constraints on the assortment size and perform numerical studies to test the performance of the heuristics.

Stability Constrained Economic Operation of Islanded Droop-Controlled DC Microgrids

This paper proposes a method to determine optimal droop values of dispatchable distributed generation (DG) units in an islanded droop-controlled dc microgrid. Objectives are to minimize fuel cost and maximize small signal stability of the system. In this paper, a detailed state-space model of an islanded droop-controlled dc microgrid including the converter, network parameters, and network interconnection is developed. The developed model is then used to obtain the dominant eigen value of the system. Particle swarm optimization is used to determine the optimal droop values of the dispatchable DG units in order to simultaneously reduce the fuel cost, increase system damping, and push the most dominant eigen value away from the imaginary axis to the left of the s -plane. The bi-objective optimization is solved using a fuzzy max–min based approach. The proposed method is validated on a sample three-bus droop-controlled DCMG test system. Comparison of the result obtained by the fuzzy max–min approach with that of controlled elitist genetic algorithm is also presented in this paper.

On the Uplink Max–Min SINR of Cell-Free Massive MIMO Systems

A cell-free massive multiple-input multiple-output system is considered using a max-min approach to maximize the minimum user rate with per-user power constraints. First, an approximated uplink user rate is derived based on channel statistics. Then, the original max-min signal-to-interference-plus-noise ratio problem is formulated for the optimization of receiver filter coefficients at a central processing unit and user power allocation. To solve this max-min non-convex problem, we decouple the original problem into two sub-problems, namely, receiver filter coefficient design and power allocation. The receiver filter coefficient design is formulated as a generalized Eigenvalue problem, whereas the geometric programming (GP) is used to solve the user power allocation problem. Based on these two sub-problems, an iterative algorithm is proposed, in which both problems are alternately solved while one of the design variables is fixed. This iterative algorithm obtains a globally optimum solution, whose optimality is proved through establishing an uplink-downlink duality. Moreover, we present a novel sub-optimal scheme which provides a GP formulation to efficiently and globally maximize the minimum uplink user rate. The numerical results demonstrate that the proposed scheme substantially outperforms the existing schemes in the literature.

A joint maxmin-lexicographic maximisation approach in fuzzy goal programming using dominance possibility and necessity criteria

In this paper, a new approach for solving fuzzy goal programming problems is introduced. The coefficients and the aspiration level of each fuzzy goal are considered either trapezoidal or triangular fuzzy numbers. Four dominance criteria (dominance possibility, strict dominance possibility, dominance necessity, and strict dominance necessity) are utilised for comparing the fuzzy numbers. The proposed approach is based on merging the maxmin approach and the lexicographic approach in a two-phase process. The first phase applies the maxmin technique by maximising the minimum achievement degree of the fuzzy goals. The second phase lexicographically maximises the achievement degrees of the fuzzy goals according to their preemptive priorities. This methodology provides the decision maker with the advantage of improving the results of his preemptive priority structure model by initially maximising the lowest achievement of the fuzzy goals, and hence guarantee that the ultimate achievement of any fuzzy goal will never be lower than a specific percentage of the achieved maxmin value. The suggested approach is illustrated by a numerical example.

A joint maxmin-lexicographic maximisation approach in fuzzy goal programming using dominance possibility and necessity criteria

In this paper, a new approach for solving fuzzy goal programming problems is introduced. The coefficients and the aspiration level of each fuzzy goal are considered either trapezoidal or triangular fuzzy numbers. Four dominance criteria (dominance possibility, strict dominance possibility, dominance necessity, and strict dominance necessity) are utilised for comparing the fuzzy numbers. The proposed approach is based on merging the maxmin approach and the lexicographic approach in a two-phase process. The first phase applies the maxmin technique by maximising the minimum achievement degree of the fuzzy goals. The second phase lexicographically maximises the achievement degrees of the fuzzy goals according to their preemptive priorities. This methodology provides the decision maker with the advantage of improving the results of his preemptive priority structure model by initially maximising the lowest achievement of the fuzzy goals, and hence guarantee that the ultimate achievement of any fuzzy goal will never be lower than a specific percentage of the achieved maxmin value. The suggested approach is illustrated by a numerical example.

A Maxmin Approach for the Equilibria of Vector-Valued Games

This paper deals with the equilibria of non-cooperative games where the preferences of the players are incomplete and can be represented by vector-valued functions. In the literature, these preferences are frequently approximated by means of additive value functions. However, other value functions can also be considered. We propose a weighted maxmin approach to represent players’ preferences, where the weights are interpreted as the relative importance of the corresponding components of the vector payoffs. We establish the relationship between the equilibria, the weak equilibria and the ideal equilibria of vector-values games and the equilibria of the scalar weighted maxmin games. The potential applicability of the theoretical results is illustrated with the analysis of a vector-valued bimatrix game where all the equilibria are generated, and it is shown how the resulting equilibrium strategies depend on the values of the parameters which represent the importance assigned to the components of the vector-valued payoffs.

Multi-surrogate-based global optimization using a score-based infill criterion

This paper presents a new global optimization algorithm named MGOSIC to solve unconstrained expensive black-box optimization problems. In MGOSIC, three surrogate models Kriging, Radial Basis Function (RBF), and Quadratic Response Surfaces (QRS) are dynamically constructed, respectively. Additionally, a multi-point infill criterion is proposed to obtain new points in each cycle, where a score-based strategy is presented to mark cheap points generated by Latin hypercube sampling. According to their predictive values from the three surrogate models, the promising cheap points are assigned with different scores. In order to obtain the samples with diversity, a Max-Min approach is proposed to select promising sample points from the cheap point sets with higher scores. Simultaneously, the best solutions predicted by Kriging, RBF, and QRS are also recorded as supplementary samples, respectively. Once MGOSIC gets stuck in a local valley, the estimated mean square error of Kriging will be maximized to explore the sparsely sampled regions. Moreover, the whole optimization algorithm is carried out alternately in the global space and a reduced space. In summary, MGOSIC not only brings a new idea for multi-point sampling, but also builds a reasonable balance between exploitation and exploration. Finally, 19 mathematical benchmark cases and an engineering application of hydrofoil optimization are used to test MGOSIC. Furthermore, seven existing global optimization algorithms are also tested as contrast. The final results show that MGOSIC has high efficiency, strong stability, and better multi-point sampling capability in dealing with expensive black-box optimization problems.

A Bi-Level Programming Model for Energy and Flexiramp Procurement in Day-ahead Market and a Fuzzy Max-Min Approach for its Solution

In this paper, we focus on solving the integrated energy and flexiramp procurement problem in the day-ahead market. The problem of energy and ramp procurement could be perfectly analyzed through Stackelberg concept, because of its hierarchical nature of the decision-making process. Such a circumstance is modeled via a bi-level programming, in which suppliers act as leaders and the ISO appear as the follower. The ISO intends to minimize the energy and spinning reserve procurement cost, and the suppliers aim to maximize their profit. To solve the proposed model, a fuzzy max-min approach is applied to maximize the players’ utilities. The objectives and suppliers’ dynamic offers, determined regarding the market clearing prices, are reformulated through fuzzy utility functions. The proposed approach is an effective and simple alternative to the KKT method, especially for problems with non-convex lower-level.

Min–max and max–min approaches to the solution of wave equation in relativistic quantum chemistry

The variation problem associated with the solution of Dirac’s relativistic electron equation is reviewed here. Derivations of the min–max and max–min theorems are discussed: A new observation is that the spurious roots of negative energy satisfy a max–min theorem. Min–max principle (MMP) for solution of Dirac equation, extendable to the Dirac–Fock case, is concisely inspected and illustrated. MMP for two-electron Dirac–Coulomb equation is also considered. The min–max relation is physically interpreted for both Dirac and Dirac–Coulomb problems. Applications to chemical systems are investigated. Mathematical advances are reviewed. Limitations of MMP are spelt out, recent mathematical developments are collated, and an outline is given of the associated theoretical and computational developments. For a complementary purpose, the case of spin-zero is briefly examined. The variation principle for Dirac fermions in nanoscience is discussed. A brief outline of future prospects is given.

Robust Waveform Optimization for MIMO-OFDM-Based STAP in the Presence of Environmental Uncertainty

The issue of robust orthogonal frequency division multiplexing (OFDM) multi-input multi-output (MIMO) radar waveform optimization is considered here with clutter uncertainty to enhance the worst-case detection probability of the MIMO-OFDM radar-based space–time adaptive processing (STAP). Robust optimization is necessary on account of the fact that waveform design is generally sensitive to the initial parameter estimation errors. According to the max–min approach, the issue of robust waveform design is constructed with the criterion of maximizing the worst-case output SINR, namely signal-to-interference-noise ratio, under the constraints of the clutter covariance matrix (CCM) errors and constant modulus to ease this sensitivity systematically, and hence the robustness of STAP’s detection to the uncertainty in the CCM estimate can be enhanced. To handle the acquired nonlinear and sophisticated issue, an iterative approach based on diagonal loading (DL) is presented, in which the inner and outer optimization issues can be recast into semidefinite programming ones by employing DL method, and thus both of them can be tackled in an efficient way. As compared to uncorrelated waveforms and the non-robust algorithm, numerical simulations reveal that the developed approach can enhance the robustness of STAP’s detection to the uncertainty in the CCM considerably.

A series-parallel inventory-redundancy green allocation system using a max-min approach via the interior point method

PurposeThe purpose of this study is to develop a novel and practical series-parallel inventory-redundancy allocation system in a green supply chain including a single manufacturer and multiple retailers operating in several positions without any conflict of interests. The manufacturer first produces multi-product and then dispatches them to the retailers at different wholesale prices based on a common replenishment cycle policy. In contrast, the retailers sell the purchased products to customers at different retail prices. In this way, the manufacturer encounters a redundancy allocation problem (RAP), in which the solution subsequently enhances system production reliability. Furthermore, to emphasize on global warming and human health concerns, this paper pays attention both the tax cost of industrial greenhouse gas (GHG) emissions of all produced products and the limitation for total GHG emissions.Design/methodology/approachThe manufacturer intends not only to maximize the total net profit but also to minimize the mean time to failure of his production system using a RAP. To achieve these objectives, the max-min approach associated with the solution method known as the interior point method is utilized to maximize the minimum (the worst) value of the objective functions. Finally, numerical experiments are presented to further demonstrate the applicability of the proposed methodology. Sensitivity analysis on the green supply chain approach is also performed to obtain more insight.FindingsThe computational results showed that increasing the number of products and retailers might lead into a substantial increase in the total net profit. This indicated that the manufacturer would feel adding a new retailer to the green supply chain strongly. Moreover, an increase in the number of machines provides significant improvement in the reliability of the production system. Furthermore, the results of the performed sensitivity analysis on the green approach indicated that increasing the number of machines has a substantial impact on both the total net profit and the total tax cost. In addition, not only the proposed green supply chain was more efficient than the supply chain without green but also the proposed green supply chain was very sensitive to the tax cost of GHG emission rather than the number of machines.Originality/valueIn summary, the motivations are as follows: the development of a bi-objective series-parallel inventory-RAP in a green supply chain; proposing a hybrid inventory-RAP; and considering the interior point solution method. The novel method comes from both theoretical and experimental techniques. The paper also has industrial applications. The advantage of using the proposed approach is to generate additional opportunities and cost effectiveness for businesses and companies that operate utilizing the green supply chain under an inventory model.

Full-Duplex Massive MIMO Multi-Pair Two-Way AF Relaying: Energy Efficiency Optimization

We consider two-way amplify-and-forward relaying, where multiple full-duplex user pairs exchange information via a shared full-duplex massive multiple-input multiple-output (MIMO) relay. Most of the previous massive MIMO relaying works maximize the spectral efficiency (SE). By contrast, we maximize the non-convex energy efficiency (EE) metric by approximating it as a pseudo-concave problem, which is then solved using the classic Dinkelbach approach. We also maximize EE of the least energy-efficient user relying on the max–min approach. We also compare SE and EE of the proposed design with existing full-duplex systems and quantify the significant improvement achieved by the proposed algorithm. We also compare EE of the proposed full-duplex system to that of its half-duplex counterparts, and characterize the self-loop and inter-user interference regimes, for which the proposed full-duplex system outperforms the half-duplex ones.

Data envelopment analysis based fuzzy multi-objective portfolio selection model involving higher moments

We study the portfolio selection problem from the perspective of incorporating more information about the non-normality of asset returns by considering the mean-variance-skewness-kurtosis framework. Using additional criteria (namely asset turnover, earnings per share, earnings per share growth rate, leverage ratio, price/earnings ratio, and beta (β) value), we employ data envelopment analysis technique to construct a benefit criterion from the efficiency scores of the assets. To incorporate the uncertainty and vagueness of real financial markets, the input data (with respect of all criteria) are assumed as (λ, ρ) interval-valued fuzzy numbers constructed using the historical data. Marginal impacts of the assets on higher moments of the portfolio are used to formulate a fuzzy multi-objective linear programming model. The constraints of the proposed model include the bounds on investment in individual assets, full utilization of the investment capital, and no short selling of assets. The signed distance ranking method is used to obtain a numeric optimization model, which is solved through a weighted max-min approach (in order to incorporate the investor’s preferences regarding investment criteria). A case of real financial market portfolio selection is presented to demonstrate the efficiency of both the proposed model and the solution method.

On the Performance of Downlink Multiuser Cognitive Radio Inspired Cooperative NOMA

This paper is considered as an application of a centralized control non orthogonal multiple access (NOMA) based cognitive radio network. Here, a base station (BS) sends simultaneously two information signals by employing the superposition coding scheme to two different types of users, i.e., group of near users and one far user. The near users, namely, the secondary users, exchange cooperatively their own received information among themselves ensuring the realization of maximal diversity gain. Besides, they are responsible for relaying information to the far user, namely, the primary user. One potential secondary user is selected to decode and forward the BS information signal to the primary user and the rest of the secondary users to reinforce the reliability, as well as, mitigate the non-decodable messages. Two equivalent cases of a relay (secondary user) selection scheme are proposed. In the first case, the selection aims at maximizing the minimum of the joint secondary to secondary (S to S) and secondary to primary (S to P) channels’ coefficients under a certain limit of interference condition. In the second case, the selection aims at maximizing the minimum of the BS to S and S to S paths while a certain quality of service of the primary user is strictly guaranteed. Assuming Rayleigh fading channels, new closed form expressions are derived for the achievable capacity associated with the two information signals. Simulation results reveal the advantage of our proposed schemes over the conventional orthogonal max–min approach and confirm the validity of our analysis.

Optimisation of a sustainable manufacturing system design using the multi-objective approach

A sustainable manufacturing system design can be defined as a process aimed at minimising the negative aspect of both economic and ecological costs. This may be partially achieved through the implementation of lean manufacturing methods in order to reduce production wastes, increase efficiency of manufacturing systems and minimise operational costs. Nevertheless, the concept of lean methods does not include environmental considerations in terms of such as energy consumption and CO2 (carbon dioxide) emissions, which are also important factors today for developing a sustainable manufacturing system. This paper addresses these issues involved in modelling a sustainable manufacturing system allowing an evaluation in energy consumption and CO2 emissions against the total cost using the multi-objective approach. In this work, a multi-objective mathematical model was developed based on a manufacturing system incorporating its economic and ecological parameters towards a minimisation of the total cost, the total energy consumption and CO2 emissions associated with relevant machines, air-conditioning units and lighting bulbs involved in each manufacturing process and material flow. The model was coded using LINGO11 to help gain optimal solutions using the e-constraint approach and the LP-metrics approach, respectively. The best solution among obtained optimal results was revealed using the max-min approach. Applicability of the proposed method was also examined using collected data from a real case study. The study concluded that the multi-objective mathematical model was useful as an aid for optimizing the manufacturing system design under the economic and ecological constraints.

Max-min fuzzy programming approach for compromise farming: a case study

<i>Max-min</i> fuzzy programming approach for compromise farming: a case study

A cost-effective decision-making algorithm for an RFID-enabled HMSC network design

PurposeThe purpose of this paper is to investigate the economic feasibility of a three-echelon Halal Meat Supply Chain (HMSC) network that is monitored by a proposed radio frequency identification (RFID)-based management system for enhancing the integrity traceability of Halal meat products and to maximize the average integrity number of Halal meat products, maximize the return of investment (ROI), maximize the capacity utilization of facilities and minimize the total investment cost of the proposed RFID-monitoring system. The location-allocation problem of facilities needs also to be resolved in conjunction with the quantity flow of Halal meat products from farms to abattoirs and from abattoirs to retailers.Design/methodology/approachFirst, a deterministic multi-objective mixed integer linear programming model was developed and used for optimizing the proposed RFID-based HMSC network toward a comprised solution based on four conflicting objectives as described above. Second, a stochastic programming model was developed and used for examining the impact on the number of Halal meat products by altering the value of integrity percentage. The ε-constraint approach and the modified weighted sum approach were proposed for acquisition of non-inferior solutions obtained from the developed models. Furthermore, the Max-Min approach was used for selecting the best solution among them.FindingsThe research outcome shows the applicability of the developed models using a real case study. Based on the computational results, a reasonable ROI can be achievable by implementing RFID into the HMSC network.Research limitations/implicationsThis work addresses interesting avenues for further research on exploring the HMSC network design under different types of uncertainties and transportation means. Also, environmentalism has been becoming increasingly a significant global problem in the present century. Thus, the presented model could be extended to include the environmental aspects as an objective function.Practical implicationsThe model can be utilized for food supply chain designers. Also, it could be applied to realistic problems in the field of supply chain management.Originality/valueAlthough there were a few studies focusing on the configuration of a number of HMSC networks, this area is overlooked by researchers. The study shows the developed methodology can be a useful tool for designers to determine a cost-effective design of food supply chain networks.

How High Should We Go? Determining Reservation Values to Negotiate Successfully for Composite Software Services

In the paradigm of service-centric computing, new value-added applications can be developed dynamically and flexibly by combining and integrating existing services. While software applications are traditionally specified and implemented as a set of functions uniform to all users, this new paradigm allows the same software service to be delivered with a different price, response time, availability, and other nonfunctional attributes to accommodate different modes of use. These nonfunctional attributes together are referred to as Quality of Service (QoS). When creating a new composite service, negotiation makes it possible for a service provider to offer the service with the QoS properties customized to the needs of a user. Automated negotiation tactics require the specification of reservation values for the QoS attributes. We present a methodology that determines the reservation values a user (or broker) should use for each component service based on the user’s minimum requirements for the composite service. Our methodology maximizes the chance of reaching a successful negotiation outcome while staying within the user’s reservation values for the composite service. We show that the problem of determining the user’s reservation values for component services can be modeled as a multiobjective optimization problem and then transformed to a single-objective optimization problem using a max-min approach. The formulation can incorporate providers' different QoS preferences to increase the chance of negotiation success. We identify problem instances for which closed-form solutions can be found for the reservation values. We show how the method of setting reservation values can be incorporated into a negotiation process that uses extant concession and trade-off tactics. Simulation experiments demonstrate the effectiveness of the proposed approach. If some providers accept offers before the negotiation process deadline, we show that dynamically changing the reservation values for the remaining providers makes the overall negotiation process more likely to succeed. The online appendix is available at https://doi.org/10.1287/isre.2016.0678 .

(No) Foundations of dominant-strategy mechanisms: a comment on Chung and Ely (2007)

This comment revisits Chung and Ely (Rev Econ Stud 74:447–476, 2007) in which robustly optimal auctions where investigated. Chung and Ely used a maxmin approach to define robust optimality. Chung and Ely provided conditions under which dominant strategy auctions are robustly optimal in their sense. This comment proposes a refinement of Chung and Ely’s criterion and shows that, with this refined criterion, dominant strategy auctions are not optimal if there are at least three bidders. According to the refinement the auctioneer should not choose dominated auctions, that is, auctions for which there exist other mechanisms that never generate lower expected revenue, and sometimes higher expected revenue. We construct such a dominating mechanism for dominant strategy auctions. The construction exploits the possibility of side bets when beliefs are not derived from common priors. Chung and Ely (Rev Econ Stud 74:447–476, 2007) admitted such beliefs.