Multi-objective Linear Programming Method Research Articles

An ε-Constraint Method for Multiobjective Linear Programming in Intuitionistic Fuzzy Environment

Effective decision-making requires well-founded optimization models and algorithms tolerant of real-world uncertainties. In the mid-1980s, intuitionistic fuzzy set theory emerged as another mathematical framework to deal with the uncertainty of subjective judgments and made it possible to represent hesitancy in a decision-making problem. Nowadays, intuitionistic fuzzy multiobjective linear programming (IFMOLP) problems are a topic of extensive research, for which a considerable number of solution approaches are being developed. Among the available solution approaches, ranking function-based approaches stand out for their simplicity to transform these problems into conventional ones. However, these approaches do not always guarantee Pareto optimal solutions. In this study, the concepts of dominance and Pareto optimality are extended to the intuitionistic fuzzy case by using lexicographic criteria for ranking triangular intuitionistic fuzzy numbers (TIFNs). Furthermore, an intuitionistic fuzzy ε-constraint method is proposed to solve IFMOLP problems with TIFNs. The proposed method is illustrated by solving two intuitionistic fuzzy transportation problems addressed in two studies (S. Mahajan and S. K. Gupta’s, “On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions,” Ann Oper Res, vol. 296, no. 1, pp. 211–241, 2021, and Ghosh et al.’s, “Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem,” Complex Intell Syst, vol. 7, no. 2, pp. 1009–1023, 2021). Results show that, in contrast with Mahajan and Gupta’s and Ghosh et al.’s methods, the proposed method guarantees Pareto optimality and also makes it possible to obtain multiple solutions to the problems.

A Closure Jacking Force Calculation Algorithm for Curved Prestressed Concrete Continuous Rigid-Frame Bridges with Asymmetric Cantilevers and Piers

Concrete continuous rigid-frame bridges are the most commonly used to cross exciting highway and railway. Before the bridge closure, the closure jacking force (CJF) needs to be determined firstly, which plays an important role in mid-span deflection controlling and the whole bridge stress state. This study proposed a closure jacking force calculation algorithm for curved prestressed concrete continuous rigid-frame bridges with asymmetric cantilevers and piers. Firstly, the mechanics method was used to analyze the relationship between the CJF and normal stress of main-girder and the pier. In addition, an actual bridge was taken as a numerical simulation case to determine the influence of the deformation and the stress induced by concrete shrinkage and creep of the main-girder and the jacking force. Finally, in order to obtain the optimal stress state of bridge piers, the multi-objective linear programming method was proposed to calculate the CJF. The results show that curved prestressed concrete continuous rigid-frame bridges usually suffered longitudinal and transverse bending and torsion under the concrete shrinkage and creep of the main-girder and the CJF, and the CJF significantly affected the normal stress of the pier with a low anti-push rigidity. The normal stress of piers was obviously linear with the CJF during whole construction and operation state of bridges at the following three states: the jacking process, bridge completion, and ten years of bridge operation. The multi-objective linear programming algorithm was an effective way to determine the optimal CJF and can fully reflect both longitudinal and transverse bending of piers.

Sustainable supplier evaluation and transportation planning in multi-level supply chain networks using multi-attribute- and multi-objective decision making

Supplier selection and transportation planning are essential tasks in managing supply chain networks (SCNs). The choice of a supplier has a significant influence on the transportation planning of SCNs, as the transportation routes will change. A limited number of studies in the literature have addressed the supplier selection and transportation planning issues simultaneously. This study proposes a two-stage multi-criteria decision-making (MCDM) approach for sustainable supplier evaluation and transportation planning in multi-level SCNs. Initially, a supplier evaluation framework is proposed using the modified indifference threshold-based attribute ratio analysis (ITARA) and the performance calculation technique of the integrated multiple multi-attribute decision-making (PCIM-MADM) to determine the supplier’s performance index. Subsequently, a multi-level SCN model is established, and the sustainable supplier selection and transportation planning problem is solved by the fuzzy multi-objective linear programming (FMOLP) method. A set of real-world data provided by a smart audio manufacturing company has been used to demonstrate the proposed approach. Decision-makers understand the importance of criteria through the weights constructed by the modified ITARA. Moreover, the results of the PCIM-MADM can effectively determine the supplier’s ranking and provide improving strategies for underperforming suppliers. The augmented max–min technique of the FMOLP can produce a single optimal solution of the transportation allocation. The results show that our approach can effectively evaluate sustainable supplier performance and optimize transportation planning in SCNs.

Assignment model with multi-objective linear programming for allocating choice ranking using recurrent neural network

Classic linear assignment method is a multi-criteria decision-making approach in which criteria are weighted and each rank is assigned to a choice. In this study, to abandon the requirement of calculating the weight of criteria and use decision attributes prioritizing and also to be able to assign a rank to more than one choice, a multi-objective linear programming (MOLP) method is suggested. The objective function of MOLP is defined for each attribute and MOLP is solved based on absolute priority and comprehensive criteria methods. For solving the linear programming problems we apply a recurrent neural network (RNN). Indeed, the Lyapunov stability of the proposed model is proved. Results of comparing the proposed method with TOPSIS, VICOR, and MORA methods which are the most common multi-criteria decision schemes show that the proposed approach is more compatible with these methods.

Optimal Land Use Structure for Sustainable Agricultural Development—A Case Study in Changsha County, South Central China

Environmental and social problems caused by overfertilization, excessive pesticides, and encroachment on farmland are increasingly serious in agricultural settings, especially in suburban agricultural areas and highly intensive agricultural areas. Hence, modern agriculture not only pursues economic benefits, but it also pays more attention to ecological functions and social stability. This paper proposes a set of methods which are designed to realize optimal agricultural benefits and sustainable development by scientifically adjusting the land use structure. Taking Changsha County in South Central China as a case study, this paper first built an index system and adopted the information entropy-TOPSIS method to assess the economic, social, and ecological benefits of agricultural land use. Next, a coupled coordination model and an obstacle model were chosen to diagnose those factors that remained as obstacles to achieving the sustainable and coordinated development of the benefits of agricultural land use. Finally, based on the analysis of the changes in the benefits and obstacles over time, socio-economic and ecological constraints were established, and the multi-objective linear programming method (MOLP) was used to determine the comprehensive benefits and optimal land use structure. The results indicate that: (1) The agricultural benefits were stably increasing from 0.20 in 1996 to 0.79 in 2016. (2) The economic benefit index is no longer the main obstacle, while the social benefit index, which includes components such as the food security index, has become the principal influencing factor. (3) The optimal land use structure and comprehensive benefits were presented by taking into consideration the economic development, environmental protection, and social needs. This study emphasizes economic development, but it also seeks coordinated development with comprehensive benefits. The results of the study could provide scientific recommendations for optimizing the agricultural land use spatial patterns and sustainable land use.

Knowledge structure-based consensus-reaching method for large-scale multiattribute group decision-making

Large-scale multiattribute group decision-making (LMGDM) requires a large number of participants with different knowledge structures. This study proposed an LMGDM consensus-reaching method in which the experts’ knowledge structures are fully considered. An information extraction mechanism is constructed to extract incomplete inference information with the form of belief distribution (BD), and the Dempster–Shafer theory of evidence is adopted to make discounting and combinations for the BDs. To reduce their number, the experts are classified into different clusters by using the extended K-means approach, and two levels of consensus measures are both calculated to determine whether the experts involved in each cluster have reached a satisfactory level of consensus. If that consensus level is not reached, a feedback mechanism is activated to advise the identified experts to adjust their assessments, which allows them to change clusters during the consensus-reaching process. Through repeating the feedback mechanism, the assessments are improved until the satisfactory consensus levels are reached. A multi-objective linear programming method is established to obtain the optimal solution that satisfies all clusters as much as possible. Finally, a numerical comparison and discussion are undertaken to demonstrate the superiority of the proposed method.

A Two-Phase Multi-criteria Fuzzy Group Decision Making Approach for Supplier Evaluation and Order Allocation Considering Multi-objective, Multi-product and Multi-period

Due to severe competition among business organizations, the selection of supplier becomes more important for business success. However, supplier selection problems are complex and unstructured since it involves a large number of criteria and some of the criteria cannot be assessed accurately. Also, supplier’s performance fluctuations and unknown information always exist in the real-world decisions. Moreover, the criteria may be qualitative or quantitative in nature as well as supplier evaluation involves a group of experts with diverse opinions. To handle such uncertain and vague information of suppliers, use of fuzzy sets is an appropriate option. Hence, for making a realistic decision, this research proposes a two-phase method to select the suitable suppliers and allocate appropriate orders to them. In first phase of the study, the ranking of supplier is performed by using fuzzy MULTIMOORA method with regard to the important criteria. Then, multi-objective linear programming (MOLP) method in fuzzy environment is proposed to allocate orders to the preferred suppliers in the second phase. The model is developed in multi-product environment by satisfying demand, lead time and capacity constraints. We used expected value method to transform the fuzzy multi objective problem into a crisp single objective problem. Appropriate orders are assigned to the preferred suppliers by considering the closeness coefficient to the fuzzy MOLP model. At last, a case study is performed in an Indian manufacturing organization to illustrate the proposed model.

The framework of an agricultural land-use decision support system based on ecological environmental constraints

Agricultural land use is a complicated systems engineering. Modern agriculture faces increasingly more risks. Managers should obtain reliable information to assist in decision-making through certain methods that allow them to achieve an economic, social, and ecological environment that is crucial in coordinating the development of agricultural land-use patterns. This study proposes a framework of an agricultural land-use decision support system (LDSS) based on ecological environmental constraints according to the DSS design philosophy, which provides a scientific basis for managers to allocate land resources. This framework of LDSS consists of a land quality assessment module, an eco-economic coupling module, and a land-use optimisation module. Firstly, it establishes a natural-society-economic land quality evaluation system to simulate the comprehensive benefit relationship of land use. Secondly, it analyses the risks of soil, water, and ecological security in the process of land use; simulates and reveals the mechanism of occurrence; and completes the correlation equation expression between natural-economic-social indictor and land use risk. Based on different scenarios design, the ecological environmental risk factors are used as constraints. Finally, the multi-objective linear programming method is employed to calculate the optimal comprehensive benefits of land use and optimal land-use structure based on the constraints of the ecological environment. Then the study takes Changsha County, a high-intensification, main grain-producing area in Central and South China, as a case area to demonstrate the feasibility of the framework of LDSS, and draws the highest comprehensive benefits and optimal structure of land use under the premise that the rural ecological environment conforms to national standards. Case study shows that the LDSS framework is feasible, easy to operate, and easy to promote. The research results can provide efficient and practical support for managers to allocate land resources and formulate sustainable land-use policies rationally.

Multi-Objective Fuzzy Linear Programming for Land Allocation under Different Crops in Bhagwanpur Distributary

A study was undertaken in Bhagwanpur distributary of Vaishali Branch Canal in Gandak Canal Command Area, Bihar to optimally allocate land area under different crops (rice and maize in kharif, wheat, lentil, potato in rabi and green gram in summer) in such a manner that maximizes net return, maximizes crop production and minimizes labour requirement employing simplex linear programming method and Multi-Objective Fuzzy Linear Programming (MOFLP) method. Maximum net return, maximum agricultural production, and minimum labour required under defined constraints (including 10% affinity level of farmers to rice and wheat crops) as obtained employing Simplex method were ` 3.7 × 108, 5.06 × 107 Kg and 66,092 man-days, respectively, whereas Multi-Objective Fuzzy Linear Programming (MOFLP) method yielded compromised solution with net return, crop production and labour required as ` 2.4 × 108, 3.3 × 107Kg and 1,79,313 man-days, respectively. As the affinity level of farmers to rice and wheat crops increased from 10% to 40%, maximum net return and maximum production as obtained from simplex linear programming method and MOFLP followed a decreasing trend and minimum labour required followed an increasing trend. MOFLP may be considered as one of the best capable ways of providing a compromised solution, which can fulfill all the objectives at a time.

Use of an optimization model for optimization of Turkey’s energy management by inclusion of renewable energy sources

In this study, Turkey’s power generation plan between 2018 and 2035 is obtained by using a new mathematical model that aims to make a generation expansion planning to have an acceptable level of a pollutant in the air considering the UN Framework Convention on Climate Change and Kyoto Protocol’s responsibilities of Turkey. The used method is a new fuzzy multi-objective linear programming (MOLP) method by considering the energy objectives of Turkey’s Ministry of Energy (MoE:MENR) and private sector’s energy targets. The multi-objective model aims are (1) energy generation cost minimization of energy production considering all energy-related costs in Turkey, (2) greenhouse gases emission’s reduction, (3) minimizing the imported energy, (4) maximizing efficiency of power plants and (5) minimizing the use of fossil fuels in power plants. By solving this mathematical model, Turkey’s 18-year power generation plan between the years 2018 and 2035 based on mainly renewable sources is formulated, and by 2035 the percentage of renewable energy sources in Turkey’s power generation is increased by 77% which includes 24.7% solar energy, 19.1% wind energy, 18.5% hydro energy, 12.3% biomass under high-demand scenario.

Comparative analysis of application efficiency of two iterative multi objective linear programming methods (MP method and STEM method)

In this paper we consider a production plan optimization problem for a company that produces textile products. The problem is solved using two iterative methods: a new method based on the cooperative game theory (MP method) and the well-known STEM method. Their application efficiency and the solutions obtained are compared. For this purpose we use four groups of criteria: (1) the general characteristics of the method (2) the criteria from the standpoint of the decision makers, (3) the criteria from the perspective of the analysts, and (4) the ‘economic’ criteria. The analysis indicates that both methods are highly efficient for solving this kind of production plan optimization problems. However, the decision-makers preferred the MP method.

Two-stage optimisation of hybrid solar power plants

Hybrid solar power plants which combine concentrated solar power (CSP) and photovoltaic (PV) systems with thermal energy storage (TES) have the potential to provide cost competitive and dispatchable renewable energy. The integration of energy storage gives dispatchability to the variable renewable generation while the combination of different generation technologies can reduce the costs. However, the design of reliable and cost competitive hybrid solar power plants requires the careful balancing of trade-offs between financial and technical performance. This is made more complicated by the dependence on a larger number of parameters compared to conventional plants and due to the integration of TES which requires that the operational profile is optimised for every design. This contribution presents a two-stage, multi-objective optimisation framework which combines multi-objective linear programming methods for the operational optimisation with multi-objective genetic algorithms for the design optimisation. The operational optimisation which is performed for every design point needs to be performed with linear programming methods. Here an automated scalarisation method is developed for the linear programming method which enables the multi-objective optimisation of the operational profile. This enables the evaluation of the trade-offs between financial and technical performance in both the design and operational optimisation, which is required to design reliable and cost competitive sustainable energy systems. The two-stage multi-objective optimisation is applied to analyse and improve the design of the hybrid solar power plant Atacama-1. It is demonstrated that balancing the trade-off between financial and technical performance is key to increase the competitiveness of solar energy and that it is possible to simultaneously increase dispatchability and decrease the levelised cost of energy. This shows that the operational and design optimisations have to be directly linked in order to exploit the synergies of hybrid systems. Thus the optimisation framework presented in this study can improve the decision making in the design of hybrid solar power plants.

Evaluation of trapezoidal fuzzy numbers on AHP based solution of multi-objective programming problems

Multi-criteria decision making method is used in such cases having multiple targets. Multi-objective linear programming problems, which deal with uncertain measurements for both objectives and constraints, are solved by fuzzy multi-objective linear programming methods. Analytic hierarchy process (AHP), one of the multi-criteria decision-making methods based on weighting of objectives, is designed to solve complex problems. In this study, multiple criteria decision making problems and decision makers’ opinions on these problems as well as their solution processes are discussed. Firstly, a fuzzy multi objective linear programming problem is solved using the approaches of Zimmerman and Hybrid. In the literature, these methods are applied using triangular fuzzy numbers based on decision makers’ opinions only in cases where the objective function is minimization. In this study, as an alternative to the triangular fuzzy numbers, trapezoidal fuzzy numbers are defined, and both minimization and maximization of objective functions are examined. Finally, a solution algorithm for a fuzzy multi-objective linear programming problem is proposed under some given conditions. The algorithm is applied to a sample problem that involves modeling of a congress organizing which aims to place attendees to accommodations under specific objectives and constraints. In the evaluation phase of the algorithm, targets are weighted by using triangular and trapezoidal fuzzy numbers. At the end, the results of the proposed algorithm are compared with the results of other methods in the literature.

Fuzzy multi-objective linear programming for a stochastic hub maximal covering problem with uncertain shipments

When there are many origins and destinations in a network, the use of hubs helps to benefit from the economies of scale in transportation and considerable reduction in the number of required links. We propose a bi-objective mathematical model for a stochastic hub maximal covering problem formulated under two real-world assumptions. In many networks, there are some risks for the shipments among the nodes. In order to consider such risks, a second objective, in addition to maximising the utility of covered nodes, is aimed at maximising the safety of weakest path in the network. Also, similar to the real world situation, it is assumed that the transport times are normal random variables. An efficient nonlinear mixed-integer model is developed and after linearising, a fuzzy multi-objective linear programming method is applied to solve it. Notably, the single objective version of our model has smaller number of variables and equations than the previous ones. The computational results by establishing 24 test problems extracted from the Turkish dataset confirm: 1) the validity of proposed formulations for single objective hub maximal covering problem; 2) the satisfying performance of proposed bi-objective model in improving the safety of designed network.

Fuzzy multi-objective optimization for closed loop logistics network design in bread-producing industries

Supply chain network design is one of the most important issues in designing supply chains because of its effect on performance and efficiency of the logistics network. Customer service improvements and environmental requirements have forced most companies to take into account the reverse flows in designing their supply chain networks. This paper employs an interactive fuzzy multi-objective linear programming (IFMOLP) method to solve fuzzy bi-objective reverse logistics network design problems. This method attempts to minimize the total cost and total delivery time of the system simultaneously. The presented method provides decision makers with an interactive framework that facilitates the process of decision making in a fuzzy environment. Moreover, this framework enables the decision makers to interactively change fuzzy data in order to reach the most satisfactory solutions. A genetic-based heuristic algorithm is presented to solve the model for larger cases. A case study in a bread-producing factory is presented to show the feasibility of the proposed approach.

Multi-objective project management by fuzzy integrated goal programming

Regulating conflicting goals with the usage of all related resources through organization is the main work of project management (PM). In this paper, the issue of evaluating the conflicting goals tradeoffs of projects was to develop a plan that the decision-maker can use to shorten their total completion time and minimize the increasing project total cost. The study showed that the total project cost minimization problem and crashing cost minimization problem with reference to direct, indirect cost and relevant constraints can be solved simultaneously via the proposed fuzzy multi-objective linear programming (FMOLP) method. Next, considering its completion time in a suitable range, we are trying to find more efficient ways of utilizing the fuzzy set to solve fuzzy multi-objective PM decision problem, and the proposed approach applies the signed distance method to transform fuzzy numbers into crisp values. The proposed approach considers the imprecise nature of the input data by implementing the minimum operator and also assumes that each objective function has a fuzzy goal. In addition, the focus of this approach is minimizing the worst upper bound to obtain an efficient solution which is close to the best lower bound of each objective function. Moreover, for attaining our objective, at the end of this paper, a detailed numerical example will be presented to illustrate the feasibility of applying the proposed approach to actual PM decision problem. Furthermore, it was believed that this approach can be utilized to solve other multi-objective decision making problems in practice. Key words: Project management, fuzzy set, fuzzy multi-objective linear programming.

Fuzzy multi-objective linear programming method applied in decision support system to control chronic disease

Diet and nutrition plays a key role to maintain good health. Based on the statistics results, it shows that bad eating habits and dietary changes, the tendency to promote people to get chronic disease is high. Fuzzy MultiObjective Linear Programming approach has been employed to obtain a complete food plan for human body. This plan will help people to obtain nutritional requirements in daily routine and to control chronic disease such as diabetes and heart attack. Decision Support System for Health was created to identify chronic and suggest food plan

Emergy-Based Adjustment of the Agricultural Structure in a Low-Carbon Economy in Manas County of China

The emergy concept, integrated with a multi-objective linear programming method, was used to model the agricultural structure of Xinjiang Uygur Autonomous Region under the consideration of the need to develop a low-carbon economy. The emergy indices before and after the structural optimization were evaluated. In the reconstructed model, the proportions of agriculture, forestry and artificial grassland should be adjusted from 19:2:1 to 5.2:1:2.5; the Emergy Yield Ratio (1.48) was higher than the average local (0.49) and national levels (0.27); and the Emergy Investment Ratio (11.1) was higher than the current structure (4.93) and that obtained from the 2003 data (0.055) in Xinjiang Uygur Autonomous Region, the Water Emergy Cost (0.055) should be reduced compared to that before the adjustment (0.088). The measurement of all the parameters validated the positive impact of the modeled agricultural structure. The self-sufficiency ratio of the system increased from the original level of 0.106 to 0.432, which indicated a better coupling effect among the subsystems within the whole system. The comparative advantage index between the two systems before and after optimization was approximately 2:1. When the mountain ecosystem service value was considered, excessive animal husbandry led to a 1.41 × 1010 RMB·a−1 indirect economic loss, which was 4.15 times the GDP during the same time period. The functional improvement of the modeled structure supports the plan to “construct a central oasis and protect the surrounding mountains and deserts” to develop a sustainable agricultural system. Conserved natural grassland can make a large contribution to the carbon storage; and therefore, it is wise alternative that promote a low-carbon economic development strategy.

An interactive robust data envelopment analysis model for determining alternative targets in Iranian electricity distribution companies

One of the primary concerns on target setting the electricity distribution companies is the uncertainty on input/output data. In this paper, an interactive robust data envelopment analysis (IRDEA) model is proposed to determine the input and output target values of electricity distribution companies with considering the existence perturbation in data. Target setting is implemented with the uncertain data and the decision maker (DM) can search the envelop frontier and find the targets based on his preference. In order to search the envelop frontier, the paper combine the DEA and multi-objective linear programming method such as STEM. The proposed method of this paper is capable of handling uncertainty in data and finding the target values according to the DM’s preferences. To illustrate ability the proposed model, a numerical example is solved. Also, the input and output target values for some of the electricity distribution companies in Iran are reported. The results indicate that the IRDEA model is suitable for target setting based on DM’s preferences and with considering uncertain data.

Target setting in the general combined-oriented CCR model using an interactive MOLP method

The traditional data envelopment analysis (DEA) model does not include a decision maker’s (DM) preference structure while measuring relative efficiency, with no or minimal input from the DM. To incorporate DM’s preference information in DEA, various techniques have been proposed. An interesting method to incorporate preference information, without necessary prior judgment, is the use of an interactive decision making technique that encompasses both DEA and multi-objective linear programming (MOLP). In this paper, we will use Zionts–Wallenius (Z–W) method to reflecting the DM’s preferences in the process of assessing efficiency in the general combined-oriented CCR model. A case study will conducted to illustrate how combined-oriented efficiency analysis can be conducted using the MOLP method.