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The ever increasing pressure to conserve the environment from global warming cannot be overemphasized. Emission from the inventory and production process contributes immensely to global warming and hence, the need to device a sustainable green inventory by the operational managers. In this paper, a multi-item multi-objective inventory model with back-ordered quantity incorporating green investment in order to save the environment is proposed. The model is formulated as a multi-objective fractional programming problem with four objectives: maximizing profit ratio to total back-ordered quantity, minimizing the holding cost in the system, minimizing total waste produced by the inventory system per cycle and minimizing the total penalty cost due to green investment. The constraints are included with budget limitation, space restrictions, a constraint on cost of ordering each item, environmental waste disposal restriction, cost of pollution control, electricity consumption cost during production and cost of greenhouse gas emission in the production process. The model effectiveness is illustrated numerically, and the solution obtained to give a useful suggestion to the decision-markers in the manufacturing sectors.

The traditionally used linear programming model for feed formulation has sought the least- cost combination of ingredients that satisfies a specific level of nutritional requirements. Together with the search for the lowest possible cost, other aspects such as maximising diet efficiency in relation to proposed production objectives and minimising any excess that may lead to unacceptable damage to the environment, are other factors that are gaining in impor- tance in the world of animal nutrition. Taking into account all these factors forces us not only to bear in mind the individual concentrations of each of the nutrients in the feed but also the ratios of these with other nutrients. In this work we show how a multi-objective fractional programming model is better adapted to current needs in feed design than the traditionally used least cost linear model. We also show how the model can be solved using the interactive multi-goal programming method with existing linear optimisation software.

The ever increasing pressure to conserve the environment from global warming cannot be overemphasized. Emission from the inventory and production process contributes immensely to global warming and hence, the need to device a sustainable green inventory by the operational managers. In this paper, a multi-item multi-objective inventory model with back-ordered quantity incorporating green investment in order to save the environment is proposed. The model is formulated as a multi-objective fractional programming problem with four objectives: maximizing profit ratio to total back-ordered quantity, minimizing the holding cost in the system, minimizing total waste produced by the inventory system per cycle and minimizing the total penalty cost due to green investment. The constraints are included with budget limitation, space restrictions, a constraint on cost of ordering each item, environmental waste disposal restriction, cost of pollution control, electricity consumption cost during production and cost of greenhouse gas emission in the production process. The model effectiveness is illustrated numerically, and the solution obtained to give a useful suggestion to the decision-markers in the manufacturing sectors. [ABSTRACT FROM AUTHOR]

In this paper, we consider a complex multi-objective fractional programming problem (CMFP). We establish the necessary optimality conditions of problem (CMFP) in the sense of Pareto optimality and derive its sufficient optimality conditions using generalized convexity. Finally, we construct the parametric dual problem to the primal problem (CMFP) and their duality theorems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the optimality conditions and saddle point theorems of a class of generalized convex (multi-objective fractional programming) problem ( MFP ) with respect to approximate weakly efficient solutions. Firstly, by using parameter method, the problem ( MFP ) is transformed into a (multi-objective non-fractional programming) problem (MNP) . Secondly, the concepts of type-I function and approximate pseudo-quasi-type-I function are introduced for the objective and constraint functions of problem ( MNP ). Finally, under assumptions of the introduced generalized convexities, the optimality conditions and saddle point theorems to problem ( MNP ) are established. [ABSTRACT FROM AUTHOR]

In this paper, we have formulated a new model of multi-objective capacitated transportation problem (MOCTP) with mixed constraints. In this model, some objective functions are linear and some are fractional and are of conflicting in nature with each other. The main objective of this paper is to decide the optimum order of the product quantity which is to be shipped from source to the destination subject to the capacitated restriction on each route. Here the two situations have been discussed for the MOCTP model. In the first situation, we have considered that all the input information of the MOCTP model is exactly known and therefore a fuzzy goal programming approach have been directly used for obtaining the optimum order quantity of the product. While in the second situation the input information of the MOCTP model are uncertain in nature and this uncertainty have been studied and handled by the suitable approaches like trapezoidal fuzzy numbers, multi-choices, and probabilistic random variables respectively. Due to the presence of all these uncertainties and conflicting natures of objectives functions, we cannot solve this MOCTP directly. Therefore firstly we converted all these uncertainties into deterministic forms by using the appropriate transformation techniques. For this, the vagueness in MOCTP defined by trapezoidal fuzzy numbers has been converted into its crisp form by using the ranking function approach. Multichoices in input information parameters have been converted into its exact form by the binary variable transformation technique. Randomness in input information is defined by the Pareto probability distribution, and for conversion into deterministic form chance constrained programming has been used. After doing all these transformations, we have applied fuzzy goal programming approach for solving this resultant MOCTP model for obtaining the optimum order quantity. A case study has been done to illustrate the computational procedure. [ABSTRACT FROM AUTHOR]

In this study the optimal cropping pattern for Piranshahr- one of western Azarbaijan’s cities- was determined using Multi-objective Fuzzy Fractional Programming with the aim of moving toward sustainable agriculture. The cross section data of 2007-2008 were used for our analysis. Data were gathered through filling out 600 questionnaires from 60 villages of Piranshahr city by using two-stage random sampling method. The results showed that the optimal cropping pattern using the Multi-objective Fuzzy Fractional Programming to achieve sustainability is different with the current cropping pattern. Moreover, the results of the stability indices (ratio of gross income to the use of chemical fertilizers and pesticides) showed the significance of this goal was built for sustainability. Sustainability indices determined for chemical pesticides fungicides, herbicides, insecticide, nitrogen, phosphate and fertilizer. Results indicate that elimination or minimization of chemical pesticides and fungicides is more important than other inputs.

First a new notion of the random exponential Hanson-Antczak type $$(\alpha ,\beta ,\gamma ,\xi ,\eta ,\rho ,h(\cdot ,\cdot ,\cdot ),\theta )$$ -V-invexity is introduced, which generalizes most of the existing notions in the literature, second a random function $$h(\cdot ,\cdot ,\cdot )$$ of the second order is defined, and finally a class of asymptotically sufficient efficiency conditions in semi-infinite multi-objective fractional programming is established. Furthermore, several sets of asymptotic sufficiency results in which various generalized exponential type $$HA(\alpha ,\beta ,\gamma ,\xi ,\eta ,\rho ,h(\cdot ,\cdot ,\cdot ),\theta )$$ -V-invexity assumptions are imposed on certain vector functions whose components are the individual as well as some combinations of the problem functions are examined and proved. To the best of our knowledge, all the established results on the semi-infinite aspects of the multi-objective fractional programming are new, which is a significantly new emerging field of the interdisciplinary research in nature. We also observed that the investigated results can be modified and applied to several special classes of nonlinear programming problems. [ABSTRACT FROM AUTHOR]

At present, water shortage has increasingly become one of the key factors restricting the sustainable development of agriculture. And during the process of agricultural production, the energy consumption has also been paid more and more attention. Therefore, in order to deal with the above problems, an improved interval linear multi-objective fractional programming model (ILMOFPM) was developed, which has the characteristic is that it not only focuses on the optimal allocation of the amount of irrigation water resources, but also on improving irrigation water resources use efficiency. To demonstrate its effectiveness and applicability, ILMOFPM was applied to a case study in Jinghuiqu irrigation district, Shaanxi Province, China. Maximum economic benefit per unit of water resource and minimum energy consumption per unit of water resource were selected as the planning objectives. A series of irrigation water resources optimal allocation schemes were obtained, which could give the decision makers the desired optimal scheme according to the actual situation. Based on the results of the established model, economic benefit and energy consumption generally decreased as the representative hydrological years changed from wet to dry, and the total crop planting area also showed a decreased trend. Moreover, a lot of water resources was saved by introducing crop evapotranspiration and effective precipitation in the developed model, 57.45%, 44.17%, 39.33% of total water resources consumption were saved under different representative hydrological years, respectively. • Energy consumption was introduced into the developed model. • Water efficiency was improved. • A series of optimal planting schemes were obtained. • It could be helpful to make decisions under different representative hydrological years. [ABSTRACT FROM AUTHOR]

The traditionally used linear programming model for feed formulation has sought the least- cost combination of ingredients that satisfies a specific level of nutritional requirements. Together with the search for the lowest possible cost, other aspects such as maximising diet efficiency in relation to proposed production objectives and minimising any excess that may lead to unacceptable damage to the environment, are other factors that are gaining in importance in the world of animal nutrition. Taking into account all these factors forces us not only to bear in mind the individual concentrations of each of the nutrients in the feed but also the ratios of these with other nutrients. In this work we show how a multi-objective fractional programming model is better adapted to current needs in feed design than the traditionally used least cost linear model. We also show how the model can be solved using the interactive multi-goal programming method with existing linear optimisation software. [ABSTRACT FROM AUTHOR]

Kaul and Kaur [7] obtained necessary optimality conditions for a non-linear programming problem by taking the objective and constraint functions to be semilocally convex and their right differentials at a point to be lower semi-continuous. Suneja and Gupta [12] established the necessary optimality conditions without assuming the semilocal convexity of the objective and constraint functions but their right differentials at the optimal point to be convex. Suneja and Gupta [13] established necessary optimality conditions for an efficient solution of a multiobjective non-linear programming problem by taking the right differentials of the objective functions and constraint functions at the efficient point to be convex. In this paper we obtain some results for a properly efficient solution of a multiobjective non-linear fractional programming problem involving semilocally convex and related functions by assuming generalized Slater type constraint qualification. [ABSTRACT FROM AUTHOR]

The minimum cost linear programming model used traditionally for feed formulation does not take account of variability of nutrients in feed ingredients. Therefore, it may be that the nutrient requirements of the animal are not adequately met. In this paper, we show how a multiobjective stochastic model that permits confronting the cost of the ration with the probabilities of meeting the nutrient requirements of the animal can enhance the process of animal diet formulation. The model presented here does not require any a priori information from the decision maker, eliciting his preferences through an interactive process. This is the main advantage in relation to other models found in the literature for treating the problem of nutrient variability, which introduce stochastic constraints in the single objective minimum cost model requiring fixing the level of probability desired for each one of the nutrients in advance. [ABSTRACT FROM AUTHOR]

Abstract: The traditionally used linear programming model for feed formulation has sought the least-cost combination of ingredients that satisfies a specific level of nutritional requirements. Together with the search for the lowest possible cost, other aspects such as maximising diet efficiency in relation to proposed production objectives and minimising any excess that may lead to unacceptable damage to the environment, are other factors that are gaining in importance in the world of animal nutrition. Taking into account all these factors forces us not only to bear in mind the individual concentrations of each of the nutrients in the feed but also the ratios of these with other nutrients. In this work we show how a multi-objective fractional programming model is better adapted to current needs in feed design than the traditionally used least cost linear model. We also show how the model can be solved using the interactive multi-goal programming method with existing linear optimisation software. [Copyright &y& Elsevier]