Computing the exact ideal and nadir criterion values is a very ‎ important subject in ‎ multi- ‎ objective LINEAR PROGRAMMING ((MOLP)) ‎ problems ‎‎. In fact ‎, ‎ these values define the ideal and nadir points as lower and ‎ upper bounds on the nondominated points ‎. ‎ Whereas determining the ‎ ideal point is an easy work ‎, ‎ because it is equivalent to optimize a ‎ convex function (LINEAR function) over a convex set which is a convex optimization problem ‎, ‎ but the problem of computing ‎ the nadir point in (MOLP) is equivalent to solving a nonconvex optimization ‎ problem whose solving is very hard in the general case ‎. ‎‎ In this paper ‎, ‎ a bi-level LINEAR PROGRAMMING problem is presented for obtaining the nadirpoint in (MOLP) problems which can be used in general to optimize a ‎ LINEAR function on the nondominated set ‎, ‎ as well ‎. Then ‎, ‎ as one of the solution methods of this problem ‎, ‎ a ‎ mixed-integer LINEAR PROGRAMMING problem is presented which obtains ‎ the exact nadir values in one stage.

Classic LINEAR Assignment method is a multi-criteria decision making method, which takes the weight of criteria into account, and each ranking is assigned to one, and only one alternative. In order to omit the need of calculating the weight of criteria, to use the priority of decision-making criteria, and to be able to assign each ranking to more than one alternative, a multi-objective LINEAR PROGRAMMING method is suggested in this paper in which an objective function is defined for each criterion, to optimize alternatives based on that particular criterion. The objective function of each criterion consists of the total related performance point variables in the LINEAR assignment model so that the best alternative, which is optimum in terms of all criteria, is chosen in the end. There are as many objective functions as criteria which all contribute to a Multi-Objective LINEAR PROGRAMMING. Then, regarding the priority of the criteria, the multi-objective LINEAR PROGRAMMING model is solved using Absolute Priorities Method. In the suggested method, measuring the weight of criteria is not required; however, the degree of the importance of each criterion is taken into account. Decision-makers exert their opinion on the final ranking by prioritizing the criteria. Constraints posed by the need to assign each ranking to only one alternative are removed in this model. Global Criterion method can be used to solve the resulting (MOLP) problem, too. In addition, when adding new criteria to the problem, as many new objective functions are added to, hence, there will be no need to recalculate the weight of criteria and do the calculations required in the LINEAR assignment method. Calculating Spearman's Correlation Coefficient shows that the proposed model in comparison with Classic LINEAR Assignment method is more consistent with the most commonly used multi-criteria decision-making methods: TOPSIS, VIKOR, and MOORA.

Background and Objectives: LINEAR PROGRAMMING (LP) is an easy way to optimize diet cost in which nutritional requirements are also met. Although LP was used for children, it has not been run to design a diet for patients with renal diseases. The aim of this study was to design a cost-optimized meal by LP for patients with kidney disease in which essential nutritional requirements are met.Materials and Methods: For this cross-sectional study, two combinatorial meals were considered, and their ingredients were chosen from the restaurant recipes. Minimum and maximum amounts for each ingredient were determined. The price of ingredients was gathered from authentic stores selling foods with approved price. The requirements for energy, protein, sodium, potassium and phosphorus expected to meet by this meal were also determined. The LP was run by Microsoft Excel. Optimized cost was calculated in while essential nutritional requirements were also met.Results: The amounts determined by LP were between the min and max, and all needs were met by the designed meals. Diet cost was also optimized.Conclusion: LP could be used to design low-cost meals for patients with kidney disease.

In this study, we discussed a fuzzy PROGRAMMING approach to bi-level LINEAR PROGRAMMING problems and their application. Bi-level LINEAR PROGRAMMING is characterized as mathematical PROGRAMMING to solve decentralized problems with two decision-makers in the hierarchal organization. They become more important for the contemporary decentralized organization where each unit seeks to optimize its own objective. In addition to this, we have considered Bi-Level LINEAR PROGRAMMING (BLPP) and applied the Fuzzy Mathematical PROGRAMMING (FMP) approach to get the solution of the system. We have suggested the FMP method for the minimization of the objectives in terms of the LINEAR membership functions. FMP is a supervised search procedure (supervised by the upper Decision Maker (DM)). The upper-level decision-maker provides the preferred values of decision variables under his control (to enable the lower level DM to search for his optimum in a wider feasible space) and the bounds of his objective function (to direct the lower level DM to search for his solutions in the right direction).