Multi-choice Knapsack Problem is a branch of regular Knapsack Problem where the objects are classified in different classes and each class has one and only one representative in final solution. Although it is assumed that each object belongs to just one class, sometimes this assumption is not valid in real Problems. In this case an object may belong to the several classes. In fuzzy multi-choice Knapsack Problem (FMCKP), fuzzy sets are applied to show that each object is a member of each class with a membership grade. In this paper we proposed two new models for fuzzy multi-choice Knapsack Problem. These models have a fuzzy constraint and so we applied the fuzzy linear programming (FLP) approach to solve them. Typical examples show the capability and efficiency of the proposed models.

Multi-choice Knapsack Problem is a branch of regular Knapsack Problem where the objects are classified in different classes and each class has one and only one representative in final solution. Although it is assumed that each object belongs to just one class, sometimes this assumption is not be respected in real Problems. In this case an object may be belongs to the several classes. We have used the fuzzy sets concept to show that each object may be a member of each class with a membership grade. We called it a fuzzy multi-choice Knapsack Problem (FMCKP). We consider two objectives for the Problem: The first one is to maximize the profit of selected combination and the second one is to minimize the total differences of the deviation of ideal relation for each class. Developing such a bi-objective model, we applied the fuzzy linear programming (FLP) approach to solve it. Experiments on a lot of typical examples show the capability and efficiency of the proposed model.

Selecting an optimum combination of items from a set of items, known as Knapsack Problems, is an important issue in the decision making domain. In this paper, a new approach is developed to solve a Multiple Attribute Knapsack Problem (MAKP) in which each combination of items is evaluated using some quantitative and qualitative attributes. The assumed qualitative attributes cannot be measured by a mathematical formulation but by a DM/expert. In this paper, a Genetic Algorithm (GA) model has been developed to generate different combinations as the sequential population of the GA model. To rate the qualitative attributes for each chromosome (or combination) in the population, a Neural Network (NN) model has been developed. The ratings (or scores) resulted from quantitative attributes (by NN) and qualitative attributes (by mathematical formulation) for each chromosome form a row of a decision matrix. Having the decision matrix and known weights of attributes, the combinations in each population are ranked by applying a MADM model. The ranks obtained for each chromosome shows the fitness of that chromosome. Using the GA model, the best combination is achieved. The results of conducted experiments show the capability of the proposed approach to deal with MAKP Problems.

The vehicle routing Problem is one of the most well-known optimization Problems, which aims to design an optimum set of routes with the lowest cost for servicing the customers in a way that is consistent with the existing constraints. The wide practical application and scope of this Problem has attracted much attention from researchers. But in return, the severity of solving has created difficulties and increased the need for heuristic and meta-heuristic solutions. This research represents a greedy heuristic method based on first categorizing then routing methods, to solve the capacitated vehicle routing Problem (CVRP) using the capabilities of Problem reduction to the Knapsack Problem. The advantages of this method include the consideration of effective criteria such as distance between customers, distance between customers to depot and demand of points in decision making, decent speed and quality of solution and the ability to utilize the benefits of reduction. Standard samples from CVRPLIB were used to evaluate the results and comparing them.

In this paper, a new approximate method is presented to solve multi-dimensional Knapsack Problem by using multiple criteria decision making (MCDM). In order to, initially efficiency values for every item is calculated then items are ranked by using MCDM. Finally, items are selected in according to their rank.

In this paper, the researchers have proposed a multi-dimensional Knapsack model for project capital budgeting Problem in uncertain situation which has been modeled through fuzzy sets. The optimistic and pessimistic situations were considered and associated deterministic models were yielded. Numerical example has been supplied toillustrate the performance of proposed model. The results were promising in the sense of helping the decision makers.

A special class of the Knapsack Problem is called the separable nonlinear Knapsack Problem. This Problem has received considerable attention recently because of its numerous applications. Dynamic programming is one of the basic approaches for solving this Problem. Unfortunately, the size of state-pace will dramatically increase and cause the dimensionality Problem. In this paper, an efficient algorithm is developed to find surrogate multipliers in each stage of dynamic programming in order to transform the original Problem to a single constraint Problem called surrogate Problem. The upper and lower bounds obtained by solving the surrogate Problem can eliminate a large number of state variables in dynamic programming and extremely reduce the duality gap according to our computational results.

This paper develops an approach to solve 0-1 multiple attribute Knapsack Problem (MAKP) in which some objectives are qualitative. The approach consists of two modules. The first module is a fuzzy expert system that evaluates the qualitative objectives. Acting as a framework, the second module generates the different combinations and finds the (near) optimal solution using a genetic algorithm model. The results of conducted experiments show the capability of proposed approach to deal with MAKP Problems.

Many portfolio optimization Problems deal with allocation of assets which carry a relatively high market price. Therefore, it is necessary to determine the integer value of assets when we deal with portfolio optimization. In addition, one of the main concerns with most portfolio optimization is associated with the type of constraints considered in different models. In many cases, the resulted Problem formulations do not yield in practical solutions. Therefore, it is necessary to apply some managerial decisions in order to make the results more practical. This paper presents a portfolio optimization based on an improved Knapsack Problem with the cardinality, floor and ceiling, budget, class, class limit and pre-assignment constraints for asset allocation. To handle the uncertainty associated with different parameters of the proposed model, we use robust optimization techniques. The model is also applied using some realistic data from US stock market. Genetic algorithm is also provided to solve the Problem for some instances.