Dynamic programming approach for fuzzy linear programming problems FLPs and its application to optimal resource allocation problems in education system

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.

The natural flow regime can sustain the ecological integrity of riverine ecosystems. Different reservoir operation polices differ in their effects on the degree of alteration of natural flow regimes. Dynamic programming plays an important role in developing operation policies. When using dynamic programming models to develop operation policies, the discrete number of storage states (DNSS), which is a key factor affecting the reservoirs operation policies, is always been determined based on computational efficiency and economic benefits. Little consideration has been given to the ecological disturbance caused by different DNSS‐based operation policies. To analyze the impact of DNSS, we built a deterministic dynamic programming model to explore the relationship among DNSS, the flow regime alteration (ecological disturbance), and the cumulative annual power generation (economic benefits) by setting a range of DNSS scenarios. We used three reservoirs with different storage coefficients (ratios of usable storage to annual average runoff) as examples and used the range of variability approach to assess the ecological disturbance under these scenarios. We compared the results with those of a stochastic dynamic programming (SDP) model and a Bayesian SDP (BSDP) model. We found that when DNSS is low, increasing DNSS improves economic benefits but causes a more severe ecological disturbance; when DNSS is high, increasing DNSS improves the economic benefits only slightly, without exacerbating the ecological disturbance; for a given DNSS, the BSDP model provides higher economic benefits than the SDP model and a similar disturbance of the riverine ecosystem; and larger reservoirs more often cause more severe disturbance of riverine ecosystems because monthly mean flows and annual extreme flows change more drastically. Our results will help to protect the riverine ecosystems and improve economic benefits if reservoir operation managers consider DNSS using dynamic programming models.

We present an approach for ordering fuzzy numbers, then we utilize it to suggest two models for solving the fuzzy linear programming (FLP) problem. In this research we present a new approach for ordering two fuzzy numbers and show some practical advantages of it over other methods. We also utilize the suggested approach to introduce two techniques to solve the FLP problem. In the first technique, we convert the fuzzy linear program into a crisp bicriteria linear program. In this new model the fuzzy objective function is replaced by two crisp objectives to be optimized, and each fuzzy constraint is replaced by two or three crisp constraints depending on the level of certainty /spl alpha/-whether /spl alpha/ has or has not been assigned. In the second technique we find the possibility distribution of the objective function over all possible feasible decisions directly.

In this paper, an efficient approach is presented to solve fully fuzzy linear fractional programming problems. In this approach, first by Alpha-cuts on the objective function and the constraints, the problem is transforming into a bi-objective linear programming problem, and then the resulting problem is converted into two crisp linear programming problems. A numerical example is presented to illustrate the proposed approach. The obtained results demonstrate the acceptable performance of the proposed solution approach.

CARE: Heuristics for two-stage multi-product inventory routing problems with replenishments

SummaryThe estimation of finite‐horizon discrete‐choice dynamic programming (DCDP) models is computationally expensive. This limits their realism and impedes verification and validation efforts. Keane and Wolpin (Review of Economics and Statistics, 1994, 76(4), 648–672) propose an interpolation method that ameliorates the computational burden but introduces approximation error. I describe their approach in detail, successfully recompute their original quality diagnostics, and provide some additional insights that underscore the trade‐off between computation time and the accuracy of estimation results.

Prescribing product upgrades, prices and production levels over time in a stochastic environment

In this paper, shortcomings and limitations of the existing methods for solving fuzzy linear programming (FLP) problems are pointed out. To overcome the limitations as well as to resolve the shortcomings, a new method is proposed for solving FLP problems. To show the advantage of the proposed method over existing methods, a FLP problem is solved by the existing methods as well as the proposed method, and the obtained results are compared.

The problem of fully fuzzy programming was one of the problems that many treatments have appeared for due to ambiguity and inaccuracy of its ambiguous coefficients. In this study, finding the optimal numerical solution for fully fuzzy linear fractional programming problems (FFLFPP) using the ranking functions was obtained. The proposed pentagonal ranking function (PPRF) multiplied by γ5 has been derived, as well as the proposed heptagonal ranking function (PHRF) multiplied by γ5 has been derived too. Two laws were obtained the first one was the novel ranking membership pentagonal and the second one was the heptagonal function then utilized to solve FFLFPP after converting the problems into FFLPP by utilizing the complementary method, then converting them to crisp linear programming problems and finding the optimal numerical solution for problems by using the proposed ranking functions. The paper included numerical example to illustrate the applied technique and the accuracy of the solution, and a comparison with available techniques were add the novelty of this study.

<abstract><p>The primary goal of optimization theory is to formulate solutions for real-life challenges that play a fundamental role in our daily lives. One of the most significant issues within this framework is the Linear Fractional Programming Problem (LFrPP). In practical situations, such as production planning and financial decision-making, it is often feasible to express objectives as a ratio of two distinct objectives. To enhance the efficacy of these problems in representing real-world scenarios, it is reasonable to utilize fuzzy sets for expressing variables and parameters. In this research, we have worked on the Fully Fuzzy Linear Fractional Linear Programming Problem (FFLFrLPP). In our approach to problem-solving, we simplified the intricate structure of the FFLFrLPP into a crisp Linear Programming Problem (LPP) while accommodating the inherent fuzziness. Notably, unlike literature, our proposed technique avoided variable transformation, which is highly competitive in addressing fuzzy-based problems. Our methodology also distinguishes itself from the literature in preserving fuzziness throughout the process, from problem formulation to solution. In this study, we conducted a rigorous evaluation of our proposed methodology by applying it to a selection of numerical examples and production problems sourced from the existing literature. Our findings revealed significant improvements in performance when compared to established solution approaches. Additionally, we presented comprehensive statistical analyses showcasing the robustness and effectiveness of our algorithms when addressing large-scale problem instances. This research underscores the innovative contributions of our methods to the field, further advancing the state-of-the-art in problem-solving techniques.</p></abstract>

The problem of designing fibre-optic networks for local-access telecommunications generates (at least) three non-trivial subproblems. In the first of these subproblems one must determine how many fibre-optic cables (fibres) are required at either end of a street. In the next subproblem a minimum-cost network must be designed to support the fibres. The network must also provide distinct paths from either end of the street to the central exchange(s). Finally, the fibre-optic cables must be placed in protective covers. These covers are available in a number of different sizes, allowing some flexibility when covering each section of the network. In this paper we describe a dynamic programming (DP) formulation for finding a minimum-cost (protective) covering for the network (the third of the subproblems). This problem is a generalised set covering problem with side constraints and is further complicated by the introduction of fixed and variable welding costs. The DP formulation selects covers along each arc (in the network), but cannot exactly model the fixed costs and so does not guarantee optimality. We also describe an integer programming (IP) formulation for assessing the quality of the DP solutions. The cost of the networks constructed by the IP model is less than those designed using the DP model, but the saving is not significant for the problems examined (less than 0.1%). This indicates that the DP model will generally give very good solutions. Furthermore, as the problem dimensions grow, DP gives significantly better solution times than IP.

A heuristic for scheduling problems, especially for scheduling farm operations

Cost benefit analysis of automatic lameness detection systems in dairy herds: A dynamic programming approach

Education system has importance in the development of any country. Without better education no country can make progress. The main objective of the study is to review the problems of education system in Pakistan. For this paper literature regarding problems of education system in Pakistan was critically analyzed. It was find out that major issues faced by our educational system were gender inequality, insufficient educational budget, outdated curriculum, poor examination system, Internal and external influence on education system, lack of quality teachers, political interference, The lack of faculty training and development, dropout ratio, lack of uniformity in education system, Education without Direction and failure to achieve millennium developmental goals. It was recommended that should take steps to decrease gender inequality. Sufficient budget should be allocated for educational purpose. Steps should be taken to minimize drop out. Internal and external interferences should be avoided to improve quality of education. Curriculum should be updated according to needs of people.