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The fixed-charge transportation problem (FCTP) is still a challenging problem in the field of mathematical programming. In this paper, we consider fixed-charge transportation problem with logarithmic objective function. In the absence of any suitable algorithm to obtain the solution of this type of nonlinear transportation problem, we discuss the advantage of polynomial approximation. There exists a major difference between the two problems that the variables in the polynomial transportation problem have no upper bound but in the logarithmic transportation problem they are bounded. Using the expansion of logarithm we show the resemblance between the structural behaviour of linear and fixed-charge transportation problems. We illustrate a numerical example in support of the developed method.

AbstractIn this paper, we introduce generalized Fibonacci n-step polynomials. Based on generalized Fibonacci n-step polynomials, we define a new class of square matrix Mh, n(x) of order n. Thereby, we state a new coding theory called generalized Fibonacci n-step polynomial coding theory. Also we obtain the relations among the code matrix elements. Next, we discuss on the correct ability of this method. It is shown that, for n= 2, the correct ability of this method is 93.33% whereas for n= 3, the correct ability of this method is 99.80%. The interesting part of this coding/decoding method is that the correct ability does not depend on the polynomial coefficients excepting the value of the nth coefficient hn(x) = 1 and increases as nincreases.

Use of sales promotion in the short-term to increase sales by revenues and market share for slow-moving items is common in the supply chain, though in such cases performances of chain members are suboptimal. On the contrary, a growing body of literature suggests that a high degree of integration is needed to optimise performances of all supply chain partners simultaneously. To address supply chain integration, when sales promotion is used, we consider a three-echelon supply chain. Demand depends on the selling price by the retailer and individual promotional mechanisms are chosen by channel members. We study game theoretical models to examine impacts of two promotional policies, e.g., mail-in-rebate (MIR) and downward direct discount (DDD) on supply chain coordination. Effectiveness and flexibility of MIR, MIR coupled with revenue sharing (RS) and DDD as coordination contracts are discussed analytically. We show that in decentralised decision making all channel members prefer manufacturers MIR and discard the possibility of distributers' MIR. Although to cut out channel conflict all chain members cooperate actively by complementing one another through contribution to invariant discount that is provided to customers, still, distributors and manufacturers prefer MIR and RS to DDD, whereas retailers prefers the reverse.

In this paper, we consider generalised bi-criterion capacitated transportation problem. A solution procedure is presented to find the optimum transportation cost. Also, an algorithm is presented to find the optimum time-cost trade of pair in a generalised bi-criterion transportation problem giving the same priority to both time and cost. The algorithm identifies efficient time-cost trade-off pairs. Among the trade-off pairs we identify ideal solution. Then we reach the optimum trade-off pair having the minimum D1-distance from the ideal solution. Finally, a numerical example is illustrated to support the algorithm.

The dual-channel supply chain model has become increasingly popular in the industry and describes a scenario in which a firm, in addition to selling through the traditional supply chain of manufacturer and retailer, opens a direct channel to the customer through Internet sales. However, in the current global business environment, corporate social responsibility (CSR) is a determining factor of choices of the customers. Based on the above important factors, this article introduces a corporate social responsibility two-echelon dual-channel supply chain. In addition to operating an online channel, the manufacturer intends to increase stakeholders' welfare by exhibiting CSR. The pricing decisions for both the cases of the decentralized and centralized scenarios are studied analytically as well as numerically. The paper also examines the effect of the degree of concern of the manufacturer regarding CSR on product compatibility and discusses feasibility of the successful operation of a dual-channel supply chain. Finally, channel coordination through all unit quantity discounts with the agreement of a franchise fee and surplus profit division through bargaining is discussed analytically.

Abstract: Maintaining load balance, in a distributed network is a complex task. It was previously assumed that the communication complexity would be for any number of nodes u which was later modified as where the number of nodes and k is prime. In this paper, we modify the communication complexity where the number of nodes and k is prime. Thereby we show that by increasing the round of message interchange by , the communication complexity is . Also by increasing infinitely the round of message interchange, the communication complexity can be made . [Copyright &y& Elsevier]

Let $\bar{d}(C)$denote the average Hamming distance of a binary constant weight code C. Aiis the distance distribution of Cand Bjis the transform quantity of Ai. Let β(n, M, w)denote the minimum average Hamming distance of a binary constant weight code with length n, size M, and weight w. In this paper, we have established the relation among Bj, $\bar{d}(C)$and Ai. We also study the problem of determining β(n, M, w)and bounds of β(n, M, w)for different values of M.

In day-to-day time-based competition, the unit selling price of a hightech product declines significantly over its short life cycle. In this paper, we introduce dynamic pricing to traditional Economic production quantity (EPQ) models for time and price sensitive products. From manufacturer/producer point of view, the model is developed to maximise total profit. We prove that the total profit is a concave function of selling price within fixed planning horizon. A solution procedure is presented to determine optimal prices, optimal number of production cycles, optimal lot size and optimal profit simultaneously. We illustrate the model with numerical examples and sensitivity analysis is also performed.

Abstract: In this paper, we introduce a new Fibonacci matrix for the m-extension of the Fibonacci p-numbers where p () is integer and m (>0). Thereby, we discuss various properties of matrix and the coding theory followed from the matrix. In this paper, we establish the relations among the code elements for all values of p (nonnegative integer) and . We also show that the relation, among the code matrix elements for all values of p and , coincides with the relation among the code matrix elements for all values of p [Basu M, Prasad B. The generalized relations among the code elements for Fibonacci coding theory. Chaos, Solitons and Fractals (2008). doi: 10.1016/j.chaos.2008.09.030]. In general, correct ability of the method increases as p increases but it is independent of m. [Copyright &y& Elsevier]

Abstract: We have considered a class of square Fibonacci matrix of order (p +1) whose elements are based on the Fibonacci p numbers with determinant equal to +1 or −1. There is a relation between Fibonacci numbers with initial terms which is known as cassini formula. Fibonacci series and the golden mean plays a very important role in the construction of a relatively new space–time theory, which is known as E-infinity theory. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. There already exists a relation between the code matrix elements for the case p =1 [Stakhov AP. Fibonacci matrices, a generalization of the cassini formula and a new coding theory. Chaos, Solitons and Fractals 2006;30:56–66.]. In this paper, we have established generalized relations among the code matrix elements for all values of p. For p =2, the correct ability of the method is 99.80%. In general, correct ability of the method increases as p increases. [Copyright &y& Elsevier]

AbstractA binary linear code in with dimension kand minimum distance dis called an [n, k, d] code. For each positive integer m, the first order Reed-Muller code R(1, m) is a linear code [2m, m+1, 2m–1]. When m=3, an extremal doubly-even self-dual code of length 23=8 exists. In this paper, it is shown that the first order Reed-Muller code R(1, m) holds a 3-(2m, 2m–1, 2m–2–1) design for m≥3. It is also shown that Steiner system holds for m=3.

Based upon the Boussinesq approximation, an initial value investigation is made of the axisymmetric free surface flows generated in an inviscid rotating stratified liquid of infinite depth by the prescribed free surface disturbance. The asymptotic analysis of the integral solution is carried out by the stationary phase method to describe the solution for large time and large distance from the source of the disturbance. The asymptotic solution is found to consist of the classical free surface gravity waves and the internal-inertial waves.