A supply chain of a coastal biomass incorporating fuzzy deterioration and freshness under dynamic unit price

Abstract

In this study, a wholesaler–retailer supply chain (SC) of a coastal biomass in an imprecise environment is developed incorporating the effect of the freshness of the units on the consumers’ demand under promotional cost sharing. Due to the imprecise nature of the rate of deterioration, the deterioration rate and the expiration time of the units are considered fuzzy numbers. As the freshness of the units depends on the expiration time, its nature is also imprecise and decreases with time. So, the seller normally decreases the unit selling price with time to boost the demand for biomass. The demand for the item is influenced by the unit selling price and freshness of the units, and hence its nature is also fuzzy. Due to the impreciseness of the rate of demand and deterioration, the model is mathematically formulated using fuzzy differential equations and fuzzy Riemann integration. The credibility measure of the fuzzy objective(average profit) with respect to a properly defined fuzzy goal is optimized for determining the marketing decision. Also, fuzzy simulation algorithms are presented for determining the said credibility measure. The models are studied in both the non-coordination scenario and the coordination scenario. In the non-coordination scenario, the retailer is the leader and the wholesaler is the follower, i.e., the retailer determines the marketing decision independently, and following the decision of the retailer the wholesaler determines his/her marketing strategy. In the coordination scenario, they make a joint decision for better performance by sharing the promotional cost incurred by the retailer by reducing the selling price to boost the demand. The model is illustrated using a set of test data and it is established that the individual profits as well as the channel profit improve if the decision is made jointly. In a particular case, the model is solved for the rough estimations of some problem parameters and an approach is proposed for the mathematical formulation and decision-making. The trust measure on a rough goal is defined for the formulation of the model and corresponding rough simulation algorithm is presented for the determination of the same. As the simulation approaches are used for the decision-making in imprecise environments, the basic particle swarm optimization algorithm is slightly modified, implemented, tested, and used for the determination of the marketing decision for the different models.

Appendix A

Particle swarm optimization: PSO is a heuristic approach used to search for an optimal or some near-optimal solutions to a single objective optimization problem(SOOP) involving continuous variables. To discuss the method, the following single objective continuous optimization problem(SOCOP) is considered:

$$\begin{aligned} \text{ Minimize }{} & {} f(X)\nonumber \\ \text{ Subject } \text{ to }{} & {} X\in S, \end{aligned}$$

(74)

where $X=(x_1,\,x_2,\,\ldots ,\,x_n)$ is the decision vector and S is the search space.

PSO was developed by Kennedy and Eberhart in 1995 by mimicking the natural behavior of folk of birds searching for their food sources (Kennedy and Eberhart 1995). The birds in a swarm search for their food sources depending upon their own experiences and the best experience among the birds. The same phenomenon is mimicked to create a PSO for solving a SOCOP like (74). In the algorithm, search area of food for the birds is compared to the search area of the solution for the target SOCOP under consideration. The position of a bird in the search space is compared with the position of a solution in the feasible region in the current iteration. The position of food in the search space of birds is compared with the position of an optimal solution in the search space of the problem.

A PSO normally starts with a set of solutions (called swarm) of the SOCOP under consideration. The birds (i.e., solution) are flown through a multi-dimensional search space, where the position of each bird is adjusted according to its own experience and experience of the best position found by the swarm so far. Let, $X_{i}(t)$ and $V_{i}(t)$ be the respective position vector and velocity vector of the i-th particle in the t-th iteration or generation. $X_{besti}(t)$ is the best position of the i-th particle in the t-th generation, i.e., at $X_{besti}(t)$, the i-th particle gives the best fitness. Also let, $X_{gbest}(t)$ be the best position vector of all particles at the current generation(t-th generation). In the $(t+1)$th generation, the position vector and velocity vector of the particles have been changed to $X_{i}(t+1)$ and $V_{i}(t+1)$ using the following rules:

$$\begin{aligned} \left. \begin{array}{l} V_{i}(t+1)\,=\omega \,V_{i}(t)\,+\,\mu _{1}\,r_{1}\,\{X_{besti}(t)-X_{i}(t)\}\\ \qquad \quad +\,\mu _{2}\,r_{2}\,\{X_{gbest}(t)-X_{i}(t)\},\\ X_{i}(t+1)\,=\,X_{i}(t)\,+\,V_{i}(t+1). \end{array}\right\} \end{aligned}$$

(75)

In (75) $r_{1},\, r_{2}$ are uniformly distributed over [0,1], and $\omega ,\, \mu _1,\,\mu _2$ are three control parameters. The implemented algorithm is presented here. In the algorithm spv and epv represents the respective lower and upper bounds of the position vectors of the potential solutions. svv and evv denote the respective lower and upper bound for the velocity vectors. The value of the objective function due to the solution $X_i(t)$ is denoted by $f(X_i(t))$. avgfit denotes the average of the function values of all the particles in the population, maxit denotes the number of maximum iterations, and $p_s$ denotes the population size. Though basic PSO was developed for unconstrained optimization, it is here slightly modified for constrained optimization using a procedure named Check_constraint$(X_i(t))$. The Check_constraint$(X_i(t))$ function returns 1 if $(X_i(t))$ satisfies the constraints of the problem otherwise returns 0. Moreover, a termination condition using convergence criteria is also incorporated in this modified version.