Determination of withdrawal schedule in single-species cultivation via genetic algorithm
Introduction
Since the development of classical EOQ model by Harris [5], a lot of research work on inventory control system (cf. Naddor [19], Whitin [21], etc.) is available in the literature. These conventional inventories are related with non-living items in industry and business sector like raw materials, finished goods, vegetables, food-grains, etc. But, presently genetic research in life-science has created a revolution and as a result, inventory of livestock are built-up for business. As a part of it, now-a-days, high breed fish cultivation for a short period with different type of species in bhery, fields, ponds, etc. are very popular in waterlogged countries like India, Bangladesh, etc. The cultivation of the species can be formed as an inventory control problem and solved easily for global optimum using the evolutionary methods such as genetic algorithm (GA).
In this area, the most of the authors have derived their models out of the work of Clark [1] who found the optimal equilibrium policy for joint harvesting of two independent species. Clark [1] assumed that each species follows a logistic growth law in absence of other species and its harvest rate is proportional to both its stock level and harvesting effort. This analysis has been extended by several authors such as Mesterton [7], [8], Wilen and Brown [22], etc. Kapur [6] discussed simple Lotka–Volterra model in case of two species population, where logistic growth of one species in the absence of other is proportional to the amount of biomass, but every species has an self-inhibiting effect in the growth rate and have an effect in the growth rate for the presence of other species. Recently Mondal et al. [16], [17] developed inventory problems of ameliorating items for price dependent demand. They uses perturbation technique to find out optimal strategies. Maiti and Maiti [10], [11] developed two two-species cultivation problems with price and biomass dependent catch rates and solved via Simulated Annealing techniques. They use numerical integration techniques to solve system of non-linear differential equations of the problem. Maiti and Maiti [14], [15] used the same technique to solve multi-item shelf-space allocation problems using genetic algorithm. Till now, none has considered the cultivation of a single species fish for finite time period and formulated it as a profit maximizing inventory model taking cost of amelioration and cultivation, etc. into consideration.
Again, the cultivation of biomass with the above realistic assumptions is so complex and non-linear that it is impossible to get the optimum solution via analytical approaches and thus we forced to apply numerical optimization techniques for approximate optimum solution. There are some inherent difficulties in the traditional non-linear numerical optimization methods used for solution of these types of problems. Normally these methods:
- (i)
are initial solution dependent,
- (ii)
get stuck to a sub-optimal solution,
- (iii)
are not efficient in handling problems having discrete variables,
- (iv)
cannot be efficiently used on parallel machines and
- (v)
are not universal rather specific problem dependent.
To overcome these difficulties, recently genetic algorithms (GA), one of the soft computing methods is used as optimization techniques for decision-making problems. These (cf. Goldberg [4] and Michalewicz [18]) are adaptive computational procedures modeled on the mechanics of natural genesis system. They exploit the historical information to speculate on new offspring with expected improved performance (cf. Goldberg [4], Pal et al. [20]). These are executed iteratively on a set of coded solutions (called population) with three operations – selection/reproduction, crossover and mutation. One iteration of these three operators is known as a generation in the parlance of GAs. Since a GA works simultaneously on a set of coded solutions, it has very little chance to get stuck at local optima. Here, the resolution of the possible search space is increased by operating on the possible solutions. Further this search space need not be continuous. Recently GAs have been applied in different areas like neural network [4], scheduling [2], numerical optimization [10], pattern recognition [3], etc. There are very few papers where GAs have been applied in the field of inventory control system. Recently Mandal and Maiti [9], Maiti and Maiti [12], [13] developed some inventory problems using GA. Michalewicz [18] proposed a genetic algorithm named contractive mapping genetic algorithm (CMGA) and proved the asymptotic convergence of the algorithm by Banach fixed point theorem.
In this paper, for the first time, a cultivation model for a single species fishes in a bhery/pond over a fixed period of time is formulated and for its solution, a non-traditional optimization method is proposed. The rates of growth and decay of fishes are assumed to be stochastically governed by two parameter Weibull distributions. Also they have some self-inhibiting effect on growth rate, which are directly proportional to the squares of the amount of the species. Initially cultivation is commenced with some amount of fishes which is to be determined. The catch rate of the fishes is a decision variable when maximum demand is known. Also the intermediate withdrawal points are determined. Selling price of the biomass is linearly dependent on time and amount of biomass withdrawal, excess over the maximum demand during a period is sold at a reduced price. To solve this system, the evolutionary method, CMGA has been developed and implemented to find out the amount of initial stock of species so that the total proceeds out of the system is a maximum. The model has been illustrated through some numerical results. An interesting result is presented for different harvesting time periods and the result is verified taking time as a decision variable.
Section snippets
Genetic algorithm
Genetic algorithms are exhaustive search algorithms based on the mechanics of natural selection and genesis (crossover, mutation, etc.) and have been developed by Holland, his colleagues and his students at the University of Michigan (cf. Goldberg [4]).
In natural genesis, we know that chromosomes are the main carriers of hereditary information from parent to offspring and that genes, which present hereditary factors, are lined up on chromosomes. At the time of reproduction, crossover and
Assumptions and notations
The proposed mathematical model of harvesting the species in a bhery/pond over a fixed time period ‘T’ is developed under the following assumption and notations:
- (i)
The harvesting process involves only one species of fishes.
- (ii)
Replenishment of species are instantaneous.
- (iii)
The deterioration and amelioration occur when the item is effectively in bhery.
- (iv)
S is the total initial biomass.
- (v)
q(t) is the amount of biomass at time t.
- (vi)
If α, β be the parameters of the Weibull distribution, then it’s probability density
Mathematical formulation
Let q(t) denote the amount of biomasses at time t then the differential equation describing the instantaneous states of q(t) is given bywith initial conditions q(0) = S and q(ti) = q(ti − 0) − K for i = 1, 2, … , N − 1, where ti − 0 represents the instant just before the time ti.
The non-linear differential equation (1) cannot be solved analytically. For the conventional problems in biomathematics with infinite time, these equations are solved by perturbation technique
Solution methodology
Average profit Z(S, K, N) is optimized by CMGA process. The process is discussed in Section 5.2. To evaluate value of Z(S, K, N) for a fixed set of values of ‘S’, ‘K’, ‘N’, different Integrals of ‘Z’ are calculated numerically by Trapezoidal rule. To evaluate the integrals of Z numerically, q(t) at different values of t are obtained by numerically solving (1) via fourth-order Runge–Kutta method. The following algorithm is used for this purpose.
Numerical results
The proposed cultivation model is now illustrated for certain numerical data. The following common parametric values are assumed to determine the initial biomass size (S), withdrawal quantity at an intermediate point (K) and number of withdrawal points of biomasses at different scenarios to maximize the profit function (Z):
Conclusion
In this paper, single species harvesting policy of fishes in a bhery for a fixed period of time has been presented. It is to be noted that till now, the existing harvesting policy available in the literature has been formulated for the infinite time horizon. Now-a-days, with the availability of high-breed species of fish, these cultivations in the third world waterlogged countries like India, Bangladesh, etc. are very popular, economically beneficial and cultivated for the fixed time interval
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