Modeling and Analysis of Supply Chain Inventory Model with two-warehouses and Economic Load Dispatch Problem Using Genetic Algorithm

Abstract In this paper a deterministic supply chain inventory model has been developed for deteriorating items having a ramp type demand with the effects of inflation with two-storage facilities and Economic Load Dispatch Problem Using Genetic Algorithm.. The owned warehouse (OW) has a fixed capacity of W units; the rented warehouse (RW) has unlimited capacity. Here, we assumed that the inventory holding cost in RW is higher than those in OW and Economic Load Dispatch Problem Using Genetic Algorithm. Shortages in inventory are allowed and partially backlogged and it is assumed that the inventory deteriorates over time at a variable deterioration rate and Economic Load Dispatch Problem Using Genetic Algorithm. The effect of inflation has also been considered for various costs associated with the supply chain inventory system. Numerical example is also used to study the behavior of the model and Economic Load Dispatch Problem Using Genetic Algorithm. Cost minimization technique is used to get the expressions for total cost and other parameters and Economic Load Dispatch Problem Using Genetic Algorithm.

point approach by traditional optimization methods. This means that GA processes a number of designs at the same time. As we have seen earlier to improve the search direction in traditional optimization methods transition rules are used and they are deterministic in nature but GA uses randomized operators. Random operators improve the search space in an adaptive manner. echelon supply chain"(We try to find the optimal sales quantity by maximizing profit, given as a nonlinear and non-convex objective function. For such complicated combinatorial optimization problems, exact algorithms and optimization commercial software such as LINGO are inefficient, especially on practical-size problems). Guchhait et. al (2013) extended "Two storage inventory model of a deteriorating item with variable demand under partial credit period" (The supplier also offers a partial permissible delay in payment even if the order quantity is less than the fixed ordered label. For display of goods, retailer has one warehouse of finite capacity at the heart of the market place and another warehouse of infinite capacity (that means capacity of second warehouse is sufficiently large) situated outside the market but near to first warehouse. Units are continuously transferred from second warehouse to first and sold from first warehouse. Combining the features of Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) a hybrid heuristic (named Particle Swarm-Genetic Algorithm (PSGA)) is developed and used to find solution of the proposed model). Bera et. al (2012) introducing "Inventory model with fuzzy lead-time and dynamic demand over finite time horizon using a multiobjective genetic algorithm" (A realistic inventory problem with an infinite rate of replenishment over a prescribed finite but imprecise time horizon is formulated considering time dependent ramp type demand, which increases with time. Lead time is also assumed as fuzzy in nature. Shortages are allowed and backlogged partially. Two models are considered depending upon the ordering policies of the decision maker). Wang et. al (2011) developed "Location and allocation decisions in a two-echelon supply chain with stochastic demand -A genetic-algorithm based solution" (Decisions include locating a number of factories among a finite set of potential sites and allocating task assignment between factories and marketplaces to maximize profit). Kannan et. al (2010) extended "A genetic algorithm approach for solving a closed loop supply chain model: A case of battery recycling" (In order to overcome this issue, it is necessary to setup a logistics network for arising goods flow from end users to manufacturers. In this study, the optimum usage of secondary lead recovered from the spent lead-acid batteries for producing new battery is presented). Yun et. al (2009) extended "Hybrid genetic algorithm with adaptive local search scheme for solving multistage-based supply chain problems" (The optimal design of supply chain (SC) is a difficult task, if it is composed of the complicated multistage structures with component plants, assembly plants, distribution centres, retail stores and so on. It is mainly because that the multistage-based SC with complicated routes may not be solved using conventional optimization methods). Farahani et. al (2008) introducing "A genetic algorithm to optimize the total cost and service level for just-intime distribution in a supply chain" (A bi-objective model is set up for the distribution network of a threeechelon supply chain, with two objective functions: minimizing costs, and minimizing the sum of backorders and surpluses of products in all periods). Nachiappan et. al (2007) extended "A genetic algorithm for optimal operating parameters of VMI system in a two-echelon supply chain" (The operational parameters to the above model are: sales quantity and sales price that determine the channel profit of the supply chain, and contract price between the vendor and the buyer, which depends upon the understanding between the partners on their revenue sharing). Altiparmak et. al (2006) developed "A genetic algorithm approach for multi-objective optimization of supply chain networks" (Supply chain network (SCN) design is to provide an optimal platform for efficient and effective supply chain management. It is an important and strategic operations management problem in supply chain management, and usually involves multiple and conflicting objectives such as cost, service level, resource utilization, etc). Maiti et. al (2006) introducing "An application of real-coded genetic algorithm (RCGA) for mixed integer non-linear programming in two-storage multi-item inventory model with discount policy" (This GA is based on Roulette wheel selection, whole arithmetic crossover and non-uniform mutation. Here, mutation is carried out for the fine-tuning capabilities of the system by non-uniform operator whose action depends on the age of the population. This methodology has been applied in solving multiple price break structure and implemented for multi-item deterministic inventory control system having two separate storage facilities (owned and rented warehouse) due to limited capacity of the existing storage (owned warehouse). Also, demand rate is a linear function of selling price, time and non-linearly on the frequency of advertisement. The model is formulated with infinite replenishment and shortages are not allowed. The stocks of rented warehouse (RW) are transported to the owned warehouse (OW) in bulk-release rule). Chan et. al (2003) introducing "Solving the multi-buyer joint replenishment problem with a modified genetic algorithm" (The joint replenishment problem (JRP) is a multi-item inventory problem. The objective is to develop inventory policies that minimize the total costs (comprised of holding cost and setup cost) over the planning horizon). Mondal et. al (2003) extended "Multi-item fuzzy EOQ models using genetic algorithm" (It uses genetic algorithms (GAs) with mutation and whole arithmetic crossover. Here, mutation is carried out along the weighted gradient direction using the random step lengths based on Erlang and Chi-square distributions. These methodologies have been applied to solve multi-item fuzzy EOQ models under fuzzy objective goal of cost minimization and imprecise constraints on warehouse space and number of production runs with crisp/imprecise inventory costs). Xie, et. al (2002) extended "Heuristic genetic algorithms for general capacitated lot-sizing problems" (The lot-sizing problems address the issue of determining the production lot-sizes of various items appearing in consecutive production stages over a given finite planning horizon. In general capacitated lot-sizing problems, the product structure can be a general acyclic network, the capacity constraints can be very sophisticated, and all the known parameters can be time-varying). Disney et. al (2000) developed "Genetic algorithm optimisation of a class of inventory control systems" (Benchmark performance characteristics. Five are considered herein and include inventory recovery to "shock" demands; in-built filtering capability; robustness to production lead-time variations; robustness to pipeline level information fidelity; and systems selectivity. A genetic algorithm for optimising system performance, via these five vectors is described). Padhy and Simon (2015) soft computing with matlab programming.

Assumptions and Notations
In developing the mathematical model of the inventory system the following assumptions are being made:

1.
A single item is considered over a prescribed period T units of time. 2. The demand rate D(t) at time t is deterministic and taken as a ramp type function of time i.e.
The replenishment rate is infinite and lead-time is zero. 4. When the demand for goods is more than the supply. Shortages will occur. Customers encountering shortages will either wait for the vender to reorder (backlogging cost involved) or go to other vendors (lost sales cost involved). In this model shortages are allowed and the backlogging rate is exp(- t), when inventory is in shortage. The backlogging parameter  is a positive constant.

5.
The variable rate of deterioration in both warehouse is taken as θ(t) = θt. Where 0< θ << 1 and only applied to on hand inventory.

6.
No replacement or repair of deteriorated items is made during a given cycle. 7. The owned warehouse (OW) has a fixed capacity of W units; the rented warehouse (RW) has unlimited capacity.
8. The goods of OW are consumed only after consuming the goods kept in RW. In addition, the following notations are used throughout this paper: The inventory level in OW at any time t. I r (t) The inventory level in RW at any time t. W The capacity of the own warehouse. Q The ordering quantity per cycle. T Planning horizon. r Inflation rate. C 1 The holding cost per unit per unit time in OW. C 2 The holding cost per unit per unit time in RW. where C 1 < C 2 C d The deterioration cost per unit. C 3 The shortage cost per unit per unit time. C 4 The opportunity cost due to lost sales.

C
The replenishment cost per order. P R Production rate which is taken as demand dependent i.e. P R = ∏D(t) R R Retailer rate which is taken as demand dependent i.e. R R = ε D(t) D R Distribution rate which is taken as demand dependent i.e. D R =δ D(t) TC 1 Transportation Cost of Manufacturer to between warehouses TC 2 Transportation Cost of warehouses to between Distribution centers The inventory levels at OW are governed by the following differential equations:

Two-warehouses inventory Model
with the boundary conditions, I 0 (0) =W and I(t 1 ) = 0 (4) The solutions of equations (1), (2) and (3) are given by 2 The inventory level at RW is governed by the following differential equations: With the boundary condition I r (0) = 0, the solution of the equation (8) is Due to continuity of I o (t) at point t = , it follows from equations (5) and (6), one has 3 3 2 2 2 2 1 1 The total average cost consists of following elements:

(i)
Ordering cost per cycle = C (11) (ii) Holding cost per cycle (C HO ) in OW (iv) Cost of deteriorated units per cycle (C D )

I (t)e dt θ t I (t)e dt θ t I (t)e dt
Shortage cost per cycle (C S ) Transportation Cost (Manufacturer to between warehouses) TC 1 = T 1 D(t)

Total Supply Chain inventory cost = Production Rate + Transportation Cost + Two-warehouses inventory + Transportation Cost + Distribution Rate + Retailer Rate
AC e δ e e r e ( δ r)e δr (δ r) C Ae e (δ r) re e (δ r) re r(δ r)

Proposed And Economic Load Dispatch Problem Using Genetic Algorithm
The objective is to find the optimal solution so that the minimum fuel cost is obtained subject to certain equality and inequality constraints. The problem may be expressed as a function which consists of the cost function and the constraints. In this work equality constraint reflects real power balance and the inequality constraint reflects the limit of real power generation.

Φ Φ Ψ
Where Φ Ψ is the fuel cost function of generating unit I and is the generation output of unit I in MW Subject to: a. Power balance constraints is given as follows Where Ψ is the total real power demand in MW b. Generating capacity constraints is given as follows Ψ Ψ Ψ for i =1, 2,…………..N Where Ψ and Ψ are the minimum and maximum output generation of unit i. The fuel cost function considering valve-point effect of the generating unit is given as follows , are the fuel cot coefficients of unit i, and and are the fuel cost coefficients of unit I with valve-point effect.
Step-by-step procedure of GA applied to ELD Problem 1. Generate the initial population of generating powers randomly. 2. Compute the total production cost of the generating power subject to the constraints in equation a. Power balance constraints is given as follows Where Ψ is the total real power demand in MW b. Generating capacity constraints is given as follows Ψ Ψ Ψ for i =1, 2,………….. The aim of this section is to understand the application of both Binary GA and Continuous GA for economic dispatching of generating power in a power system satisfying the power balance constraint for system demand and total generating power as well as the generating power constraints for all units. Therefore a simple three generating unit test system is considered and the details of the test system are given in

Conclusion
This study incorporates some realistic features that are likely to be associated with the inventory of any material. Decay (deterioration) overtime for any material product and occurrence of shortages in inventory are natural phenomenon in real situations and Economic Load Dispatch Problem Using Genetic Algorithm. supply chain inventory shortages are allowed in the model. In many cases customers are conditioned to a shipping delay, and may be willing to wait for a short time in order to get their first choice. Generally speaking, the length of the waiting time for the next replenishment is the main factor for deciding whether the backlogging will be accepted or not. The willingness of a customer to wait for backlogging during a shortage period declines with the length of the waiting time and Economic Load Dispatch Problem Using Genetic Algorithm. Thus, supply chain inventory shortages are allowed and partially backordered in the present chapter and the backlogging rate is considered as a decreasing function of the waiting time for the next replenishment. Demand rate is taken as exponential ramp type function of time, in which demand decreases exponentially for the some initial period and becomes steady later on. Since most decision makers think that the inflation does not have significant influence on the supply chain inventory policy, the effects of inflation are not considered in some inventory models and Economic Load Dispatch Problem Using Genetic Algorithm. However, from a financial point of view, an inventory represents a capital investment and must complete with other assets for a firm's limited capital funds. Thus, it is necessary to consider the effects of inflation on the supply chain inventory system. Therefore, this concept is also taken in this model. From the numerical illustration of the model, it is observed that the period in which inventory holds increases with the increment in backlogging and ramp parameters while inventory period decreases with the increment in deterioration and inflation parameters. Initial inventory level decreases with the increment in deterioration, inflation and ramp parameters while inventory level increases with the increment in backlogging parameter. The total average cost of the system goes on increasing with the increment in the backlogging and deterioration parameters while it decreases with the increment in inflation and ramp parameters and Economic Load Dispatch Problem Using Genetic Algorithm.. The proposed model can be further extended in several ways. For example, we could extend this deterministic model in to stochastic model. Also, we could extend the model to incorporate some more realistic features, such as quantity discount or the unit purchase cost, the inventory holding cost and others can also taken fluctuating with time and Economic Load Dispatch Problem Using Genetic Algorithm.