 A Soft Computing Techniques Application of an Inventory Model in Solving TwoWarehouses Using Cuckoo Search Algorithm
 Introduction
 Inventory Models with Two Warehouses
 Cuckoo Behavior and Lévy Flights
 Related Works
 Assumption and Notations
 Mathematical Formulation of Model and Analysis
 Cuckoo Search Algorithm
 Numerical Analysis
 Sensitivity Analysis
 Conclusions
 References
A Soft Computing Techniques Application of an Inventory Model in Solving TwoWarehouses Using Cuckoo Search Algorithm
Ajay Singh Yadav
SRM Institute of Science and Technology
Anupam Swami
Government Post Graduate College
Navin Ahlawat
SRM Institute of Science and Technology
Srishti Ahlawat
SRM Institute of Science and Technology
Introduction
Inventory Models with Two Warehouses
Classic inventory models are usually developed using a single warehouse system. In the past, researchers have done a lot of research on management and control. Warehouse management and control systematically respond to demand and supply chains. To do this, production units (product manufacturers and terraces), suppliers, and retailers must supply raw materials and finished products for demand and supply in the market and for consumers. With the conventional model, it is assumed that the demand and the cost of storage are constant and that the goods are delivered directly if needed as part of an endless revision policy. However, over time, many researchers assume that demand can change over time from other prices and factors, and operating costs can change over time and depend on other factors. Many models have been developed without bottlenecks by understanding some of the different requirements over time. For all models that consider declining demand due to inventory, it is assumed that inventory costs remain constant throughout the inventory cycle. Unlimited storage capacity is usually considered when testing an existing model. However, in hightraffic markets such as markets, corporate markets, etc., storage space for goods can be limited. In other cases, adequate storage can be carried out if the decision is made to take up a large quantity of paper. This could be due to the attractive volume discounts available, or if the cost of purchasing the item is higher than other storage costs, or if the demand for goods is too high, or if it is a seasonal product such as reduced production or if there are frequent supply issues. In this case, these items cannot be stored in an existing warehouse (your warehouse alone, abbreviated as OW). In order to save the writing width, additional warehouses (rented warehouses, abbreviated RW) need to be set up, which can be located a short distance W or short distances there, as there is no warehouse. Not close. Rent out
Cuckoo Behavior and Lévy Flights
Cuckoo cast birds not only because of the beautiful sounds they produce but also because of their aggressive breeding strategy. Some species, such as ani and guira cuckoo, lay eggs in community nests, although they may take eggs from other species to increase their chances of infecting their own eggs. Many species combat reproductive parasitism by laying eggs in other host bird nests (often other species). There are three types of brood parasitism: intraspecific brood parasitism, cooperative financing, and breeding. Some host birds can conflict directly and attack cuckoos. If the bird provides that the egg does not belong, it throws this foreign egg or simply leaves its nest and builds a new nest somewhere else. Some cuckoo species, such as the New World parasitic reproductive pile, have been developed so that female parasitic cuckoos are often specialized in mimicking the color and pattern of eggs of the selected host species. This reduces the likelihood that they will lay eggs and further increase their reproductive capacity. In addition, the timing of several species that fertilize the egg is remarkable. Parasitic cuckoos often prefer the nest where the host bird has just laid eggs. In general, cuckoo eggs are just ahead of host eggs. Once shaving the first hat, the first step is to chase the host’s egg by gradually throwing the egg out of the nest, boosting cuckoo chicken visits on the host bird.
On the other hand, various studies have shown that the behavior of flight animals and insects is characteristic of Levy flight. Recent studies by Reynolds and Frye indicate that fruit flies or Drosophila melanogaster explore your landscape in a series of 90° direct flight paths with a turn of 90°. This resulted in an intermittent and scalable Levyflight research model. Studies of human behavior such as the Ju/’hoansi investigation pattern for hunters and gatherers also indicate the characteristics of Levys flights. Even light can be connected to Levy flights. Such behaviors are then used for optimization and optimal research, and preliminary results indicate that they are promising.
Related Works
It is generally assumed that the cost of holding in the RW is higher than that because of the additional maintenance costs, material handling, etc. To reduce inventory costs, it will be costeffective to consume RW goods first. Barthelemy et al. (2008). Levy flight for lights. Chocolate et al. (2007). Levy flights at DobeJu/’hoage pattern. Pandey, et al. (2019). Marble Industry Optimization Analysis Based on Optical Genetic and Optical Algorithms. Pavlyukevich (2007a). Levy flights, nonlocal searches and simulations of violence and Levy flight abatement. Payne, Sorenson, and Klitz (2005). The Cuckoos. Reynolds and Frye (2007). The freeodor tracker in Drosophila is consistent with the search for the optimal intermittent free scale, PLoS. Shlesinger (2006). Search for research. Shlesinger et al. (1995). (Eds), Flight Levy and Topics in Phyics. Yadav and Swami (2018a). The productioninventory backlogging model is a sizeable measurement with timeconsuming cost and broken Weibull and the integrated Supply Chain Model for Producing Goods and the Hanging Stocks Symbol Under Impression and Inflation Environment. Yadav and Swami (2019a). The Double Flexible Model and the Demand for Upward and Inflation Costs and the inventory model for goods that are not easily interrupted by variable holding costs in two storage areas. Yadav, et al. (2019a) Supply Chain Inventory Model To Regulate Goods With Warehouse & Inflation Distribution Center. Yadav, et al. (2019b) Chemical Industry Manufacturer’s Chain For Warehouse And Distribution Center Using Bee Artificial Algorithms. Yadav, et al. (2017a) Inflation Inventory Model for Moving Items in Tw'o Storage Systems. Yadav, et al (2016) Multi Object Optimization for Component Electron Inventory Models & Improve Items and Doubles using Genetic Algorithms. Yadav, et.al (2017b). Two WarehouseBased Inventory Models for Immediate and Delayed Input Devices. Yadav, et al. (2020). Electronic components provide electronic Industry development management for storage and their impact on the environment using the Particle Swarm Optimization Algorithm. Yadav, et al. (2017c). Chain Supply Inventory Model for Two Warehouses and Soft Computing Optimization Yadav, et al. (2017d). Influence of inflation on inventory model of two warehouses for deteriorated goods with varying times of demand and supply.
Assumption and Notations
Notations & Assumption
Ordering cost = Woe Ability of OW = Z,
Ability of RW = Z_{2}
The length of replenishment cycle = T„
Maximum inventory level per cycle to be ordered = (Max),,
The time up to which inventory' vanishes in RW = 7j
The time at which inventory level reaches to zero in OW and shortages begins = T„ Definite time up to which holding cost is constant = К The holding cost in OW = OW„
The holding cost in RW = RW,,
The shortages cost = SCw
The opportunity cost = LCw
The level of inventory in RW = П„(()
The level of inventory' in OW = П,„ (l)
Determine the inventory level at time t in which the product has shortages = П,(г) Deterioration rate in RW = D„
Deterioration rate in OW = (D, + 1)
Purchase cost per unit of items = PCw Maximum amount of inventory backlogged = W_{/B }Amount of inventory lost = W„
Cost of purchase = P
The present worth cost of shortages = S
The present worth cost of lost sale = L
The present worth cost of holding inventory = H
The total relevant inventory' cost per unit time of inventory system = T^{c} [T. T, ]
(5 +1)if t > 0
Demand => D(l) =
{(£+ l) + Ф/Jif I > 0
Mathematical Formulation of Model and Analysis
Now the level of inventory at different time intervals is given by solving the equations above (8.1)—(8.4) in the boundary condition
Therefore, differential Equation (8.1) gives
Now at / = 0, Пя = Z_{2}, therefore, Equation (8.5) yields
Maximum amount of inventory backlogged during shortages period (at t=T) is given by
Amount of inventory lost during shortages period The maximum inventory to be ordered is given as
Now continuity at t = t shows that P"(t_{[}) = P"(t_{]}) therefore, from Equations (8.6) and (8.7), we have
where
which is quadratic in t_{2} and further can be solved for t_{2} in terms of t_{h} i.e., where
And
Next, the total relevant inventory cost per cycle includes following parameters:
Holding cost for Case 1
Case  2: When к > 7
Holding cost for Case  2
The present worth of shortages cost
The present worth opportunity cost / Lost sale cost Present worth purchase cost
Therefore, the total relevant inventory cost per unit per unit of time is denoted and given by Case 1:
Case 2:
Cuckoo Search Algorithm
Cuckoo Search (CS) is a new heuristic algorithm inspired by parasite reproduction behaviors that are mandatory in certain cuckoo species that lay eggs in nest nests. Some cuckoos specialize in mimicking the color and pattern of the eggs of several selected hosts. This reduces the likelihood of leaving the egg. If the host bird detects a foreign egg, it is either left behind or eliminated. Parasitic cuckoos prefer a nest where the host bird lays eggs. Cuckoo eggs hatch early than host eggs, and when absorbed, they chase host eggs away from the nest. For example, cuckoo chickens receive a lot of food, and sometimes, they mimic the sound of a rooster in order to eat more. Most of the time, cuckoos search for a simple, random street that becomes a Markov chain, the next position based on the current position, and the possible transition from the next. The use of Levy flights instead of simple random routes improves search capabilities. Levy’s flight is a random walk on stage after spreading heavy probability. Each cuckoo is a possible solution to the problem under consideration. The main goal is to come up with a new and possibly better (cuckoo) solution to replace it with a less efficient solution. Every nest has eggs, but as the problem progresses, some eggs can be used to give a number of solutions. There are three basic rules customized for CS. The first rule is that every cuckoo lays eggs and throws them at random nests. The second rule states that the nest with the longest physical form is transmitted to the next generation, while the latter rule indicates that the number of host nests is recorded and that the eggs that have been hatched by the cuckoo are found by the host bird with a probability of m [0, 1], and according to m, the host bird throws its eggs or leaves. It is assumed that only m fraction of the nest is replaced by the new nest. Cuckoo hunters have been implemented on the basis of three rules. In order to generate a new solution P'^{+}' for the cuckoo clock, a Levy flight is performed. This step is called Global Random Walk and is given by
The local random walk is given by:
where P' is the previous solution, 8 > 0 is the step size with respect to the scales of the problem, and
Because of the recent effects of the new solution on Levy flights, local browsers are accelerating. Here, some of the solutions that must be generated by remote field randomization, which prevents the system from optimally functioning, are gamma functions, m is the probability switch. € is a random number and (1 < v < 3). The stride length in the cuckoo hunt is very limited, and every big step is possible due to largescale randomization.
The Pseudocode for CS is Given in Algorithm 1
Algorithm: Pseudo code of Cuckoo Search (CS) algorithm.
Begin:
 —о —о Initialize cuckoo population: n o—oDefined dimensional objective function,/(.v) do Until iteration counter < maximum number of iterations
 —о —о —oglobal Search:
 o—ogenerate new nest P,'^{+l} using Eq. (A)
 —о —о evaluate fitness of P,'*^{[}
 —о —о choose a nest j randomly from n initial nests.
 —о —о —oif the fitness of P/^{+l} better than that of Pf —о —о —^replace j by P'^{+l}
 —o —o ^>end if
 —о —о local search:
 —о —о abandon some of the worst nests using probability switch.
 —о —о create new nest using Eq. (B)
 —о —о Evaluate and find the best.
 oend until —oupdate final best End
Numerical Analysis
The following data were randomly selected on units that have been used to find the optimal solution and model validation for the threeplayer manufacturers,
TABLE 8.1
The Computational Optimal Solutions of the Models are Total Relevant Cost and CS
S.N. 
Cost Function 
Case1 
Case2 
Algorithm 
Case1 
Case2 
T^{C}(T„T„) 
T^{C}(T„T„> 
T^{C}(T„T„) 
T^{C}(T„T„] 

1 
T,* 
2.47477 
7.31247 
CS of T,* 
2.47477 
7.30247 
2 
T_{2}* 
22.7053 
35.7420 
CS of T_{2}* 
42.7053 
35.7440 
3 
T_{n}* 
74.2487 
37.9892 
CS of T* 
74.2487 
37.9894 
4 
Total relevant cost 
43.5249 
21.0422 
CS 
035249 
40042.2 
distributors, and retailers. The data are given as (<; +1) =500, C= 1500, Z =2000, Ф = 0.50, = 60, RW_{H} = 75, PC_{W} = 1500, Ц, =0.013, (D, + 1) = 0.014, SC,,, =250,
k= 1 =0.06, and LC_{W} = 100. The value of the decision variable is calculated for the model for the two separate cases. The optimal solution of model computing is shown in Table 8.1.
Actual values should be called by CS specifically through experience and trialanderror. However, some standard settings are reported in the literature.
 • Population Size of Cuckoo Search =150
 • Number of generations of Cuckoo Search = 3000
 • Crossover type of Cuckoo Search = two Point
 • Crossover rate of Cuckoo Search =1.8
 • Mutation types of Cuckoo Search = Bit flip
 • Mutation rate of Cuckoo Search = 0.003 per bit
If single crossover points instead of Cuckoo Search two cutpoint crossover of Cuckoo Search employed by rate of crossover of Cuckoo Search can be down to a maximum of 1.50.
Sensitivity Analysis
TABLE 8.2
Sensitivity Analysis with Respect to Parameter Rate
t; 
t; 
t; 
T^{C}(T,,T„) 

(S+l) 
550 
4.49638 
64.3930 
74.6398 
340800 
450 
4.3944 
54.4334 
74.3336 
336654 

Ф 
0.55 
4.46488 
59.4764 
73.6847 
343436 
0.45 
4.50943 
90.9446 
98.3064 
300468 

Woe 
3650 
4.47477 
64.7064 
74.4498 
337644 
750 
4.47475 
64.7009 
74.4433 
337634 

C_{s} 
475 
4.47605 
63.0333 
73.6067 
338484 
345 
4.46435 
59.5406 
80.9334 
333457 
TABLE 8.2 (continued)
Sensitivity Analysis with Respect to Parameter Rate
t; 
t; 
t; 
T^{C}(T,,T„) 

LC_{W} 
330 
4.47468 
64.6849 
74.4604 
337574 
50 
4.47543 
64.8369 
74.3899 
337868 

К 
3.773 
4.54073 
68.6383 
83.7033 
353730 
0.805 
4.33043 
39.5857 
44.5368 
84565.8 

Cp 
3650 
4.59484 
63.9583 
75.4053 
340438 
750 
3.76330 
55.3799 
67.8856 
343983 

ow„ 
66 
4.46344 
59.6488 
73.9630 
343077 
30 
4.53653 
88.4048 
95.3834 
303374 

RW„ 
84.5 
4.37334 
64.3004 
76.3954 
344304 
37.5 
3.39647 
54.6439 
60.8334 
309304 

D_{0} 
0.0343 
4.48564 
63.0470 
74.3373 
333744 
0.0065 
4.33759 
78.6439 
94.4048 
374563 

(D, + 1) 
0.0354 
4.46534 
66.0434 
76.9065 
334683 
0.007 
4.54740 
44.0303 
58.7006 
375946 
TABLE 8.3
Sensitivity Analysis with Cuckoo Search (CS) Algorithm
Function 
Algorithm 
Best 
Worst 
Mean 
Standard Deviation 
Woe 
CS 
0.49608 
64.0900 
74.6098 
040800 
RW„ 
CS 
0.46488 
59.4764 
70.6807 
043436 
ow_{H} 
CS 
0.47477 
60.7060 
74.0498 
037604 
sc_{w} 
CS 
0.47468 
60.6809 
74.0600 
037570 
LC_{W} 
CS 
0.47605 
63.0333 
73.6067 
038080 
PC_{W} 
CS 
0.50073 
68.6083 
80.7003 
050730 
Conclusions
According to the scope of the research, the current studies can be divided into two categories: the first category uses the twoinventory system to model economic order volumes at a constant demand ratio using soft computing techniques in a single firm, and the second category studies the inventory of deteriorating items in the twowarehouse inventory management system for the economic order quantity model with a constant demand ratio, using soft computing. In terms of quantity, there are far fewer studies in the second category than in the first category. The study of inventory problems of deteriorating items is a new area of research compared to the research of inventories of ordinary objects, so that the total amount of research is much smaller than that of traditional ones.
In this paper, we propose a twostorey inventory model for determinants of broken goods with linear timedependent demand and a variety of costs to call cycle lengths with the goal of minimizing total inventory costs using the cuckoo search algorithm: Lack of permission and partial withdrawal using the cuckoo search algorithm. Two different cases have been discussed with one variable holding cost during the cycle period and another with the cost of holding for a total cycle length, and it is noted that while the variable captures the total cost of inventory more than in other cases using the cuckoo search algorithm. Further, the proposed model is very useful for deteriorating goods, as it increases in both the warehouses where the total cost of inventory decreases using the cuckoo search algorithm. This model can be improved by combining other setbacks, probabilistic demand patterns, and other realistic combinations using the cuckoo search algorithm.
References
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