A Soft Computing Techniques Application of an Inventory Model in Solving Two-Warehouses 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 high-traffic 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 Levy-flight 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 cost-effective 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 free-odor 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 production-inventory backlogging model is a sizeable measurement with time-consuming 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 Warehouse-Based 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 = Z2

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 = Tc [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 = Z2, 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 t2 and further can be solved for t2 in terms of th 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 is the multiplication based on the input. Here, Pj and Р/ are solutions chosen at random and PlKi, is the best solution for the moment. In this work, the length of random meals in Levy flights because of the more efficient exploration of the search space by Levy flights is considered and derived from the Levy distribution with limited variants and meanings.

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 large-scale 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 three-player manufacturers,

TABLE 8.1

The Computational Optimal Solutions of the Models are Total Relevant Cost and CS

S.N.

Cost Function

Case-1

Case-2

Algorithm

Case-1

Case-2

TC(T„T„)

TC(T„T„>

TC(T„T„)

TC(T„T„]

1

T,*

2.47477

7.31247

CS of T,*

2.47477

7.30247

2

T2*

22.7053

35.7420

CS of T2*

42.7053

35.7440

3

Tn*

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, RWH = 75, PCW = 1500, Ц, =0.013, (D, + 1) = 0.014, SC,,, =250,

k= 1 =0.06, and LCW = 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 trial-and-error. 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 cross-over points instead of Cuckoo Search two cut-point cross-over 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;

TC(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

Cs

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;

TC(T,,T„)

LCW

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

D0

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

owH

CS

0.47477

60.7060

74.0498

037604

scw

CS

0.47468

60.6809

74.0600

037570

LCW

CS

0.47605

63.0333

73.6067

038080

PCW

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 two-inventory 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 two-warehouse 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 two-storey inventory model for determinants of broken goods with linear time-dependent 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

Barthelemy, P., Bertolotti, J., and Wiersma, D. S. (2008). A Levy flight for light, Nature, 453, 495-498.

Pandey, T., Yadav, A.S., and Malik, M. (2019). An analysis marble industry inventory optimization based on genetic algorithms and particle swarm optimization. International Journal of Engineering and Advanced Technology, 7(6S4), 369-373.

Pavlyukevich, I. (2007a) Levy flights, non-local search and simulated annealing, Journal of Computational Physics, 226, 1830-1844.

Pavlyukevich, I. (2007b). Cooling down Levy flights, Journal of Physics A: Mathematical and Theoretical, 40, 12299-12313.

Payne, R. B., Sorenson, M. D., and Klitz, K. (2005). The Cuckoos, Oxford University Press, Oxford.

Reynolds, A. M., and Frye, M. A. (2007). Free-flight odor tracking in Drosophila is consistent with an optimal intermittent scale-free search, PLoS One, 2, e354.

Shlesinger. M. F. (2006). Search research. Nature, 443, 281-282.

Shlesinger. M. F.. Zaslavsky, G. M. and Frisch, U. (Eds.) (1995). Levy Flights and Related Topics in Phyics, Springer, Berlin.

Yadav, A. S., and Swami, A. (2018a). A partial backlogging production-inventory lot-size model with time-varying holding cost and weibull deterioration. International Journal Procurement Management, 11(5), 639-649.

Yadav, A. S., and Swami, A. (2018b). Integrated supply chain model for deteriorating items with linear stock dependent demand under imprecise and inflationary environment, International Journal Procurement Management, 11(6), 684-704.

Yadav, A. S., and Swami, A. (2019a). A volume flexible two-warehouse model with fluctuating demand and holding cost under inflation. International Journal Procurement Management, 12(4), 441-456.

Yadav, A. S., and Swami, A. (2019b). An inventory model for non-instantaneous deteriorating items with variable holding cost under two-storage. International Journal Procurement Management, 12(6), 690-710.

Yadav, A. S., Bansal, К. K., Kumar, J. and Kumar, S. (2019a). Supply chain inventory model for deteriorating item with warehouse & distribution centres under inflation. International Journal of Engineering and Advanced Technology, 8(2S2), 7-13.

Yadav, A. S., Kumar, J., Malik, M., and Pandey, T. (2019b). Supply chain of chemical industry for warehouse with distribution centres using artificial bee colony algorithm, International Journal of Engineering and Advanced Technology, 8(2S2), 1-6.

Yadav, A. S., Mahapatra, R. P, Sharma, S., and Swami, A. (2017a). An inflationary inventory model for deteriorating items under two storage systems. International Journal of Economic Research, 14(9), 29-40.

Yadav, A. S„ Mishra, R., Kumar, S., and Yadav, S. (2016). Multi objective optimization for electronic component inventory model & deteriorating items with two-warehouse using genetic algorithm. International Journal of Control Theory and applications, 9(2), 15-35.

Yadav, A. S., Sharma, S„ and Swami, A. (2017b). A fuzzy based two-warehouse inventory model for non instantaneous deteriorating items with conditionally permissible delay in payment. International Journal of Control Theory and Applications, 10(11), 107-123.

Yadav, A.S., Swami, A., Ahlawat, N., Bhatt, D. and Kher, G. (2020). Electronic components supply chain management of Electronic Industrial development for warehouse and its impact on the environment using Particle Swarm Optimization Algorithm International Journal Procurement Management. Optimization and Inventory Management (Book Chapter), Springer.

Yadav, A. S., Swami, A., Kher, G. and Sachin Kumar, S. (2017c). Supply chain inventory model for two warehouses with soft computing optimization. International Journal of Applied Business and Economic Research, 15(4), 41-55.

Yadav, A. S., Taygi, B., Sharma, S. and Swami, A. (2017d). Effect of inflation on a two-ware- house inventory model for deteriorating items with time varying demand and shortages. International Journal Procurement Management, 10(6), 761-775.

 
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