# Journal of Management Information and Decision Sciences (Print ISSN: 1524-7252; Online ISSN: 1532-5806)

Review Article: 2021 Vol: 24 Issue: 6

# An inventory model with price dependent demand rates as power law form using ANT colony optimization

Satish Kumar, SRM Institute of Science and Technology

Ajay Singh Yadav, SRM Institute of Science and Technology

Veenita Sharma, Chaudhary Charan Singh University

Navin Ahlawat, SRM Institute of Science and Technology

Govindarajan Arunachalam, SRM Institute of Science and Technology

Citation Information: Kumar, S., Yadav, A. S., Sharma, V., Ahlawat, N., & Arunachalam, G. (2021). An inventory model with price dependent demand rates as power law form using ANT colony optimization. Journal of Management Information and Decision Sciences, 24(6), 1-10.

### Abstract

The main object of keeping inventories is to meet market demands using Ant Colony Optimization for Traveling Salesman Problem. Typically, retailers face a wide range of demands for different types of goods. These demands are not under the control of the organization using Ant Colony Optimization for Traveling Salesman Problem. In the present paper, demand taken as price dependent as power law form as well as time. The rate of deterioration and holding cost are considered also time dependent using Ant Colony Optimization for Traveling Salesman Problem. Shortages are allowed with partial backlogged using Ant Colony Optimization for Traveling Salesman Problem. Results are illustrated along with sensitivity analysis.

### Keywords

Inventory; Price dependent demand rates; Ant colony optimization; Traveling salesman problem.

### Introduction

Normally every businessman, maintain stock of goods for smooth running of business operation. The EOQ models are most successful models because they are very simple to understand and apply. But the situation of demand is different in today market where demand is always varies with time and selling price etc. So, in the present inventory model, the authors decided to take variable demands rate i.e. time and price dependent demand rate, which better match with real market situations.

### Assumptions and Notations

Assumptions and notations for present model-

1. = the purchase cost/unit taken as constant of the item.

2. ‘K’= the cost of order per cycle.

4. is the initial holding cost and

5. ‘p’= the selling price for every unit item.

6. be the scale and β be the shape parameter of demand curves.

7. is the decay rate.

8. ‘T’ is the cycle length.

9. QT be the highest stock level.

10. Shortages with backlogged rate are permitted defined by where.

11. T1 Shortage starting time.

12. ‘s’ shortage cost/unit/year.

13. 'π ' per unit opportunity cost for lost sales.

### The Model and Solution

In this section, a perishable item replenishment policy with partial backlog is considered. The Figure 1 shows the behavior of the inventory system.

Figure 1 The Behavior of the Inventory System

Let I(t) be the level of inventory at any time t. The change in inventory w.r.to t defined as

(1)

Under condition, I(T1) = 0 (2)

Now, from (1),

(3)

And

(4)

Where I(0) is consider as initial stock. Ignoring the power more than one of a, b, c and μ . Taking Z(t) be the stock which are loss due to deterioration of items during the interval [0, t].

(5)

Using equation (2) in equation (5), we have

(6)

Also the total demand in the interval

(7)

Therefore, ordered quantity per cycle is

QT = Total decay + Total demand in the interval [0, T1]

(8)

Since

Therefore, from equation (3), we can

(9)

Therefore

(10)

The average system cost is given by

Also, the order rate is

(12)

Now, the behavior of the order rate w.r.to a, b, c and μ is determined by

Also with respect to price ‘p’ is

(13)

Thus, the order rate increase if increases in a, b, c and μ and decrease with increase in p. Now, our objective is to obtain values of T1, T and p which make C(T1, T,p) as minimum. The necessary condition for C(T1, T,p) as minimum are

(15)

(16)

Solving (14), (15) and (16) simultaneously and find T1, T & p for which function C(T1, T,p) will be minimum.

Working of ACO for TSP

Initially, each ant is placed at random on a city. When developing a viable solution, the ants select the next city to visit using a probabilistic decision rule. When an ant k declares in city i and constructs the partial solution, the probability of moving to the next neighboring city j i is given by

(17)

Where is the intensity of trails between edge (i,j) and is the heuristic visibility of the edge (i, j), and Is the influencing factor of pheromones,β3 is the influence of the local node, and Jk (i) is a set of cities that remain to be visited when the ant is in city i. Once each ant has completed their turn, the amount of pheromones on each path will be adjusted with the following equation.

(13)

is pheromone evaporation coefficient and ρ where

(14)

(18)

(1-ρ ) is the decay parameter of pheromones (0<ρ<1) where it represents the evaporation of the track when the ant chooses a city and decides to move. Lk is the length of the turn for each formed per ant k and m is the number of ants. Q is the pheromone deposition factor.

Case Study

In this section, we consider numerical examples with data same as reference research with appreciate units to illustrate the models numerically.

Then the optimal result of the model is

Sensitivity Study

The aim of present section is to identify parameters to the changes of which the solution of the model is sensitive.

 Table 1 Sensitivity Study α Parameter Value % Change T T1 QT C 80000 -20 1.08385 0.44729 62.65580 367.87000 90000 -10 1.02208 0.42422 66.76390 390.11200 110000 +10 0.92488 0.38734 74.73440 431.14800 120000 +20 0.88567 0.37226 77.88210 450.25900
 Table 2 Sensitivity Study for β Parameter Value % Change T T1 QT C 1.20 -20 0.51416 0.22352 138.22600 776.26300 1.35 -10 0.70617 0.30173 99.09240 564.89400 1.65 +10 1.33248 0.53726 50.00220 299.26400 1.80 +20 1.83340 0.70511 35.04770 217.78700
 Table 3 Sensitivity Study For μ Parameter Value % Change T T1 QT C 0.08 -20 0.97146 0.40670 70.76290 410.74800 0.09 -10 0.97064 0.40559 70.70650 410.94700 0.11 +10 0.96903 0.40340 70.59550 411.34100 0.12 +20 0.96824 0.40233 70.54100 411.53600

### Observations

From above Tables 1,2 & 3, we observed that:

• It is observed that duration of the cycle length have negative correlation corresponding to the parameters α and μ but have positive correlation corresponding to the parameters β Also, model is more sensitive for the parameters β comprising to other parameters

• The average cost of the system has negative correlation with respect to the parameter β but have positive correlation w.r.to α and μ.Also it is noted that the model is more sensitive for the parameters β comprising to other parameters.

• From the above points, we can say that sufficient care should be about parameter β in conducting model.

### Conclusions

This model developed under considering the demand rate as time and price during available inventory and price dependent during shortages with backlog rate depend on waiting time using Ant Colony Optimization for Traveling Salesman Problem. Demand function have negative derivative with respect to price i.e. demand is a decreasing when selling price of items is increasing using Ant Colony Optimization for Traveling Salesman Problem. Variable holding cost is taken in account. To find approximate results cost minimization technique is applied using Ant Colony Optimization for Traveling Salesman Problem. The problems are illustrated numerically. In, future studies the present model will be more realistic after considering such as stock dependent demand rate, life time items, trade credit policy, lead time, Weibull distribution and considering inflation etc.

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