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2
A PRODUCTION INVENTORY MODEL FOR DETERIORATING ITEMS WITH TIME AND PRICE RELIANT DEMAND USING FLOWER POLLINATION
OPTIMIZATION
Amit kumar1, Ajay Singh Yadav1'*, Dharmendra Yadav2
•
department of Mathematics, SRM Institute of Science and Technology,
Delhi-NCR Campus, Ghaziabad, India, 201204 2Department of Mathematics Vardhman College, Bijnor, India, 246701 [email protected], [email protected], [email protected]
* Corresponding author
Abstract
Effective management of production inventory for deteriorating items with dynamic demand patterns is crucial for businesses operating in today's competitive markets. This paper proposes a comprehensive model that addresses the complexities arising from the dual storage locations, item deterioration, and demand dependencies on both time and selling price. To optimize the decision variables associated with production and inventory control, we employ the Flower Pollination Optimization (FPO) algorithm, a nature-inspired meta-heuristic known for its ability to efficiently navigate complex search spaces. The two-storage production inventory model integrates the dynamics of item deterioration over time, capturing the real-world challenges faced by supply chain managers. The demand for items is modeled to be sensitive to both temporal variations and changes in selling prices, reflecting the intricate nature of market dynamics. Our approach leverages the FPO algorithm to explore and exploit the solution space, allowing for the identification of optimal or near-optimal strategies for production quantities, order quantities, and inventory levels. The FPO algorithm mimics the pollination process in nature, striking a balance between exploration and exploitation to efficiently search for solutions in a highly dynamic and nonlinear environment. The proposed model and optimization approach are validated through extensive simulations and sensitivity analyses. The results demonstrate the effectiveness of the FPO algorithm in finding robust solutions that enhance inventory management, mitigate deterioration-related losses, and adapt to varying demand scenarios. This research contributes to the field of supply chain optimization by offering a novel perspective on tackling the challenges associated with dual storage, item deterioration, and demand dependencies. The findings provide valuable insights for practitioners seeking advanced strategies for optimizing their production inventory systems in the face of evolving market conditions.
Keywords: Production Inventory Model, Deteriorating Items, Two-Storage, Shortages, Time and Selling Price Dependent Demand, Flower Pollination Optimization.
1. Introduction and related work
In the realm of supply chain management, the effective control and optimization of inventory systems play a pivotal role in ensuring the success and sustainability of businesses. As markets become increasingly dynamic and customer demands evolve, the complexities associated with
managing production inventory for deteriorating items intensify. This is particularly true in scenarios where the demand for items is not only influenced by temporal variations but also by changes in selling prices. To address these challenges, we propose a two-storage production inventory model that accounts for the intricacies of dual storage locations, item deterioration, and demand dependencies on time and selling price. The management of deteriorating items presents a unique set of challenges due to the perishable nature of certain goods over time. Incorporating this deterioration factor into inventory models is crucial for avoiding unnecessary losses and ensuring that products reaching customers are of the highest quality. Moreover, the consideration of time-dependent demand recognizes the influence of various temporal factors, such as seasonality or market trends, on the overall demand pattern. Adding an additional layer of complexity, our model acknowledges the impact of selling prices on demand. Price elasticity is a well-established concept in economics, and understanding how changes in selling prices affect the demand for items is vital for making informed decisions in a competitive marketplace.To address the optimization problem inherent in this multifaceted inventory management model, we turn to the Flower Pollination Optimization (FPO) algorithm. FPO, inspired by the pollination process in flowers, offers a nature-inspired meta-heuristic that excels in navigating complex and dynamic search spaces. By mimicking the pollination behavior of flowers, the FPO algorithm strikes a balance between exploration and exploitation, making it well-suited for finding optimal or near-optimal solutions in intricate and non-linear environments.
This research aims to contribute to the field of supply chain optimization by proposing a novel approach to managing production inventory in the face of deteriorating items with time and selling price dependent demand. Through extensive simulations and sensitivity analyses, we evaluate the effectiveness of the FPO algorithm in optimizing production quantities, order quantities, and inventory levels. The outcomes of this study provide valuable insights for practitioners seeking advanced strategies to enhance their production inventory systems, adapt to changing market conditions, and minimize losses associated with item deterioration.
Supply chain management can be defined as: "Supply chain management is the coordination of production, storage, location and transport between players in the supply chain to achieve the best combination of responsiveness and efficiency for a given market. Many researchers in the inventory system have focused on a product that does not overcome spoilage. However, there are a number of things whose meaning doesn't stay the same over time. Yadav et al. [1-10] developed the deterioration of these substances plays an important role and cannot be stored for long. Yadav, et al. [11-20] studied deterioration of an object can be described as deterioration, evaporation, obsolescence and loss of use or restriction of an object, resulting in less inventory consumption than under natural conditions. When raw materials are put in stock as a stock to meet future needs, there may be a deterioration of the items in the arithmetic system which could occur for one or more reasons, etc. Storage conditions, weather or humidity. Yadav, a. al. [21-53] explore the inach generally states that management has a warehouse to store the purchased warehouse. However, for various reasons, management may buy or lend more than it can store in the warehouse and call it OW, with an extra number in a rented warehouse called RW near OW or just off it. Yadav and swami. [54-61] developed an inventory costs (including maintenance costs and depreciation costs) in RW are generally higher than OW costs due to additional costs of running, equipment maintenance, etc. Reducing inventory costs will cost-effectively utilize RW products as quickly as possible. Actual customer service is only provided by OW, and to reduce costs, RW stock is cleaned first. Such arithmetic examples are called two arithmetic examples in the shop. Yadav and Kumar [62] established the management of the supply of electronic storage devices and integration of environmental and nerve networks. Yadav, A.S. [63-65] analysize of seven supply chain management measures to improve inventory of electronic storage devices by submitting a financial burden using GA and PSO and supply chain management analysis to improve inventory and inventory of equipment using genetic computation and model design and chain inventory analysis from bi inventory and economic difficulty in transporting goods by genetic computation. Swami, et. al. [66-68] developed inventory policies for inventory and inventory needs and miscellaneous inventory costs based
on allowable payments and inventory delays An example of depreciation of various types of goods and services and costs by keeping a business loan and inventory model with pricing needs low sensitive, inventory costs versus inflationary business expense loans. Gupta, et. al. [69- 70] established the objectives of the Multiple Objective Genetic Algorithm and PSO, which include the improvement of supply and deficit, inflation and a calculation model based on a genetic calculation of the scarcity and low inflation of PSO. Singh, et. al. [71, 72] studied an example with two stock depreciation on assets and inventory costs when updating particles and an example with two inventories of property damage and inventory costs in inflation and soft computer techniques. Kumar, et. al. [73-75] delayed control of alcohol supply and particle refinement and green cement supply system and inflation by particle enhancement and electronic inventory system and distribution center by genetic computations. Chauhan and Yadav [76-77] depreciation example at two stores and warehouses based on inventory using one genetic stock and one vehicle stock for demand and inflation inventory with two distribution centers using genetic stock. Pandey, et. al. [78] analysize of marble Improvement of industrial reserves based on genetic technology and improvement of multiple particles. Ahlawat, et. al. [79] studied the white wine industry in supply chain management through nerve networks. Singh, et. al. [80] examines the best policy to import damaged goods immediately and pay for conditional delays under the supervision of two warehouses. The study by Yadav et al. [81] focuses on enhancing inventory management for degrading commodities within the framework of green technology investment, accounting for factors including selling price, carbon emissions, and time-sensitive demand. It emphasizes sustainable practices in inventory models by fusing conventional economic principles with environmental factors. Using an interval number technique, Yadav, Yadav, and Bansal [82] examine a two-warehouse inventory management model for degrading commodities in order to take demand and cost uncertainty into consideration. Their analytical optimization methods demonstrate how spending money on preservation technology can save waste and increase inventory efficiency. With an emphasis on a two-warehouse system to maximize inventory levels, Yadav, Yadav, and Bansal [83] offer an inventory model that tackles the deterioration of commodities during storage. Their strategy emphasizes how crucial it is to successfully control degradation costs in order to raise total inventory efficiency.
2. Assumptions and notations 2.1. Assumptions
The following assumptions were used in the formulation of the mathematical model.
1. The unit production cost is a function of the rate of production.
2. The rate of production is considered to be decision variable.
3. The demand rate is a function of time and selling price, which is D(t, p) = (a + fit —
Yt2)p—x) a > 0, fi e [0,1), Y e [0,1).
4. The rate of deterioration is constant and different for both the warehouses.
5. The OW has limited capacity of W units and the RW has unlimited capacity.
6. Deterioration units can't be repaired or replaced.
7. The RW is located near the OW and thus the transportation cost between them is negligible.
8. The inventory cost (including carrying cost and deterioration cost) in RW is higher than that in OW.
9. Shortages are not allowed.
10. The holding cost is constant for both the warehouses.
Amit kumar, Ajay Singh Yadav and Dharmendra Yadav
A PRODUCTION INVENTORY MODEL FOR DETERIORATING ITEMS RT&A, No 1 (82) WITH TIME AND PRICE RELIANT DEMAND USING FPO_Volume 2°, March 2025
2.2. Notations
Table 2 is provided a description of the notations utilised for the constructed mathematical model.
Table 1: Notations
Notation Units Description
a Constant Coefficient of demand function
ß Constant Coefficient of demand function
Y Constant Coefficient of demand function.
di Constant Deterioration rate in OW.
02 Constant Deterioration rate in RW.
P - Production rate.
p $/Units Selling price of product.
U capability constraint The owned warehouse capacity
SUCi $/unit Set-up cost
Cd $/Units Deterioration cost.
h $/Units Holding cost.
TVC $/Units The function for total inventory cost.
3. Mathematical Model Formulation
The mathematical model for a production inventory system handling decaying products with two-storage includes time-dependent and selling price-dependent demand. Assume that I(t) reflects the inventory level at time t and that there are two storage facilities: one for immediate sales and another for buffer stock. The demand function D(p(t),t) is influenced and altered by the selling price p(t). The degradation rate affects the inventory's usefulness, therefore manufacturing costs, storage costs, and potential revenue loss due to spoiling must all be balanced. The objective is to minimize the total cost, which includes holding costs, production expenses, and lost revenue due to deterioration, subject to constraints on inventory levels, demand, and production rates. (See Fig. 1).
Inventory Level
Ia(t)f U(0
hit
Lit) VI f 1 /
Time
li2 li3 li4 /5
Figure 1: A graphical depiction of the two-warehouse production inventory model.
The inventory level is characterized by the following differential equations:
dIii(t) dt
+ 01 Ia(t) = P - (a + et - Yt2)p-X; t G [0, til]
with the boundary condition (B.C.) Ii1 (0) = 0.
dig (t) dt
+ 02Ii2(t) = P - (a + et - Yt2)p-X; t G [til, ti2]
with the boundary conditions (B.C.) Ii2(ti1) = 0.
dIg(t) dt
+ 02Ii3(t) = -(a + et - Yt2)p ; t G [ti2, ti3]
with the boundary conditions (B.C.) Ii3(ti3) = 0.
dIg(t) dt
+ 0iIn(t) = -(a + et - Yt2)p-X; t G [ta, ti4]
with the boundary conditions (B.C.) Ii4(ti4) = 0.
dIi5 (t)
dt
+ 0iIi5 (t)= 0; t G [tu, ta]
with the boundary conditions (B.C.) Iis(tii) = W.
The solutions of the differential Eqs. (1) -(5) are (6) -(10), respectively:
(2)
(3)
(4)
(5)
In(t)
1 f 2y + p02_2Oi + ^ 1 03 v px +101 px + px ) 03
+ p02_2OI + №
pX + 01 pX + pX
+
Yt2 t(2Y + №1)
pX01
PX02
(6)
Ii2(t)
03 V PX
K21 + P02 - 2001 + Ж
2 pX pX I
+
_ e-t02 e-tn02
+ P02 -L03V px 2
Yt21 tn (2 y + в02 )
2 2a02 в02
pX + pX
PX02
PX02
+
Yt2 t(2Y + №2) 1 PX02 PX02 J
(7)
I (t) (-a0\ + в02 + 2y) + Yt2 t(2Y + №2)
Ii3(t) =---+
PX03
PX02
PX02
_ e02(ti3-t)
(-a02 + в02 + 2y)+ Yt23 ti3 (2y + №2 )
PX023
PX02
PX022
(8)
Ij4(t)= (-a02 + в01 + 2y) + Yt2 - t(2Y + №1)
px03
PX01
PX02
_ e01(ti4-t)
(-a02 + № + 2y)+ Yt24 _ ti4(2y + §01 )
PX03
PX01
PX02
(9)
Ii5(t) = Wedl(ti1-t)
Therefore, the relevant costs of the production inventory system are as follows. 1. Set up costs for the cycle :
SUQ = Csu
(10)
(11)
2. Holding costs in RW for the cycle :
HCrw — h
l'ti2 f ti3
I Ii2 (t)dt + / Ii3 (t)dt L J til Jti2
HC
RW
he-02ti2
604
127e°2ti1 - 12ye02ti2 - 6a.d2e02ti1 + 6a.d2e02ti2 + 6p02e02ti1
- 6p02e02ti2 + t21e02ti1 - 3^03121 e02ti2 + 6j02t21 e02ti1 - 6j0^t'21e02ti1 - 2703131 e02ti1 + 2703t31e02ti1 - 12y02t31 e02ti1 + 12^0^e02ti1 + 6P02PXe02ti1 - 6P02PXe02ti1 + 6a0\ti1e02ti2 - 6a^2,ti1 e02ti2 - 6^0^ti1e02ti1 + 6^0^t^e02ti1
he-02ti2 r
+ 12ye02ti2 - 12ye02ti3 - 6x0^e02ti2
- 6P03PXti1 e02ti1 + 6P03PXti1 e02ti1
6PA0|
+ 6a02e02ti3 + 6fi02e02ti2 - 6p02e02ti3 + 3J00|t22e02ti2 - 3J00|t2e02ti3 + 6y02t22e02ti2 - 6702123e02ti3 - 2703132e02ti2 + t33e02ti3 - 12y02t32e02ti2 + 12Y02t33e02ti3 + 6P02PXe02ti2 - 6P02PXe02ti3 + 6a0^ti2e02ti2 - 6a0^ti3e02ti3
- 6fi02ti2e02ti2 + 6p02ti3e02ti3 - 6P03PXti2e02ti2 + 6P0%PXti3e02ti3
3. Holding costs in OW for the cycle :
HCqw — h
ti1 ti3 ti4
I In (t)dt + Ii5 (t)dt + Ii4(t)dt
_ M0 Jti1 J ti3
(12)
HC
qw —
he-01 ta
6PA04
12je01 ti4 - 12Ye01ti4 - 6a02e01ti3 + 6a02e02ti4 + 6^01 e02ti3
- 6fi01e02ti4 + 3^0?t23e01ti4 - 3^03ti3e01ti3 + 6j0^t23e01 ti3 - 6j02t24e02ti4
- 2703133e01ti3 + 2y03t3i4e01 ti4 - 12Y01t33e01 ti3 + 12^0^e01ti4 + 6P02PXe01ti3 - 6P02Pxe01 ti4 + 6a03ti3e01 ti3 - 6a0{t4e01ti4 - 6^0^ti3e01 ti4 + 6^0^ti3e01 ti4
- 6P03Pxti3e01ti3 + 6P03Pxti4e01 ti4
+K^+P02 - pa
2a02 p6±
' e01 (ti1 ti3) — 1 \ ( e01 (ti1) _ 1
- m-—,—1) + P
01
00-1
t21 (27 + №1) 2 pX02
+
Y31
3 pX01
a(e-01 tn - 1) + fi(e-01fi1 - 1) + 2j(e-01tn - 1)
pX02
PX03
pX04
4. Deterioration costs in RW for the cycle :
dcrw — CD 02
ti1
ti2 f ti3
Ii2 (t)dt + Ia(t)dt
(13)
DCrw = CD 02 e-02
127ee2tn - 127е02*а - 6а0^e02ti1 + 6а0^e02ti2
+ 6ß02e02ti1 - 6ß02e02ti2 + 2y0%t3i2e02ti2 - Пув^г02ti1 + 3ß0f t22e02ti2 - 3ß03ti2e02ti2 + 6y02t21e02ti1 - 6y02122e02ti2 - 2j0^t31e0l2t'2 + 12j02t32e02ti2 + 6P02PXe02ti1 - 6P02PXe02ti2 + 6а0\ ti 1 e02ti1 - 6а03ti2e02ti2 - 6ß0f 11 e02ti1
+
6PA0|
127e1
02ti2
+ 6ß02ti2e01 ti2 - 6P03PXti1 e0lti1 + 6P03Pxti2e02ti42
- 127e02ti3 - 6а02e02ti2 + 6x0%e02ti3 + 6ß02e02ti2 - 6ß02e02ti3
- 2703132e02ti2 + 2j03t33e02ti3 - 12j02t32e02ti2 + 12j02t%e02ti3
+ 6а0\ti2e02ti2 - 6а0\ti3e02ti3 - 6ß0^ti2e02ti2 + 6ß0^ti3e02ti3
(14)
5. Deterioration costs in OW for the cycle :
DCOW = CD 01
fti1 fti3 fti4
I Ia(t)dt + Ii5(t)dt + Ii4(t)dt
t0 ■'ti1 ■'ti3
( e-01 ti3
DCOW = CD 01S
6Px014
12Te01 ti3 - 12ye01ti4 - 6а02e01ti3 + 6а02e01ti4
+ 6ß0ie01ti3 - 6ß01e01 ti4 + 3ß03123e01ti3 - 3ß03ti4e01ti4 + 6j0^t23e01 ti3 - 6j02t24e01ti4 - 2703t33e01 ti3 + 2703ti4e01ti4 - 12y01 t%e01Îi3 + 12Y01t34e01 ti4 + 6а01 ti3e01ti3 - 6а01 ti4e01ti4 - 6ß02 ti3e01ti4
+ 6ß02tö e01ti4
+
t*(2Y + P02 - ^ + - w
03 ^ PX + P01 PX + PX I W
e01 (ti1 -ti3) — 1
+ 'e01^ - П - fj1(27 + ß01 ) + ^
0а1
2 pX02
3pX01
a(e-01 tn - 1) + ß(e-01tn - 1) + 2j(e-01tn - 1)
pX02
pX0\
pX04
6. Production cost for the cycle :
PCi = no (P)
r ti1 r ti2
Pdt + Pdt ¡0 Jti1
2
PCi = P2 noti2
Therefore, the present worth of total variable cost for the cycle 1
TVC = t [SUCi + hcrw + hcow + dcrw + dcow + PCi ] Note that for the detailed version of Equation (17), see Appendix A.
(15)
(16)
(17)
4. Flower Pollination Optimization Methodology
Flower Pollination Optimization (FPO) is a naturalistic approach to solving complex optimization issues by modeling flower pollination. FPO takes inspiration from the biological mechanisms of pollination, which involve the transport of pollen from flowers to pollinators and the subsequent reproduction of plants, to effectively explore and exploit the search space (see Fig. 2). In many
fields, including artificial intelligence, finance, and engineering, this method excels at determining the optimal solutions to continuous and discrete optimization problems. The algorithm details of the FPO technique which were brought al. multi-purpose optimization level (Darwin, [83]) after gaining the first literature (Yang, [82]) and Investigation of Artificial Intelligence Based Optimization Algorithms. Okula, et, al., [84] are as follows:
Algorithm:_
Step 1: (Installation Phase): Randomly distribute N-flower particle (potential solution variables) in solution space. Assign algorithm values, specify the transition probability parameter (go). Perform the necessary arrangements for the problem to be solved.
Step 2: Calculate the objective function value (fitness) according al. position of the flowers -particles (potential solution variables). Find out what's best.
Step 3: Repeat the following steps throughout the iterative process (eg until you reach a certain number of iterations or until you reach a desired value in the objective function): (For each particle; for each purpose function size).
Step 3.1: (Global - Local Pollination Phase): Generate a random value. If the value produced is less than the value of equation and Levy Flights (step vector: L). If the value produced is equal to or greater than the value of go, uniform distribution in the range [0,1]. Run the local pollination process in the context.14
Step 3.2: Calculate the purpose function value (fitness) according al. updated position of flowers -particles (potential solution variables).
Step 3.3: Update the global best value (and hence the variable position) if the best objective at that time is found to be better than the function value.
Step 4: Iteration - At the end of the cycle the value (s) obtained according al. global best position is considered to be the optimum value (s)._
Stop
Figure 2: Flowchart for the Flower Pollination Optimization.
5. Graphical representation
The primary objective of the graphical representations of the suggested model is a flowchart listing the key components. To demonstrate how to manage inventory for things that are deteriorating, it begins with two storage systems. The diagram highlights the relationship between time, selling price, and demand, showing how these factors dynamically change inventory levels. The flowchart also demonstrates the Flower Pollination Algorithm approach, which shows how potential solutions are developed and improved to cut costs and boost the efficacy of inventory management. All things considered, the complex relationships revealed in the model and the optimization technique employed are well communicated by this visual aid.
Figure 3: Relation between Demand rate and time using different values of p.
20
-160 -1-1-'-1-
0 5 10 15 20 25
Time (t)
Figure 4: Relation between Demand rate and time using different values of a.
Figure 5: Relation between Demand rate and time using different values of f.
Figure 6: Relation between Demand rate and time using different values of y.
Following are a few insights drawn from the graphical representation's observations.
1. If the selling price parameter p are increased, the total average inventory cost (TVC) rises due to the higher demand rate (see Fig. 3).
2. If the parameters a and f are increased, the total average inventory cost (TVC) rises quickly due to the extended running time of the warehouses. Simultaneously, both the cycle length and the product price decrease (see Fig. 4 and Fig. 5).
3. If the demand rate parameter (7) increases, the total average inventory cost (TVC) rises rapidly due to the higher amount of product waste (see Fig. 6).
6. Conclusion
In this study, we presented a comprehensive approach to address the challenges associated with managing a two-storage production inventory model for deteriorating items with time and selling price dependent demand. The complexities of dual storage locations, item deterioration, and dynamic demand patterns were considered, reflecting the real-world intricacies faced by supply chain managers. To optimize decision variables and navigate the complex solution space, we employed the Flower Pollination Optimization (FPO) algorithm, a nature-inspired metaheuristic known for its efficacy in solving complex optimization problems.
The proposed model demonstrated its relevance by integrating the impact of item deterioration over time, allowing for a more accurate representation of inventory dynamics. The consideration of time-dependent demand and selling price dependencies further enriched the model, capturing the nuances of market fluctuations and consumer behavior. Our choice of the FPO algorithm proved effective in finding solutions that strike a balance between exploration and exploitation. By simulating the pollination process in flowers, the FPO algorithm efficiently explored the solution space, leading to robust strategies for production quantities, order quantities, and inventory levels. The adaptability of FPO to dynamic and non-linear environments was crucial in addressing the intricate nature of the proposed inventory model. Through extensive simulations and sensitivity analyses, we validated the effectiveness of our approach, showcasing its ability to enhance inventory management, mitigate deterioration-related losses, and adapt to varying demand scenarios. The findings contribute valuable insights for supply chain practitioners seeking advanced strategies to optimize their production inventory systems in the face of evolving market conditions. As we conclude, it is important to emphasize the practical implications of our research. The proposed model and optimization approach offer a forward-looking perspective on addressing the challenges in dual storage inventory systems. The integration of FPO provides a powerful tool for decision-makers to refine their inventory strategies, ultimately improving overall supply chain efficiency and resilience. While this study has provided significant contributions, there are opportunities for further research. Future investigations could explore the applicability of the proposed model in different industry contexts and evaluate the performance of other meta-heuristic algorithms for comparison. Additionally, incorporating more nuanced factors such as supply chain disruptions or sustainability considerations could enrich the model further.
In conclusion, this research contributes to advancing the field of supply chain optimization by proposing an innovative solution to a complex inventory management problem. The integration of a two-storage production inventory model with FPO optimization provides a robust framework for addressing real-world challenges and paves the way for more resilient and adaptive supply chain strategies.
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[18] Yadav, A.S., Navyata, Ahlawat, N. and Pandey, T. (2019) Reliability Consideration based Hazardous Substance Storage Inventory Model for decaying Items using Simulated Annealing. International Journal of Advance Research and Innovative Ideas in Education, Volume 5 Issue 9,1134-1143.
[19] Yadav, A.S., Swami, A. and Kher, G. (2019) Blood bank supply chain inventory model for blood collection sites and hospital using genetic algorithm. Selforganizology, Volume 6 No.(3-4), 13-23.
[20] Yadav, A.S., Swami, A. and Ahlawat, N. (2018) A Green supply chain management of Auto industry for inventory model with distribution centers using Particle Swarm Optimization. Selforganizology, Volume 5 No. (3-4)
Amit kumar, Ajay Singh Yadav and Dharmendra Yadav
A PRODUCTION INVENTORY MODEL FOR DETERIORATING ITEMS RT&A, No 1 (82)
WITH TIME AND PRICE RELIANT DEMAND USING FPO Volume 20, March 2025
[21 [22 [23 [24 [25 [26
[27
[28 [29 [30 [31 [32 [33 [34 [35 [36
Yadav, A.S., Ahlawat, N., and Sharma, S. (2018) Hybrid Techniques of Genetic Algorithm for inventory of Auto industry model for deteriorating items with two warehouses. International Journal of Trend in Scientific Research and Development, Volume 2 Issue 5, 58-65. Yadav, A.S., Swami, A. and Gupta, C.B. (2018) A Supply Chain Management of Pharmaceutical For Deteriorating Items Using Genetic Algorithm. International Journal for Science and Advance Research In Technology, Volume 4 Issue 4, 2147-2153.
Yadav, A.S., Maheshwari, P., Swami, A., and Pandey, G. (2018) A supply chain management of chemical industry for deteriorating items with warehouse using genetic algorithm. Selforganizology, Volume 5 No.1-2, 41-51.
Yadav, A.S., Garg, A., Gupta, K. and Swami, A. (2017) Multi-objective Genetic algorithm optimization in Inventory model for deteriorating items with shortages using Supply Chain management. IPASJ International journal of computer science, Volume 5, Issue 6, 15-35. Yadav, A.S., Garg, A., Swami, A. and Kher, G. (2017) A Supply Chain management in Inventory Optimization for deteriorating items with Genetic algorithm. International Journal of Emerging Trends & Technology in Computer Science, Volume 6, Issue 3, 335-352. Yadav, A.S., Maheshwari, P., Garg, A., Swami, A. and Kher, G. (2017) Modeling & Analysis of Supply Chain management in Inventory Optimization for deteriorating items with Genetic algorithm and Particle Swarm optimization. International Journal of Application or Innovation in Engineering & Management, Volume 6, Issue 6, 86-107. Yadav, A.S., Garg, A., Gupta, K. and Swami, A. (2017) Multi-objective Particle Swarm optimization and Genetic algorithm in Inventory model for deteriorating items with shortages using Supply Chain management. International Journal of Application or Innovation in Engineering & Management, Volume 6, Issue 6,130-144.
Yadav, A.S., Swami, A. and Kher, G. (2017) Multi-Objective Genetic Algorithm Involving Green Supply Chain Management International Journal for Science and Advance Research In Technology, Volume 3 Issue 9, 132-138.
Yadav, A.S., Swami, A., Kher, G. (2017) Multi-Objective Particle Swarm Optimization Algorithm Involving Green Supply Chain Inventory Management. International Journal for Science and Advance Research In Technology, Volume 3 Issue, 240-246. Yadav, A.S., Swami, A. and Pandey, G. (2017) Green Supply Chain Management for Warehouse with Particle Swarm Optimization Algorithm. International Journal for Science and Advance Research in Technology, Volume 3 Issue 10, 769-775.
Yadav, A.S., Swami, A., Kher, G. and Garg, A. (2017) Analysis of seven stages supply chain
management in electronic component inventory optimization for warehouse with economic
load dispatch using genetic algorithm. Selforganizology, 4 No.2, 18-29.
Yadav, A.S., Maheshwari, P., Swami, A. and Garg, A. (2017) Analysis of Six Stages Supply
Chain management in Inventory Optimization for warehouse with Artificial bee colony
algorithm using Genetic Algorithm. Selforganizology, Volume 4 No.3, 41-51.
Yadav, A.S., Swami, A., Gupta, C.B. and Garg, A. (2017) Analysis of Electronic component
inventory Optimization in Six Stages Supply Chain management for warehouse with ABC
using genetic algorithm and PSO. Selforganizology, Volume 4 No.4, 52-64.
Yadav, A.S., Maheshwari, P. and Swami, A. (2016) Analysis of Genetic Algorithm and Particle
Swarm Optimization for warehouse with Supply Chain management in Inventory control.
International Journal of Computer Applications, Volume 145 "No.5,10-17.
Yadav, A.S., Swami, A. and Kumar, S. (2018) Inventory of Electronic components model for
deteriorating items with warehousing using Genetic Algorithm. International Journal of
Pure and Applied Mathematics, Volume 119 No. 16, 169-177.
Yadav, A.S., Johri, M., Singh, J. and Uppal, S. (2018) Analysis of Green Supply Chain Inventory Management for Warehouse With Environmental Collaboration and Sustainability Performance Using Genetic Algorithm. International Journal of Pure and Applied Mathematics, Volume 118 No. 20,155-161.
Yadav, A.S., Ahlawat, N., Swami, A. and Kher, G. (2019) Auto Industry inventory model for deteriorating items with two warehouse and Transportation Cost using Simulated Annealing
Algorithms. International Journal of Advance Research and Innovative Ideas in Education, Volume 5,Issue 1, 24-33.
[38] Yadav, A.S., Ahlawat, N., Swami, A. and Kher, G. (2019) A Particle Swarm Optimization based a two-storage model for deteriorating items with Transportation Cost and Advertising Cost: The Auto Industry. International Journal of Advance Research and Innovative Ideas in Education, Volume 5, Issue 1, 34-44.
[39] Yadav, A.S., Ahlawat, N., and Sharma, S. (2018) A Particle Swarm Optimization for inventory of Auto industry model for two warehouses with deteriorating items. International Journal of Trend in Scientific Research and Development, Volume 2 Issue 5, 66-74.
[40] Yadav, A.S., Swami, A. and Kher, G. (2018) Particle Swarm optimization of inventory model with two-warehouses. Asian Journal of Mathematics and Computer Research, Volume 23 No.1, 17-26.
[41] Yadav, A.S., Maheshwari, P.„ Swami, A. and Kher, G. (2017) Soft Computing Optimization of Two Warehouse Inventory Model With Genetic Algorithm. Asian Journal of Mathematics and Computer Research, Volume 19 No.4, 214-223.
[42] Yadav, A.S., Swami, A., Kumar, S. and Singh, R.K. (2016) Two-Warehouse Inventory Model for Deteriorating Items with Variable Holding Cost, Time-Dependent Demand and Shortages. IOSR Journal of Mathematics, Volume 12, Issue 2 Ver. IV, 47-53.
[43] Yadav, A.S., Sharam, S. and Swami, A. (2016) Two Warehouse Inventory Model with Ramp Type Demand and Partial Backordering for Weibull Distribution Deterioration. International Journal of Computer Applications, Volume 140 -No.4, 15-25.
[44] Yadav, A.S., Swami, A. and Singh, R.K. (2016) A two-storage model for deteriorating items with holding cost under inflation and Genetic Algorithms. International Journal of Advanced Engineering, Management and Science, Volume -2, Issue-4, 251-258.
[45] Yadav, A.S., Swami, A., Kher, G. and Kumar, S. (2017) Supply Chain Inventory Model for Two Warehouses with Soft Computing Optimization. International Journal of Applied Business and Economic Research, Volume 15 No 4, 41-55.
[46] Yadav, A.S., Rajesh Mishra, 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, Volume 9 No.2, 881-892.
[47] Yadav, A.S., Gupta, K., Garg, A. and Swami, A. (2015) A Soft computing Optimization based Two Ware-House Inventory Model for Deteriorating Items with shortages using Genetic Algorithm. International Journal of Computer Applications, Volume 126 - No.13, 7-16.
[48] Yadav, A.S., Gupta, K., Garg, A. and Swami, A. (2015) A Two Warehouse Inventory Model for Deteriorating Items with Shortages under Genetic Algorithm and PSO. International Journal of Emerging Trends & Technology in Computer Science, Volume 4, Issue 5(2), 40-48.
[49] Yadav, A.S. Swami, A., and Kumar, S. (2018) A supply chain Inventory Model for decaying Items with Two Ware-House and Partial ordering under Inflation. International Journal of Pure and Applied Mathematics, Volume 120 No 6, 3053-3088.
[50] Yadav, A.S. Swami, A. and Kumar, S. (2018) An Inventory Model for Deteriorating Items with Two warehouses and variable holding Cost. International Journal of Pure and Applied Mathematics, Volume 120 No 6, 3069-3086.
[51] Yadav, A.S., Taygi, B., Sharma, S. and Swami, A. (2017) Effect of inflation on a two-warehouse inventory model for deteriorating items with time varying demand and shortages. International Journal Procurement Management, Volume 10, No. 6, 761-775.
[52] Yadav, A.S., R. P. Mahapatra, Sharma, S. and Swami, A. (2017) An Inflationary Inventory Model for Deteriorating items under Two Storage Systems. International Journal of Economic Research, Volume 14 No.9, 29-40.
[53] Yadav, A.S., Sharma, S. and Swami, A. (2017) 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, Volume 10 No.11, 107-123.
Amit kumar, Ajay Singh Yadav and Dharmendra Yadav
A PRODUCTION INVENTORY MODEL FOR DETERIORATING ITEMS RT&A, No 1 (82)
WITH TIME AND PRICE RELIANT DEMAND USING FPO Volume 20, March 2025
[54 [55 [56 [57 [58 [59 [60 [61 [62 [63 [64 [65 [66
[67 [68
[69
Yadav, A.S. and Swami, A. (2018) Integrated Supply Chain Model for Deteriorating Items With Linear Stock Dependent Demand Under Imprecise And Inflationary Environment. International Journal Procurement Management, Volume 11 No 6, 684-704. Yadav, A.S. and Swami, A. (2018) A partial backlogging production-inventory lot-size model with time-varying holding cost and weibull deterioration. International Journal Procurement Management, Volume 11, No. 5, 639-649.
Yadav, A.S. and Swami, A. (2013) A Partial Backlogging Two-Warehouse Inventory Models For Decaying Items With Inflation. International Organization of Scientific Research Journal of Mathematics, Issue 6, 69-78.
Yadav, A.S. and Swami, A. (2019) An inventory model for non-instantaneous deteriorating items with variable holding cost under two-storage. International Journal Procurement Management, Volume 12 No 6, 690-710.
Yadav, A.S. and Swami, A. (2019) A Volume Flexible Two-Warehouse Model with Fluctuating Demand and Holding Cost under Inflation. International Journal Procurement Management, Volume 12 No 4, 441-456.
Yadav, A.S. and Swami, A. (2014) Two-Warehouse Inventory Model for Deteriorating Items with Ramp-Type Demand Rate and Inflation. American Journal of Mathematics and Sciences Volume 3 No-1,137-144.
Yadav, A.S. and Swami, A. (2013) Effect of Permissible Delay on Two-Warehouse Inventory Model for Deteriorating items with Shortages. International Journal of Application or Innovation in Engineering & Management, Volume 2, Issue 3, 65-71.
Yadav, A.S. and Swami, A. (2013) A Two-Warehouse Inventory Model for Decaying Items with Exponential Demand and Variable Holding Cost. International of Inventive Engineering and Sciences, Volume-1, Issue-5, 18-22.
Yadav, A.S. and Kumar, S. (2017) Electronic Components Supply Chain Management for Warehouse with Environmental Collaboration & Neural Networks. International Journal of Pure and Applied Mathematics, Volume 117 No. 17, 169-177.
Yadav, A.S. (2017) Analysis of Seven Stages Supply Chain Management in Electronic Component Inventory Optimization for Warehouse with Economic Load Dispatch Using GA and PSO. Asian Journal Of Mathematics And Computer Research, volume 16 No.4, 208-219. Yadav, A.S. (2017) Analysis Of Supply Chain Management In Inventory Optimization For Warehouse With Logistics Using Genetic Algorithm International Journal of Control Theory And Applications, Volume 10 No.10, 1-12 .
Yadav, A.S. (2017) Modeling and Analysis of Supply Chain Inventory Model with two-warehouses and Economic Load Dispatch Problem Using Genetic Algorithm. International Journal of Engineering and Technology, Volume 9 No 1, 33-44.
Swami, A., Singh, S.R., Pareek, S. and Yadav, A.S. (2015) Inventory policies for deteriorating item with stock dependent demand and variable holding costs under permissible delay in payment. International Journal of Application or Innovation in Engineering & Management, Volume 4, Issue 2, 89-99.
Swami, A., Pareek, S., Singh S.R. and Yadav, A.S. (2015) Inventory Model for Decaying Items with Multivariate Demand and Variable Holding cost under the facility of Trade-Credit. International Journal of Computer Application, 18-28.
Swami, A., Pareek, S., Singh, S.R. and Yadav, A.S. (2015) An Inventory Model With Price Sensitive Demand, Variable Holding Cost And Trade-Credit Under Inflation. International Journal of Current Research, Volume 7, Issue, 06,17312-17321.
Gupta, K., Yadav, A.S., Garg, A. and Swami, A. (2015) A Binary Multi-Objective Genetic Algorithm & PSO involving Supply Chain Inventory Optimization with Shortages, inflation. International Journal of Application or Innovation in Engineering & Management, Volume 4, Issue 8, 37-44.
Gupta, K., Yadav, A.S., Garg, A., (2015) Fuzzy-Genetic Algorithm based inventory model for shortages and inflation under hybrid & PSO. IOSR Journal of Computer Engineering, Volume 17, Issue 5, Ver. 1, 61-67.
[71] Singh, R.K., Yadav, A.S. and Swami, A. (2016) A Two-Warehouse Model for Deteriorating Items with Holding Cost under Particle Swarm Optimization. International Journal of Advanced Engineering, Management and Science, Volume -2, Issue-6, 858-864.
[72] Singh, R.K., Yadav, A.S. and Swami, A. (2016) A Two-Warehouse Model for Deteriorating Items with Holding Cost under Inflation and Soft Computing Techniques. International Journal of Advanced Engineering, Management and Science, Volume -2, Issue-6, 869-876.
[73] Kumar, S., Yadav, A.S., Ahlawat, N. and Swami, A. (2019) Supply Chain Management of Alcoholic Beverage Industry Warehouse with Permissible Delay in Payments using Particle Swarm Optimization. International Journal for Research in Applied Science and Engineering Technology, Volume 7 Issue VIII, 504-509.
[74] Kumar, S., Yadav, A.S., Ahlawat, N. and Swami, A. (2019) Green Supply Chain Inventory System of Cement Industry for Warehouse with Inflation using Particle Swarm Optimization. International Journal for Research in Applied Science and Engineering Technology, Volume 7 Issue VIII, 498-503.
[75] Kumar, S., Yadav, A.S., Ahlawat, N. and Swami, A. (2019) Electronic Components Inventory Model for Deterioration Items with Distribution Centre using Genetic Algorithm. International Journal for Research in Applied Science and Engineering Technology, Volume 7 Issue VIII, 433-443.
[76] Chauhan, N. and Yadav, A.S. (2020) An Inventory Model for Deteriorating Items with Two-Warehouse & Stock Dependent Demand using Genetic algorithm. International Journal of Advanced Science and Technology, Vol. 29, No. 5s, 1152-1162.
[77] Chauhan, N. and Yadav, A.S. (2020) Inventory System of Automobile for Stock Dependent Demand & Inflation with Two-Distribution Center Using Genetic Algorithm. Test Engraining & Management, Volume 83, Issue: March - April, 6583 - 6591.
[78] Pandey, T., Yadav, A.S. and Medhavi Malik (2019) An Analysis Marble Industry Inventory Optimization Based on Genetic Algorithms and Particle swarm optimization. International Journal of Recent Technology and Engineering Volume-7, Issue-6S4, 369-373.
[79] Ahlawat, N., Agarwal, S., Pandey, T., Yadav, A.S., Swami, A. (2020) White Wine Industry of Supply Chain Management for Warehouse using Neural Networks Test Engraining & Management, Volume 83, Issue: March - April, 11259 - 11266.
[80] Singh, S. Yadav, A.S. and Swami, A. (2016) An Optimal Ordering Policy For Non-Instantaneous Deteriorating Items With Conditionally Permissible Delay In Payment Under Two Storage Management International Journal of Computer Applications, Volume 147 -No.1,16-25.
[81] Yadav, A.S., Kumar, A., Yadav, K.K. et al. Optimization of an inventory model for deteriorating items with both selling price and time-sensitive demand and carbon emission under green technology investment. Int J Interact Des Manuf (2023). https://doi.org/10.1007/s12008-023-01689-8
[82] Yadav, K.K., Yadav, A.S. & Bansal, S. Interval number approach for two-warehouse inventory management of deteriorating items with preservation technology investment using analytical optimization methods. Int J Interact Des Manuf (2024). https://doi.org/10.1007/s12008-023-01672-3
[83] Krishan Kumar Yadav , Ajay Singh Yadav, and Shikha Bansal. "Optimization of an inventory model for deteriorating items assuming deterioration during carrying with two-warehouse facility" Reliability: Theory & Applications, vol. 19, no. 3 (79), 2024, pp. 442-459. doi:10.24412/1932-2321-2024-379-442-459
Appendix A
TVC = -T
(Csu + p2noti2 )
+
he-02 fi2 604
127e02ti1 - 12ye02t'2 - 6a02e02ti1 + 6a02e02ti2 + 6fi02e02ti1
- 6рд2ев2ti2 + Зрд?, t21e02ti1 - 3рв3i2ie02ti2 + 670^ e02ti1 - 6yd2, t21e02ti1 - 2jd3131 e°2ti1 + 2jd3t31 e02ti1 - 12j02t?31e02ti1 + 12j02t\e02ti1 + 6P0^Pxe02ti1 - 6P0\Pxe02ti1 + 6a0^ti1 e02ti2 - 6a03ti1 e02ti2 - 600^e02ti1 + 6J802ti1e02ti1
he-®'2^2 Г
+ 12ye02ti2 - 12ye02ti3 - 6x0^e02ti2
- 6P®3Pxti1e92ta + 6P&3PX ti1 e®2ti1
6PA04
+ 6a0fe02ti3 + 6p02e02ti2 - 6p02e02ti3 + 3^3122e02ti2 - 3^2e02Î''3 + 6y02t22e02ti2 - 6j0^t23e®2ti3 - 2703132e02ti2 + 2j0^t33e02ti3 - 12j02t32e02ti2 + 12jd2t33e02ti3 + 6P02Pxe02ti2 - 6P0jPxe02ti3 + 6a0^ti2e02ti2 - 6a.0\ ti3e02ti3
- 6в®2 ti2e02ti2 + 6в®2 ti3e02ti3 - 6P03Pxti2e02ti2 + 6P®3Pxti3e02ti3
• he-01
+ h
6Px04
12?e01 ti4 - 12je01ti4 - 6a02e01ti3 + 6a02e02ti4 + 6^01 e02ti3
- 6p01e02ti4 + 3^03t23e01 ti4 - 3^03ti3e01ti3 + 6j0^t23e01ti3 - 670?t24e02ti4
- 2703133e01ti3 + 2703134e01 ti4 - 12j01t33e01ti3 + 12^01134e01ti4 + 6P0fPxe01 ti3 - 6P02Pxe01 ti4 + 6a03ti3e01ti3 - 6a03ti4e01ti4 - 6p02ti3e01ti4 + 6p02ti3e01 ti4
- 6P03Pxti3e01ti3 + 6P03Pxti4e01ti4
+ tjL(2Y + Pe2_2al + + 03 V Px + 1 Px + px
e01(ti1-ti3) 1 e01(ti1) 1
- m-—,—M + p
01
0a2
(27 + в01 )
2 Px02
a(e-01 *'1 - 1) + e(e-01ta - 1) + 2j(e-01 - 1)
+
Y31
3 px01
p\02
px03 px04
127e02ta - 12je02ti2 - 6a0^e02ti1 + 6a0^e02ti2
+ 6p02e02ti1 - 6p02e02ti2 + 2y03t32e02ti2 - 12j02t31 e02ti1 + 3^0^t22e02ti2 - 3e&3Lt2e02ti2 + 6y02t21e02fi1 - 6702122e02ti2 - 2j0^t31 e012ti2 + 12j02t32e02ti2 + 6P02Pxe02ti1 - 6P02Pxe02ti2 + 6a02ti1e02fi1 - 6a0^ti2e02ti2 - 60ti1e02fi1
+ 6fi02ti2e01 ti2 - 6P03Pxti1 e01ti1 + 6P03Pxti2e02ti42
+
6Px04
12ye02ti2
- 12ye02ti3 - 6a0fe02ti2 + 6a0^e02ti3 + 6p02e02ti2 - 6p02e02ti3 + 3в0Щ2e0212 - 3^03t23e02ti3 + 6702122e02ti2 - 6j0^ti33e02ti3
- 2703132e02ti2 + 2y03t33e02ti3 - 12j02132e02ti2 + 12j021%e02ti3
+ P
e01 (t'1) - 1\ t21 (27 + в01)
0a1
2 px0f
+
t(e-01ti1 -1)
рЩ
+ e(e-01ta - 1) + 2y(e-01ta - 1)
Y31
3 px01
px03
px04
+ P2 noti2
(18)
1
