Mathematical Modeling of Storage and Inventory Control Policies for Cost Minimization in Deteriorating Item Warehouse System

Mathematical Modeling of Storage and Inventory Control Policies for Cost Minimization in Deteriorating Item Warehouse System

Somen Guha

Research Scholar, Department of Mathematics, YBN University, Ranchi-834004,

Dhrub Kumar Singh

Research Supervisor and Head, department of Mathematics, YYBN University, Ranchi-834010,

Abstract – Inventory control is an important part of supply chain management, especially when products are stored in different warehouses and lose value over time. Many items such as food, medicines, and electronic products deteriorate due to spoilage or become outdated. Because of this, companies must choose proper storage and ordering policies to reduce total cost and avoid shortages.

This research develops a simple mathematical model for managing deteriorating items in a multi-warehouse system. The system includes one owned warehouse and two rented warehouses with different costs and deterioration rates. One rented warehouse has low rent but high deterioration, while the other has high rent but low deterioration. This situation is commonly seen in real business environments where firms use both cheap and high-quality storage facilities.

The main aim of the model is to find the best order quantity and the best way to distribute inventory among the warehouses so that the total average cost is minimized. The total cost includes holding cost, deterioration cost, and shortage cost. Demand is assumed to be known and changes according to location, such as village, city, and micro-city areas. Replenishment is instant, shortages are allowed and fully backlogged, and warehouse capacities are assumed to be unlimited.

The total cost function is developed and optimized using basic mathematical methods. A numerical example is used to compare two-warehouse and three-warehouse systems. The results show that the three-warehouse system gives a lower total cost.

The study concludes that using an additional low-deterioration warehouse, even with higher rent, helps reduce overall cost and improves inventory management

Keywords: Inventory control, Deteriorating items, Multi-warehouse system, Cost minimization, EOQ model, Supply chain management.

1. Introduction:

   Inventory control constitution one of the most significant and practical challenges in contemporary logistics and supply chain management, particularly where goods must be preserved and managed across distant and diverse location. Effective inventory management demands the careful formulation of appropriate storage policies for warehouse, ensuring optimal preservation availability of goods. Model based solution to inventory problems generally address two primary objectives, first, the continuous availability of off seasonal goods over a specified time horizon; and second, the substantial retention of product quality over time.

   Warehouse are typically classified into two categories:

   1. Low rent warehouse characterized by high deterioration rates.

   2. High rent warehouse characterized by low deterioration rate.

      For example, rented warehouse may be either airconditioned (high rent with low deterioration) or non airconditioned (low rent with high deterioration). A business may strategically hire a combination of two type of warehouses according to its operational requirement. Typically, two operational model includes their warehouses; One owned warehouse (denoted as ow) and two rented warehouses (denoted as RW1 and RW2).

      In this configuration, RW2 is associated with high rental costs but exhibits low deterioration rates, whereas RW1 incurs low rental costs but experiences higher rates of deterioration.

      The primary objective of this model is to investigate two optional inventory policies in terms of Economic order quantity (EOQ) and End of life (EOL) management within such an environment, particularly when multiple locations incorporate a mixture of these to warehouse categories. Each warehouse is assumed to operate independently, determining its optimal policies as though functioning as an autonomous entity. This modality approach mirrors practical scenarios encountered in real world logistics system.

      Organization with multiple warehouses often manage them centrally, particularly when it comes to inventory ordering. While centralization simplifies decision making, it also introduces complexity in optimizing inventory policies for each warehouse location.

      Deterioration items are products that lose value over time due to physical decay or market obsolescence. They can be classified into two categories;

      Physical Deterioration: Items like meat, vegetables and medicine, spoil decay or get damaged.

      Value obsolescence: Items like electronics or fashion goods that lose value due to new technological developments or changing consumer preferences.

      Despite its importance deterioration is often overlooked in warehouse planning and modeling.

      Warehouse are classified based on location using factors such as population density, resource availability, and business activity. The classification is as follows:

      Location type	Index (i)	Population Range	Notation
      Village Level	i =1	0 % to u%	0 S u
      City Level	i = 2	u% to v%	u S v
      Micro City Level	i = 3	v% to 100%	v S 100

      Constant u and v are determined based on official population records of the practical experience of supply chain experts and planners.

      Demand

      Location 1 Village

      Parameter

      Location 2 City

      Location 3 Micro City

      Fig.-i: Warehouse Classification based on location

2. A Review:

   Chen and Kang (2009-a) [1] formulated the integrated inventory models under the two-level trade credit policy with price sensitive demand and negotiation scheme. A review article on trade credit can be reviewed by Cheng et al (2009) [2] incorporated the concept of vendor and buyer integration and order size dependent trade credit. Different other researchers [7-10] have developed inventory models assuming demand as constant, time dependent, stock dependent or price dependent. X. Wang, Tang, and Zhao [6] worked on fuzzy economic order quantity inventory model without backordering.

   Teng and Chang (2009) [5] gave the models discussing Vendor-buyer inventory models with trade credit financing under both non-cooperative and integrated environments. Jagi (1995)

   discussed the effect of deterioration of units when delay in payments is permissible. Teng et al (2005) [3] developed retailers optimal ordering and pricing policy for the assumption of deterministic and constant demand.

3. Symbols and Assumptions:

   The proposed model is based on the following assumption:

   Item demand is deterministic, and the demand rate varies by location and over the given time horizon. The demand rate at the ith location is given by Ri = Qi/T, where Qi is the order quantity and T is the total schedule time.

   • The schedule time T is constant.

   • Replenishment occurs instantaneously (infinite replenishment rate)

   • The order quantity for the ith location is Qi. The variable ai, bi and ci represent the quntities stored in warehouse OWi , RW1i, RW2i, respectively. Here ai, is the primary storage preference, followed by bi and finally ci.

   • The deterioration rates of items stored in OWi, RW1i, RW2iat the ith location are denoted by xi, yi, and zi respectively.

   • The holding costs per unit for items stored in OWi, RW1i, RW2i at the ith location are represented by B1i, B2i, and B3i respectively.

   • Thus, the order quantity is Qi = ai+ bi + ci and the deterioration adjusted value is

     ai + bi + ci

     xi yi zi

   • Although the warehouse capacities OWi, RW1i, and RW2i are considered unlimited, the optional stock level for each is determined by comparing their holding cost and deterioration rates to minimize overage cost.

   • Di represents the shortage cost per unit time.

4. Proposed Inventory Model:

   The total lot size Qi + Si enter the system where Si represent any leftover shortages from the previous cycle and Qi is the inventory for the current period T. Out of Qi the inventory is distributed as follow: ai units in OWi bi unit in RW1i and ci units in RW2i all stored simultaneously.

   The consumption process being with goods Zi from RW2i which are consumed and cleared by time t2. Then from time t2 to t1 ,yi from RW1i is consumed. Finally, from time t1 to t0, consumption of a from OWi starts. Deteriorated items are discarded once that inventory is depleted.

   Due to deterioration, shortage occur and are associated with cost coefficients xi, yi, zi for OWi, RW1i, RW2i respectively. The total shortage cost over the period T is given by xiai + yibi + zici . The objective is to

   determine the optimal inventory level aO for OWi and bO for RW1i such that the total average cost is minimized

   i i

   across different location.

   Fig.-ii: Three-warehouse set-up

5. Mathematical Formulation:

   As per assumption for demand

   (1-zi)ci = (1-yi)bi = (1-xi)ai

   = xiai + yibi + zici = Qi = R (1)

   t2 t1-t2

   tO-t1

   T-tO T i

   Average holding cost of warehouse RW2i

   = 1 { c t z + 1-yi c t }A

   T i 2 i 2 i 2 3i

   Average holding cost of warehouse RW1i is

   = 1 {b t + 1-yi (t – t )b + yb (t -t )}A

   T i 2

   2 1 2 i

   i 1 2 2i

   Average holding cost of warehouse OWiis

   = 1 {a t + 1-xi (t -t )a + xa (t -t )}A

   T i 1

   Average shortage cost over time T

   2 O 1 i

   i O 1 1i

   = 1 (x a + y b + z )(T to D )

   2T i i i i i i

   So, the total average cost TCi is: –

   1 1 – zi 1 – yi

   Tc = [[c t z + c t }A + [b t + b (t -t ) + y b (t – t )}A ]

   i T i 2 i

   2 i 2 3i i 2

   2 i 1 2

   i i 1 2 2i

   1

   + [[a t

   + 1 – xi (t

   – t )a

   + a x (t

   – t )}A ]

   T i 1

   2 O 1 i

   i i O 1 1i

   + 1 [(x a + y b + z c )(T – t )]D (2)

   2T i i i i i i O i

   Tc = 1 [[1 + zi c t }A + [b t + 1 – yi b (t – t )}A ]

   i T 2

   i 2 3i i 2

   2 i 1 2 2i

   1

   + [[a t

   1 + xi

   + a (t -t ) + a (t

   -t )}A ]

   T i 2 i 1 2 2 i O 1 1i

   + 1 1

   T 2

   [ (xiai + yibi + zici )(T – tO)Di] (3)

   Tc = 1 [[1 + zi c t }A + [b t + 1 + yi b (t – t )}A ]

   i T 2

   i 2 3i i 2

   2 i 1 2 2i

   1

   + [[a t

   1 + xi

   + a (t -t ) + a (t

   -t )}A ]

   T i 2 i 1 2 2 i O 1 1i

   + 1 1

   T 2

   [ (xiai + yibi + zici )(T – tO)Di] (4)

   Also , from the fig we have Zi = Qi – ai – bi and after replacing the proportional value of

   t2, (t1 – t2), (tO – t1)and(T – tO) from equation (1) we get

   2 i

   i 2

   1 1 – z2 1 – y2

   Tc = [(Q – a – b ) A + {(Q – a – b )b (1-z ) + y }A ]

   i Qi

   i i 2 3i

   i i i i i

   2 i 2i

   1

   + [{a b (1- y ) + (Q – a

   2

   a

   – b )a (1- z ) + i (1 – x2)}A ]

   Qi i i i

   i i i i 2

   i 1i

   + 1 1 {

   }2 (5)

   [

   Qi 2

   aixi + biyi + zi(Qi – ai – bi) Di]

   On differentiating Tciw.r. to ai and bi we have

   aTci = 1 [ – (Q – a – b )(1- z2)A -b (1 – z )A ]

   aai Qi

   i i i

   i 3i i

   i 2i

   + 1 [{b (1 – y ) + (1 – z )(Q – 2a – b ) + a (1 – x2)}A ]

   Qi i i

   i i i i i

   i 1i

   + 1 [{a x + b y + z (Q – a – b )}(x – z )D ] = 0 (6)

   Qi i i

   i i i i

   i i i i i

   And, aTci = 1 [ – (Q – a – b )(1- z2)A + {(Q – a – 2bi)(1 – z ) +b (1 – y2)}A ] +

   ai Qi

   i i i

   i 3i i i

   i i i 2i

   1 [{a (1- y ) – a (1-z )}A + {a x + b y + z (Q – a – b )}(y – z )D ] = 0

   Q i i i

   i 1i

   i i i i i i

   i i i i i

   – (7)

   Now above two equations can be written as

   F1iai + F2ibi = F3i (8)

   And

   Where,

   F2iai + F4ibi = FSi (9)

   F1i = [ (1 – z2)A3i – (1 – 2zi + x2)A1i + (xi – zi)2Di]

   i i

   i

   i

   F2i = [ (1 – z2)A3i – (1 – zi)A2i + (zi – yi)A1i + (yi – zi)(xi – zi)Di] F3i = [ ziQi(zi – xi)Di + Qi(1- z2)A3i – Qi(1 – zi)A1i]

   F4i = [ (1 – z2)A3i – (1 – 2zi + y2)A2i + (yi – zi)2Di]

   i i

   i

   FSi = [ ziQi(zi – yi)Di + Qi(1- z2)A3i – Qi(1 – zi) A2i]

   Solving equation (8) and (9) we have optimum result aO and bO as

   i i

   Also consider, r = a2Tci = 1 [(1 – z2)A – (1 – 2z + x2)A + (x – z )2D ] (10)

   i

   i aa2 Qi

   i 3i

   i i 1i

   i i i

   s = a2Tci = 1 [(1 – z2)A – (1- z )A + (z -y )A + (y – z )(x – z )D ]

   i aaiai Qi

   i 3i

   i 2i

   i i 1i

   i i i i i

   – (11)

   t = a2Tci = 1 [(1 – z2)A – (1 – 2z + y2)A + (y -z )2D ] (12)

   i

   i a2 Qi

   i 3i

   i i 2i

   i i i

   Here A1i < A2i < A3i < Di and xi, yi, zi are probabilities of deterioration lies between 0 to 1.

   Now Hessian matrix H for Tci (ai, bi)is:

   H= [ri si] si ti

   To confirm a minimum, the Hessian must be positive definite:

   1. ri > 0 ,

      t

   2. Determinant of it riti – s2 > 0

   Here r = a2Tci (13)

   i

   i aa2

   = 1 [(1- z2)A – (1 – 2z + x2)A + (x -z )2D ] > 0

   Qi i 3i

   i i 1i

   i i i

   t

   And also since, riti – s2 > 0 (14)

   Clearly from (13) and (14) we observe that TC is minimum for aO, bO

   i i

   and cO = Qi – aO – bO

   i i i

6. Example and Comparison:

   The monthly demand for the item remains uniform at 500 units. Order size is remained constant for a year and shortages are backlogged. The firm has storage facility they are own warehouse and two rented warehouses. Holding cost for the storage.

   Capacities are rupee 10,12,15 per unit time respectively deteriorations are 30,20 and 10 on accordance and shortage is at the rate Rs.2 per unit time.

   Given data: –

   • Demand = 5000 unit / month

   • Time Horizon (T) = 1 year

   • Total Annual Demand (Q0) = 5000 × 12 = 60000 unit

   • Capacities of warehouse.

     Own warehouse (a1) = 10, Rented warehouse (a2) =12, (a3) =15

   • Deterioration rate: 0.3, 0.2,0.1 respectively, Shortage Cost Rs.2 per unit time

   • For two warehouses:

     a1 = 42353 unit b1 = 17647 unit

     Average cost (TC)=Rs.313324.6

   • For three warehouses:

   a1 = 19917 unit, a2 = 38265 unit, a3 = 1818 unit Average cost (TC) = Rs.308888.9

   Clearly Ai =Rs. (313324.6 308888.9) =Rs. 4435.7 which is positive

7. Sensitivity analysis:

   Sensitivity with Respect to Holding Cost

   Table-1. Sensitivity Table: Holding cost: Holding costs are varied by ±10% and ±20%.

   Change in Holding

   Cost

   Two-Warehouse TC (RS.)	Three-Warehouse TC (RS)
   20%	301,120	296,540
   10%	307,215	302,610
   Base	313,324.60	308,888.90
   10%	320,540	316,120
   20%	328,910	324,780

   Total cost increases almost in a straight line as holding cost increases. The three-warehouse system always gives lower total cost, showing it works better. When holding cost is higher, firms keep less inventory and use shorter cycle times.

   Sensitivity with Respect to Deterioration Rate: Deterioration rates are increased and decreased by ±10% and ±20%.

   Table-2: Sensitivity Table: Deterioration Rate

   Change in Deterioration Rate	Two-Warehouse TC (RS)	Three-Warehouse TC (RS)
   20%	305,420	300,610
   10%	309,380.00	304,780.00
   Base	313,325	308,889
   10%	318,910	314,540
   20%	325,840	321,720

   Total cost changes a lot when the deterioration rate changes. The impact is stronger in the two-warehouse system because items deteriorate more there. The third warehouse, with low deterioration but higher rent, helps reduce losses by acting as a buffer.

   Sensitivity with Respect to Shortage Cost: As, shortage cost is varied by ±10% and ±20%.

   Table-3: Sensitivity Table: Shortage Cost:

   Change in Shortage Cost	Two-Warehouse TC (RS)	Three-Warehouse TC (RS)
   20%	309,110.00	305,340.00
   10%	311,215	307,120
   Base	313,325	308,889
   10%	316,890	312,440
   20%	325,840	321,720

   When shortage cost increases, total cost increases but not too much. The three-warehouse system reduces shortages better because inventory is managed more efficiently.

   Table-4: Comparative Sensitivity Summary:

   Parameter	Sensitivity Level	Key Insight
   Holding cost	High	TC rises sharply; incremental storage is beneficial
   Deterioration rate	Very High	Major driver of cost; justifies multi-warehouse policy
   Shortage cost	Moderate	Less influential than holding and deterioration

   Conclusion of Sensitivity analysis

   The three-warehouse system stays the best choice even when conditions change, showing it is very reliable. Firms that handle fast-deteriorating items gain a lot from using warehouses with low deterioration, even if rent is high. The sensitivity results show that investing in better storage is more important than just choosing the cheapest rent.

8. Conclusion: – This indicates that the three-warehouse setup provides cost saving compared to the two- warehouse system.

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