A MULTI-ITEM INVENTORY MODEL WITH VARIOUS DEMAND FUNCTIONS CONSIDERING DETERIORATION AND PARTIAL BACKLOGGING

Article History: Inventory management is an important thing to be considered in order to run the activities of a company smoothly. By considering the deterioration factor, partial backlogging policy


INTRODUCTION
Inventories are stored goods for future use or sale. Good inventory management is needed by companies to maintain the continuity of their business through managing inventory costs. According to [1], inventory costs generally consist of purchasing costs, ordering costs, holding costs, and shortage costs. In addition, determining the optimal order time and order quantity are two things that need to be given special attention. Both of these are related to the availability of goods to fulfill demands from customers. If there is too much inventory, then the holding costs will increase. Conversely, if there is too little inventory, there can be a shortage of goods, which will result in lost sales and a loss of potential profits.
Other than demand factors, the deterioration of goods is one of the important factors that need to be considered. Deterioration is a decrease in the quality of an item that results in the item being unusable or having no selling value. Deterioration will occur if goods are stored too long in the warehouse, especially medicines, food, vegetables, and fruit. If the inventory stays in the warehouse for too long, there can be a reduction in quality, loss, or damage to the goods when they arrive to consumers. Therefore, companies shouldn't store deteriorating goods for a long period and in large quantities, as it will be detrimental to the company itself. Several mathematical models have been developed involving deterioration factors (see, for example, [21], [28], [29], [30], [31], [32]).
Inventory management generally aims to fulfill consumer demand. In reality, there are conditions where consumer demand cannot be fulfilled. When the demand comes, the company's inventory has run out. In these conditions, there are a number of consumers who want to wait until the goods arrive (backorder), but there are also those who do not want to wait and look for other companies (lost sales). Related to this, companies can take a policy called partial backlogging. Partial backlogging is a policy of fulfilling a certain amount of consumer demand even though inventory has run out. It is assumed that not all consumers are willing to wait until the inventory is available again so that only part of the consumer demand is fulfilled. Several studies have included this partial backlogging in the inventory model developed, including [5], [8], [23], [31], [32].
[31], [32] have developed an inventory model for deteriorating goods with time-dependent demand and holding costs and considering partial backlogging. This paper is an extension of the model in [31], [32] by considering three types of goods with different demand functions for constant-deterioration goods and considering partial backlogging. Of the three types of goods, the ordering policy that minimizes the total cost of inventory will also be determined. This ordering policy is related to multi-item inventory management, such as joint order policy, individual order policy, or a combination of the two policies.
There are three models developed, namely: (1) Model I, which is an inventory model with the inventory-dependent demand function, (2) Model II, which is an inventory model with a time-dependent demand function, (3) Model III, which is an inventory model with an exponentially decreasing demand function. The decision variable of the developed model is to determine the time when to reorder and the time when the inventory runs out. From these decision variables, the optimal order quantity and total inventory cost will be determined by choosing the right ordering policy.

RESEARCH METHODS
The mathematical model in this paper is developed on the basis of the following notations and assumptions.

Notations
The notations used in the development of this model are: : holding cost per cycle, : deterioration cost per cycle, : shortage cost per cycle, : ordering cost per order, : lost sales cost per cycle, : deterioration cost per unit for the i th item, : shortage cost per unit per cycle for the i th item, : holding cost per unit per cycle for the i th item, : ordering cost per unit per cycle for the i th item, : lost sales cost per unit for the i th item, : percentage of deteriorating item, : percentage of partial backlogging item, : reorder time,

Assumptions
The following assumptions are used in this model. 1.
There is no lead time for ordering items, meaning that inventory will be replenished immediately when inventory runs out and an order is placed.

2.
The deterioration factor is constant, ∈ (0,1), and there is no replacement or repair for deteriorated items.

3.
Inventory-dependent demand function is expressed as follows: where is the initial demand and is the demand during backlogging, with > 0, > 0, and 0 < < 1.

4.
Time-dependent demand function is expressed as follows: where is the initial demand and is the demand during backlogging, with > 0, > 0, and 0 < < 1.

5.
The demand function for an exponentially decreasing item is expressed as follows: where is the initial demand and ℎ is the demand during backlogging, with > 0, ℎ > 0, and 0 < < 1.

6.
When the inventory runs out but the demand is still there or there is a shortage of items, at time ∈ [ 1 , ], a partial backlogging policy will be used. The partial backlogging function is expressed as follows [31], [32]: where 0 < < 1. is an illustration of an inventory model with a partial backlogging factor. At the beginning of the cycle ( = 0) there is an inventory of units. Inventory will decrease due to demand and deterioration factors and finally the inventory will run out at 1 . When the inventory runs out but there is still demand, partial backlogging policy will be applied until the items are available for the next cycle.

Development of Model I
The inventory decreases due to demand and deterioration factors in the interval [0, 1 ] can be modeled through the following differential equation: where ( ) is as in (1). By using the boundary condition 1 ( 1 ) = 0 in Equation (5), we obtain By using the initial condition 1 (0) = in equation (6), the maximum inventory level ( ) can be determined as follows: At the time = 1 , the items run out, so there is a shortage in the interval [ 1 , ]. Thus, the inventory level is expressed as follows: By using the boundary condition 2 ( 1 ) = 0 in Equation (7), we obtain Let = in Equation (8), we obtain the maximum amount of backlogged demand per cycle ( ) as follows: So, the order quantity per cycle is given by There are five cost components for the total cost of inventory for one year, that is deterioration cost (DC), shortage cost (SC), holding cost (HC), ordering cost (OC), and lost sale cost (LSC), which each are given below.

Deterioration Cost (DC)
Shortage Cost (SC) Holding Cost (HC) Ordering Cost (OC) Lost Sale Cost (LSC) Total Cost of Inventory for One Year To find the values of 1 and that minimize the total inventory cost for one year, Equation (15) must satisfy the following conditions: 1.
The first partial derivative test, to find the stationary point ( 1 , ), is:

Development of Model II
Using the same approach as the development of Model I, the maximum inventory level ( ) and the order quantity per cycle ( ) for Model II are given as follows: The deterioration cost (DC), shortage cost (SC), holding cost (HC), ordering cost (OC), and lost sale cost (LSC) for Model II are given below.

Total Cost of Inventory for One
Year The same conditions as of the Model I are also applied in order to find the values of 1 and that minimize the total cost of inventory for one year in Equation (22).

Development of Model III
In the same way as the development of Model I, the maximum inventory level ( ) and the order quantity per cycle ( ) for Model III are as follows: There are five cost components for the total cost of inventory for one year are given below.

Total Cost of Inventory for One Year
Equation (29) must satisfy the same conditions as the Model I in order to find the optimal values of 1 and that minimize the total cost of inventory for one year.

Ordering Policy for Multi-Item Inventory Model
In ordering items, there are three types of replenishment policies that can be applied, namely individual order policy, joint order policy, and the combination of the two policies.

Individual Order Policy
The individual order policy is a policy where the company places orders for each type of item separately. Suppose is the number of types of items and is the total cost of inventory for the i th item, then the total cost of the individual order policy is

Joint Order Policy
The joint order policy is a policy where the company places orders for all types of items together from the same supplier. This means that the order is only made once so the ordering cost is only charged once. Suppose is the number of types of items and is the total cost of inventory for the i th item, then the total cost of the joint order policy is

Combined Order Policy between Individual and Joint Order
This policy is a combination of individual and joint order policies and can only be used if the company orders more than two types of items. Suppose, the company places an order for three types of items, then there are three alternatives for the replenishment policy. a. Order the first item and the second item together, while the third item separately. The total cost for the first alternative is b. Order the first item and the third item together, while the second item separately. The total cost for the second alternative is = 0 1,3 + ∑ ( + + + ) + ( 2 + 2 + 2 + 2 + 2 ) c. Order the second item and the third item together, while the first item separately. The total cost for the third alternative is = 0 2,3 + ∑( + + + ) + ( 1 + 1 + 1 + 1 + 1 ) 3 =2 (34)

Results
Suppose a company sells three types of items with parameter values shown in Table 1 and the ordering cost of joint order policy and combined order policy between individual and joint order given in Table 2.
If the company orders items separately (individual orders), then using equation (30), the time when the inventory runs out ( 1 ), the reorder time ( ), the order quantity ( ), and the total cost ( ) for each of the three items is shown in Table 3.
If the company orders items for the three types of items individually, the total cost that must be incurred by the company for 1 year is IDR 416,383.  Furthermore, the total cost incurred if joint order policy is applied will be analyzed. To analyze the total cost, is selected based on the optimal value of each of the three items and the optimal value of the joint order policy.
If the company orders items jointly (joint order), then using Equation (31), the total cost incurred for each selected is shown in Table 4, while the time when the inventory of the three items runs out and the order quantity for each of the three items of the minimum total cost is shown in Table 5.
From Table 4, the minimum total cost with a joint order policy is when = 0.3, which is IDR 374,043. From Table 5, it is obtained that the company must place an order quantity for the first item is 42 units, while the second and third items are 41 units each, with the time when the inventory of the three items runs out being 0.2262, 0.2383, and 0.2489 years (about 83, 87, and 91 days), respectively.  Table 5. Values of , , and , , for = .

Notation Value
Time when the inventory runs out Next, we will analyze the total cost incurred for a combination of individual and joint orders, where this policy has three alternatives.
If the company orders items in combination, then using equations (32), (33), (34), the total cost incurred for the first and third alternatives is shown in Table 6 and for the second alternative is shown in Table 7. The time when the inventory runs out, the reorder time, and the orders quantity for each of the three items of the minimum total cost is shown in Table 8.
From Table 6 and Table 7, the minimum total cost obtained by using the first alternative is IDR 386,966. From Table 8, the company must place an order for the first and second items every 0.2902 years (about 106 days), while the third item every 0.4282 years (about 156 days), with the time until the inventory of the first, second, and third items runs out being 0.1696, 0.1780, and 0.3081 years (about 62, 65, and 112 days), respectively. The orders quantity for each of the three items are 33, 32, and 49 units, respectively.

Discussion
Based on the numerical example above, the total cost generated by the joint order policy is cheaper than the total cost generated by the individual order policy or the combined individual and joint order policies. This makes sense because the ordering cost when ordering all three types of items at once from the same supplier is only charged once; this can save the company expenses.
Furthermore, a sensitivity analysis will be carried out to determine the effect of changes in ordering costs for the joint order policy and the combined policy between individual and joint orders on the time when the inventory runs out, the reorder time, and the total cost.
From Table 9, it is obtained that the greater the ordering cost for the joint order ( ) policy, the longer the time when the inventory of the three items runs out ( 1 , 2 , 3 ) and the longer the reorder time ( ). This happens because of the increase in the order quantity of the three items ( 1 , 2 , 3 ) so that the time required during the sales period is getting longer. In addition, the greater the ordering cost, the greater the total cost ( ) that must be incurred because the order quantity is increasing. From Table 10 − Table 12, it is obtained that for the three alternative combined policies, the greater the ordering cost for items ordered together and the ordering cost ordered separately, the longer the time when the inventory of the three items runs out ( 1 , 2 , 3 ) and the longer the reorder time ( ). This happens because of the increase in the order quantity of the three items ( 1 , 2 , 3 ) so that the time required during the sales period is getting longer. In addition, the greater the ordering cost, the greater the total costs ( ) that must be incurred because the order quantity is increasing. This has the same effect as the joint order policy. From the sensitivity analysis results, it is found that when determining ordering policies for three types of items, the ordering cost ( ) is very influential in contributing to the total cost ( ), that is, the greater the ordering cost, the greater the total cost.

CONCLUSIONS
In this paper, three inventory models have been developed with demand functions depending on inventory, time, and the following exponential function. There are three alternative ordering policies available for the company to choose from in order to minimize the total inventory cost. Based on our numerical examples and sensitivity analysis, we conclude that: a. Compared to other policies, the joint order policy gives the minimum total cost. b. When the ordering cost for the joint order policy and combined policies is higher, then the time the inventory runs out becomes longer and also the reorder time. c. The ordering cost has a substantial contribution to the total cost compared to other costs for the three ordering policies.
Using a suitable distribution for demand and considering discount factors from the supplier are some directions for further research.