Three-Echelon Inventory Model with Permissible Delay in Payments under Controllable Lead Time and Backorder Consideration

This paper proposes a three-echelon inventory model with permissible delay in payments under controllable lead time and backorder consideration to find out the suitable inventory policy to enhance profit of the supply chain. In today’s highly competitive market, the supply chain management has become a critical issue in both practice and academic and supply chain members have to cooperate with each other to bring more benefits. In addition, the inventory policy is a key factor to influence the performance of the supply chain. Therefore, in this paper, we develop a three-echelon inventory model with permissible delay in payments under controllable lead time and backorder consideration. Furthermore, the purpose of this paper is to maximize the joint expect total profit on inventorymodel and attempt to discuss the inventory policy under different conditions. Finally, with a numerical example provided here to illustrate the solution procedure, we may discover that decision-makers can control lead time and payment time to enhance the performance of the supply chain.


Introduction
Inventory policy and supply chain are closely linked to each other; inventory policy can easily influence the performance of supply chain [1].Jammernegg and Reiner [2] pointed out that the efficient inventory management can enhance performance of supply chain.To enhance the performance of the supply chain, firms play the different role in the supply chain, and they should have a good partnership and share the information with each other.Cachon and Fisher [3] established a platform for firms in the supply chain; it could share information, built the forecast model, and reduced the risk from inventory model.Olson and Xie [4] proposed that vendor and buyer should use the same inventory system to cooperate with each other, and inappropriate inventory policy might cause firms to bear great loss.With the above discussion, the proper inventory policy is a critical part in the supply chain.We would like to develop an integrated inventory model to obtain the optimal solutions and attempt to discuss the inventory policy, and next we have to review some previous literatures to define the research gaps.
Stock is a major part in inventory policy.Excess stocks cause higher cost, and shortage cannot satisfy customer immediately.Frankel [5] believed that how to control stocks was a vital practice in supply chain.Das Roy et al. [6] extended Maddah and Jaber's [7] model to discuss an optimal shipment strategy for defective items in shortage situation.Sana [8] researched two-warehouse systems in EOQ model with limited capacity of own warehouse.Due to the stochastic demand, they regarded lost sale as penalty in shortage conditions.Nasiri et al. [9] emphasized that firms should keep safety stocks to satisfy customer under instable demands; hence, how to keep adequate stocks is an extremely critical issue in inventory policy.Shortage often causes a great sales loss, normally; firms adopt backorder method to lower sales loss; backorder means the buyer does not receive cargos in time and allows cargos delivery late.Lo et al. [10] indicated that backorder was an important factor in their inventory 2 Mathematical Problems in Engineering model; Pan et al. [11] added backorder into their inventory model that considered lead time and safety factor.When shortage occurs, in addition to using backorder method to handle unsatisfied customers, firms can also offer few discounts to save customers.Ouyang et al. [12] discussed the impact of backorder discount in the inventory model under backorder consideration.Pan and Hsiao [13] established an integrated inventory model with controllable lead time and backorder discount considerations.Therefore, backorder consideration should be mentioned in inventory model.
In addition to backorder method, compressed lead time can also alleviate shortage condition.Lead time includes preparative time for send orders, set-up time of facilities, productive time, transportation time, and so on.When the firms face the uncertain demands, they usually compress the lead time.Chandra and Grabis [14] indicated that short lead time could enhance the service level and lower inventory level effectively, but short lead time also caused highly order cost.The previous researches have regarded lead time as a constant, but more and more researches have regarded lead time as a variable or controllable parameter in recent years; Ben-Daya and Raouf [15] added lead time as a decision variable in the inventory model.Xia and Huang [16] established an inventory model with quantity discount and controllable lead time when demand distribution was unknown.And Song et al. [17] studied the effects of controllable lead time in the inventory model; they indicated that shorter lead time can not only satisfy customer's demand, but also lower inventory level effectively.In 2008, Lo and Yang [18] proposed a notable idea.They showed the relationship between lead time and delay payment time, and their cost functions in the model would be divided into two cases by different payment times.But they considered lead time as a constant.If the lead time is regarded as a variable, it is interesting to know whether we can find out the better inventory policy by controlling lead time and payment time.
Permissible delay in payments is a common commercial strategy in real life; the purpose of this strategy is to enhance elasticity of capital.In other words, firms can obtain additional interest income from unpaid payments, but equally firms may pay the additional opportunity cost in some conditions.Lo and Yang [18] indicated that firms should let payment time be longer than lead time to obtain additional interest income.Chen and Kang [19] established an integrated inventory model with permissible delay in payments.Zhu et al. [20] considered permissible delay in payments in their inventory model with limited storage capacity and shortage.Huang et al. [21] proposed an inventory policy between buyer and vendor and studied the relationship between permissible delay in payments and order-processing time reduction.Sana and Chaudhuri [22] proposed a deterministic EOQ model with delays in payments and price-discount offers; they considered various types of deterministic demands for retailer to operate their model.They all have added delay payment method in their inventory models.To fit the real life, inventory model should take delay payment into account.
There are many issues that can be discussed in inventory model, and we reviewed the literatures about backorder, controllable lead time, and permissible delay in payment.Shortage is an extremely common condition in real life and always accompanies sales loss.Hence, in the second paragraph, we simply defined backorder.To lower the sales loss, the backorder method is widely used in years.After reviewing Ouyang et al. [12] and Pan and Hsiao's [13] research, we would like to regard backorder as a major condition in our inventory model.
In the third paragraph, we knew that compressed lead time was another method to alleviate shortage condition.Recently researches have indicated that the controllable lead time is a popular issue in inventory model.Lead time consists of many components, and these components such as productive process and transportation process can be compressed by renewing facilities or using fast vehicles.Short lead time can respond to demand quickly and enhance the service level.With high service level, firms can gain more revenue from their customers but compressing lead time will cause additional cost to firms.With inspiration from Lo and Yang's [18] research, it is interesting to know whether we can find out better inventory policy by controlling lead time and payment time.
In the fourth paragraph, we discussed the permissible delay in payment, a common commercial strategy in business; the strategy aims to enhance the elasticity of capital.Vendor always benefit from interest income by delaying payment time, but buyer would pay an additional opportunity cost.Generally, supply chain can benefit from delay strategy; hence, how to maximize the total profit by adjusting payment time is a potential issue to discuss.
To simulate supply chain completely, we would like to discuss the above-mentioned ideas in three-echelon inventory model.There are several researches that have studied inventory problems in three-echelon models.Roy et al. [23] built a three-echelon supply chain model involving newsboy problem to deal with the inventory control problems.They also claimed that uncertain demand may cause shortage condition in their integrated inventory model.Sana [24] investigated an EOQ/EPQ model in three-echelon supply chain management; he focused on the volume flexibility and replenishment lot size problem in a collaborating system.Pal et al. [25] established a three-echelon inventory model; they considered multi-items and multisuppliers to make model more realistic.Sana et al. [26] proposed a three-echelon supply chain model with multiple suppliers, manufacturers, and retailers for multiple items.They aimed to compare the expected profit between the collaborating system and Stakelberg game structure.Three-echelon inventory model has clearly shown how supply chain works in the above researches.
With the above discussion, we would like to combine the concepts of Pan and Hsiao [13] and Lo and Yang's [18] researches to develop a three-echelon integrated inventory model with delay payment, controllable lead time, and backorder consideration.In this paper, we can determine our proposed inventory model's optimal solutions and sensitivity analysis of decision variable.Finally, with appropriate backorder condition, we can attempt to determine better inventory policy by controlling lead time and payment time.

Notations and Assumptions
Tersine [27] believed that lead time was the time between the placement of an order and delivery of cargos.And lead time can be divided into preparative time for send orders, set-up time of facilities, productive time of products, transportation time, and so on.Lead time always includes transportation and productive process, but supplier just supplies the raw materials to the manufacturer.Hence we may suppose that manufacturer's lead time only includes transportation process, and retailer's lead time includes both.Lead time can be reduced by paying additional cost, and this additional cost was called "crashing cost" [12,20,21].According to Pan and Hsiao's [13]

Assumptions
(1) This is a single supplier, a single manufacturer, and a single retailer for a single product.
(2) Shortages are not allowed at supplier and manufacturer.
(3) Lead time of manufacturer includes transportation process, and lead time of retailer includes transportation and productive process.
(4) The lead time of retailer consists of n mutually independent components; the jth components have a normal duration   and a minimum duration   ,  = 1, 2, . . ., .
(5) Let   be the length of lead time with components 1, 2, . . .,  crashed to their minimum values, and   can be expressed as 6) The relationship between the supplier's purchasing cost   , manufacturer's purchasing cost   , retailer's purchasing cost   , and retailer's selling price  is  ≥   ≥   ≥   .
(7) We regard imperfect production as basic condition in our proposed model, and inspecting costs are equal to    and    and repair costs are equal to      and      [18].(8) The imperfect products will completely and immediately fix [18].
(9) The supplier offers the manufacturer a permissible delay period .During this permissible delay period, the manufacturer sells the goods and uses the sale revenue to earn interest at a rate of   .At the end of this time period, the manufacturer pays the purchasing cost to the supplier and the rest of the goods in the warehouse bring a capital opportunity cost at a rate of   .
(10) The time horizon is infinite.
(13) Inventory is continuously reviewed and replenishments are made whenever the inventory level falls to the reorder point .

Building Model
In this section, we discuss and develop the supplier's model, manufacture's model, and retailer's model in following section and combine the above models to an integrated inventory model.To build the proposed model, we extended Lo and Yang's [18] research to compute relevant opportunity cost and interest income and referred to Pan and Hsiao's [13] research to compute relevant shortage and crashing cost.
As for the holding cost, it could be computed by using integration in Appendix A's Figures 1, 2, and 3 [28].
3.1.The Supplier's Total Annual Profit.In each production run, the supplier's costs include ordering cost, inspecting cost, repair cost, holding cost, and opportunity cost.And the supplier's revenues include sales revenue and interest income.With permissible delay in payments, we knew that, if payment time was longer than lead time, then it would bring additional interest income to the manufacturer; in addition, the supplier must pay equally opportunity cost.
If payment time was shorter than lead time, then it would bring additional opportunity cost and fewer interest income to manufacturer; also, the supplier could earn interest income and pay fewer opportunity cost.Owing to the fact that the supplier's profit function has that two following cases, based on length of lead time   and payment time , the different parts between two possible cases are as follows.
Thus, TP 1 and TP 2 are given by TP 1 = sales revenue − ordering cost − inspecting cost (1)

The Manufacturer's Total Annual Profit.
In each production run, the manufacturer's costs include ordering cost, inspecting cost, repair cost, holding cost, transportation cost, and opportunity cost.And the manufacturer's revenues include sales revenue and interest income.We have already known the relationship between length of lead time and payment time from Section 3.1.Owing to the fact that the manufacturer's profit function has the two following cases, based on length of lead time   and payment time , the different parts between two possible cases are as follows.
Thus, TP 1 and TP 2 are given by TP 1 = sales revenue − ordering cost − inspecting cost (2)
Step 3.With the different duration, substitute   and   into (11) to find  1 .

The Sensitivity Analysis of Decision
Variable.First, we need to know how   and   influence the EJTP  (  ,   ,   ) and individual profit, and, according to function (4), we developed four different models, and we could reach the optimal EJTP  (  ,   ,   ) by applying the algorithm; in addition, we take the EJTP 1 ( 1 ,   ,   ) as an example here; the results are shown in Table 2.
The results are shown in Table 2; we could observe how lead time of retailer   and backorder discount   influenced the EJTP 1 ( 1 ,   ,   ) and individual profit.The total profit of EJTP 1 ( 1 ,   ,   ) decreased with longer lead time of retailer   , and the profit of retailer increased with longer lead time   .We might indicate that performance of EJTP 1 ( 1 ,   ,   ) could be enhanced by compressing the lead time but it also caused lower profit for the retailer.The profit of EJTP 1 ( 1 ,   ,   ) and retailer could reach the optimal results when   = 3; it is suggested that backorder discount should be offered appropriately to the buyer.Owing to the fact that the expected joint total annual profit was greater than individual's profit (supplier + manufacturer + retailer) and shorter lead time was more benefit for EJTP 1 ( 1 ,   ,   ).It seems that compressing the lead time was an available policy in our proposed inventory models.And we continue discussing the more detailed inventory policy in the following sections.

The Permissible Delay Period 𝑋 and EJTP.
Let us observe the value of profit with  0 = 0.8 by varying the permissible delay period , and, with applying the proposed procedure, the solution results are contained in Figure 4 (see Appendix B).
To compare which condition is more beneficial to our proposed inventory model, we have more detailed results in Table 3.
According to Section 3.1, we knew that, if payment time was longer than lead time, then it would bring additional interest income to the buyer.On the other hand, if payment time was shorter than lead time, it would cause buyer to bear additional opportunity cost.From the above discussion, we had four possible cases in EJTP  (  ,   ,   ), for  = 1, 2, 3, 4, and considered the different duration of   , for  = 0, 1, 2. Hence, we had twelve possible values of profit.According to the data from Table 3, we could indicate that the optimal profit occurred in EJTP 4 ( 4 ,   ,   ) with normal duration condition, and the worst profit occurred in EJTP 3 with normal duration condition.
The Hessian matrix H at the optimal solution  * No matter which payment time we varied, the optimal profit still occurred in the same condition.Hence, through controllable lead time and payment time, we could find the optimal policy in our proposed integrated inventory model.

Conclusions and Future Works
In this paper, we discussed the issue of a three-echelon inventory model with permissible delay in payments under controllable lead time and backorder consideration to find out the suitable inventory policy to enhance profit of the supply chain.Owing to information revolution, the community can get anything they want easily.It causes that any goods in the market always have a stable price due to the inspection by community.Hence, the enterprises are hardly raising the price on their product.If they want to enhance the profit, to control the relevant cost and reduce it efficiently is a good approach to enhance profit for them.
In the manufacture part, the firms could delay the payment time to obtain additional interest income, and the proper payment time could be a great benefit for the firms.In the customer part, length of lead time usually influenced the price of products.Fast lead time could respond to customer's demand quickly, and firms could request customers for high price.Inevitably, firms must pay the additional cost for reducing lead time.In the stock part, firms usually adopted backorder approach to minimize sales loss and offered some proper discount to save customers.
From the above discussion, we proposed our inventory model and analyzed its profits, in a three-echelon inventory model case.First, we observed that integrated profit is greater than individual's profit.Also we concluded that the compressed lead time and appropriate backorder discount are most benefit factors for entire supply chain in Section 5.1.Second, we fixed the backorder ratio and observed the optimal profit by varying payment time in Sections 5.2 and 5.3.With the different duration and four cases in EJTP  (  ,   ,   ), we had twelve possible values of profit, and the optimal profit occurred in EJTP 4 ( 4 ,   ,   ) with normal duration.Finally, we discover that the profit of supply chain will be different by compressing lead time and varying payment time.
In the supply chain, with the real data as input parameters, our proposed model can help decision-makers enhance the performance of the supply chain by controlling lead time and payment time.As far as the authors' knowledge goes, no such   type of three-echelon inventory model involving controllable lead time has yet been discussed in relevant literatures.These are the new major contributions of the present paper.Finally, it is our hope that this work will encourage future work in this area and related areas.And we will improve our further research in more real-world complexities, such as the concept of present value, multiple vendor and buyer, and apply the fuzzy approach into the proposed models, and we will cooperate real case to get the actual data that can illustrate real numerical examples to our future research.

B. Profit Variation
See Figure 4.

C. Proof
The following test can be applied at a nondegenerate critical point .

𝐶
: the crashing cost at the jth component in   (  = 1, 2)   : the jth component in normal duration ( = 1, 2)   : the jth component in minimum duration ( = 1, 2) 0(): probability density function (): probability distribution function (): the total crashing cost per production run TP  : supplier's total annual profit TP  : manufacturer's total annual profit TP  : retailer's total annual profit JTP: the joint total annual profit EJTP: the expected joint total annual profit.

𝐼 𝑟𝑝 : retailer's capital opportunity cost per dollar per year 𝐼 𝑟𝑒 : retailer's interest earned per dollar per year 𝑛 𝑟 : the number of shipments per production run from retailer to customer
research, retailer's lead time consisted of  mutually independent components; every component has a normal duration   and a minimum duration  ,  = 1, 2, . .., , given () =   ( −1 −   ) + ∑ =1   (  −   ) for   ∈ [  ,  −1 ]to compute the crashing cost.To establish the inventory model, the following notations and assumptions are used.
If   < , then (i) interest income =   (   −   /2).If   ≥ , then (i) opportunity cost =     (  −   ) 2 /2  ; (ii) interest income =   (  ) 2 /2  .Thus, TP 1 and TP 2 are given by TP 1 = sales revenue − ordering cost − transportation cost − holding cost − shortage cost revenue and interest income.We have already known the relationship between length of lead time and payment time from Section 3.1.Owing to the fact that the retailer's profit function has the two following cases, based on length of lead time   and payment time , the different parts between two possible cases are as follows.Mathematical Problems in EngineeringCase 1.

Table 1 :
Lead time data of numerical example.
( 4,   ,   ) = 20678.1.5.3.The Permissible DelayPeriod  and EJTP.In this section, we observe the value of profit by varying the permissible delay period .The detailed results have been shown in Table4.This time, we varied the permissible delay period .We noticed that the optimal profit still occurred in EJTP 4 ( 4 ,   ,   ) with normal duration condition.But the worst profit was different from previous result; it occurred in EJTP 4 ( 4 ,   ,   ) with minimum duration condition.( 4 ,   ,   ) = 20313.7.

Table 3 :
The value of profit in different conditions by varying .

Table 4 :
The value of profit in different conditions by varying .