A SINGLE-CONSIGNOR MULTI-CONSIGNEE MULTI-ITEM MODEL WITH PERMISSIBLE PAYMENT DELAY, DELAYED SHIPMENT AND VARIABLE LEAD TIME UNDER CONSIGNMENT STOCK POLICY

This article proposes a two-level fuzzy supply chain inventory model, in which a single consignor delivers multiple items to the multiple consignees with the consignment stock agreement. The lead time is incorporated into the model and is considered a variable for obtaining optimal replenishment decisions. In addition, crashing cost is employed to reduce the lead time duration. This article investigates four different cases under controllable lead time to analyze the best strategy, focusing on two delays such as delay-in-payments and delay-in-shipment. In all four cases, all associated inventory costs are treated as a trapezoidal fuzzy number, and a signed distance method is employed to defuzzify the fuzzy inventory cost. An efficient optimization technique is adopted to find the optimal solution for the supply chain. Four numerical experiments are conducted to illustrate the four cases. Any one of these experimental results will provide the best solution for the ideal performance of the business under controllable lead time in the consignment stock policy. Finally, the managerial insights, conclusion and future direction of this model are provided. Mathematics Subject Classification. 90B05. Received July 8, 2020. Accepted July 25, 2021.


Controllable lead time
Jha and Shanker [18] developed the production inventory model by considering the crashing cost for multibuyer. Jamshidi et al. [17] considered a flexible inventory model with controllable lead time. Sarkar et al. [26] examined the effects of quality improvement and price discounts in the context of controllable lead time. Shin et al. [34] developed an inventory model following a continuous review methodology with variable lead time. Sarkar et al. [27] developed a model between single-vendor multi-buyer with varying production rate and controllable lead time. Ganguly et al. [11] designed the supply chain model with the influence of controllable lead time. Ahmad and Benkherouf [2] investigated an inventory model with replenishment decisions under partial backorder. Sharma et al. [33] analyzed the supply chain model with deteriorating products under varying lead time. Sarkar et al. [30] illustrated the deteriorating products-inventory model with varying demand and lead time.

CS policy under fuzzy environment
Ouyang and Yao [22] examined the distribution free inventory model with fuzzy demand. Björk [4] proposed an Economic Order Quantity (EOQ) model by considering the lead time, inventory level, and demand as a triangular fuzzy number, and defuzzification is done by using the signed distance method. Kazemi et al. [21] developed an inventory model by considering the inventory cost as the trapezoidal fuzzy number. Ali and Nakade [3] have developed a framework for examining the disruption of the supply chain in uncertain situations. Rani et al. [23] illustrated a model with carbon emission depended demand under a fuzzy environment. Sarkar et al. [29] suggested the supply chain model by assuming the inventory associated cost as a triangular fuzzy number under the signed distance method. Karthick and Uthayakumar [19] investigated the imperfect production inventory model with triangular fuzzy demand under the signed distance method. Karthick and  Uthayakumar [20] developed a VMI-consignment stock policy model with multiple items and trapezoidal fuzzy number under the graded mean integration method.

The literature gap in previous research
From the above discussion, we observed that the CS policy model plays an essential role in business management. Braglia and Zavanella [5], Valentini and Zavanella [36], and Huang and Chen [16] developed a CS policy model for industrial purpose. Zavanella and Zanoni [40] extend the work of [5,16,36] by considering the single buyer to multiple buyers. Yi and Sarker [37] examined the inventory model with CS agreement with the incorporation of variable lead time. Also, Zahran et al. [39] have analyzed the CS policy model with permissible payment delay. However, Zahran et al. [39], Yi and Sarker [37], Braglia and Zavanella [5], Valentini and Zavanella [36] and Huang and Chen [16] do not consider their models with multiple buyers with multiple products. Nevertheless, Zahran et al. [39] does not discuss how their model operates with controllable lead time. In trading, lead time plays a significant role in avoiding shortages, so lead time reduction is considered necessary. Furthermore, there is no inventory model in the literature for dealing with CS policy between a single consignor and multiple consignees with multiple products in a fuzzy environment. Based on that, in addition, this study presents four special cases associated with two delays: delay in shipping and delay in payment.
Contributions of various study articles from the existing literature are given in Table 1. The rest of the paper has been comprised as follows: In Section 3, notations and assumptions are given to develop the model. In Section 4, four different cases are developed under controllable lead time in the fuzzy environment. The defuzzification process for the fuzzified total profit function is developed in Section 5. In Section 6, the solution procedure has been derived to find optimal solutions. Moreover, in this paper, all basic inventory cost is treated as a trapezoidal fuzzy number, and the defuzzification process is done using the signed distance method. Four numerical examples are considered for each case to validate this model in Section 7. Numerical discussions and managerial insights are given in Sections 8 and 9, respectively. Finally, the conclusion is given in Section 10.

Problem definition
The consignor produces a certain quantity of goods and transfers them equally to each consignee. Once the goods are withdrawn from the consigned inventory, the consignee pays the consignor an equal payment in an equal interval scheme (see, for instance, [39]). Also, if the consignee's warehouse reaches the maximum stock level, the shipments will be delayed. This aside, lead time plays a crucial role in the supply chain, so lead time crashing cost is incorporated to reduce lead-time length. This study analyzes the consequences of delayed deliveries and delayed payments in four different cases with uncertain supply chain costs. In the first and second cases, the shipment is considered without delay, whereas it is considered delayed in the third and fourth cases. Similarly, in the second and fourth cases, the payment (with interest charges) to the consignor is considered with delay and in the first and third cases without delay.
In this paper, we develop a mathematical model using the notations and assumptions listed below.

Notations
The following notations will be used to develop the model.

Indices
i The index of items and 1 ≤ i ≤ z, where z is the total number of items j The index of consignee's and 1 ≤ j ≤ y, where y is the total number of consignee c The index of cases and c = 1, 2, 3, 4

Assumptions
The following assumptions are considered while developing the model.
(2) The production rate of ith item per year is finite, and it should be greater than the demand rate of the ith item for jth consignee (i.e., p i > d ij ) to avoid shortages. (3) The system inventory is continuously reviewed, and the shortage is not allowed. (4) The cycle time is common for both the consignor and consignee. (5) The holding cost of the consignor is divided into two parts, namely financial and physical. Therefore, consignor's holding cost of the ith item for the jth consignee, h mij = h f mij + h p mij , unit holding cost of the ith item for the jth consignee in transit, h dij = h p dij + h f mij , and consignee's unit holding cost of the ith item for the jth consignee, h rij = h p rij + h f rij (refer, [37]). (6) The consignee incurs only the physical holding cost for ith item. (7) For the jth consignee, the lead time l j consists of n j components which are mutually independent. The kth component has a minimum duration m j,k , normal duration n j,k and a crashing cost per unit time e j,k and assume that e j,1 ≤ e j,2 ≤ . . . ≤ e j,nij . The lead time components are to be crashed one at a time beginning from the least component of e i and so on. (8) Let l j,0 = nij k=1 n j,k and l j,f is the length of the lead time components 1, 2, 3, . . . , f crashed to their minimum duration, then expression of l j,f is given by l j,f = l j,0 − f j=1 (n j,k − m j,k ), where f = 1, 2, . . . , n ij and crashing cost for the lead time per cycle is given by (see, for instance, [27])

Mathematical model
In this section, a trapezoidal fuzzy number and signed distance method are provided for a preliminary purpose, then a mathematical formulation is developed, including four cases. Basic costs related to inventory and production are unpredictable due to various factors, i.e., inflation, the global energy crisis, fuel prices, and oil prices. Failing to consider these unforeseen circumstances results in an unstable supply chain model. For this reason, all the specific costs associated with the consignor and the consignee are considered to be fuzzy costs (Trapezoidal fuzzy number) in the proposed model. The signed distance method is used to solve fuzzy parameters.

Signed distance method
For any t ∈ R, d(t, 0) = t is named as the signed distance from t to 0. If t > 0, then the distance from t to 0 is t = d(t, 0); if t < 0, the distance from t to 0 is −t = −d(t, 0). Therefore, d(t, 0) = t is known as the signed distance from t to 0.

Mathematical formulation
All four cases in this study consider the model between a single consignor and multiple consignees with multi-item based on the CS policy.
The cost associated with CS policy model are considered as the trapezoidal fuzzy number 4.1, which are given in the following: , and Consigor's physical holding cost: , ϕ ctijc , ϕ cpric , ϕ c bic , ϕ ccic and ϕ cpic , i = 1, 2, 3; j = 1, 2, 3; c = 1, 2, 3, 4, are arbitrary positive numbers under the following conditions: The cost formulation of the consignor and consignees are described as follows.

Consignor's cost formulation
The costs associated with the consignor for y consignees and z items are derived as following: Setup cost. Setup cost is the cost of purchasing and maintaining the equipment needed for the production stage before manufacturing the products, Raw material cost. Spare parts are required to make a finished product, so the cost of purchasing those spare parts (raw materials) is known to be a raw material cost, Production cost. Production costs refer to the cost of producing or manufacturing an item. Also, this includes direct labour costs, direct material and overhead costs for production, Lead time crashing cost. Lead time is the interval between when an order is placed to fill the goods and when the order is received. However, to reduce the length of lead time, the crashing cost is used as

Consignee's cost formulation
The costs associated with y consignees for z items are derived as follows: Purchasing cost. Purchase cost refers to the cost of purchasing products from the consignor, Ordering cost. The cost required by y consignees to process the order from the consignor is said to be an ordering cost, Transaction cost. The commission paid by y consignees for transaction per cycle is calculated as The total cost of the supply chain (without inventory holding cost of the consignor and y consignees) is derived by adding equations (4.3)-(4.9).
C total = SC + RMC + PC + LTCC + PRC + OC + TRC The consignor produces q ij of items in each n ij batches within cycle with fixed setup cost S vi at finite production rate p i . In order to avoid the shock out, the production rate is assumed to be greater than the demand rate. The consignor utilizes the consignee's warehouse space to store the manufactured items; this seems to be more advantageous for the consignor to keep fewer items in the warehouse. Moreover, there is an advantage for the consignee by holding the maximum stock level to avoid stockouts. The inventory pattern of the consignor, transit, consignees and financial behaviour of y consignees can be seen in Figure 1, and the average inventory of the system is calculated as The average inventory of y consignees are derived by dividing the area The average inventory in transit is calculated as.
The average inventory of the consignor I consignor is derived by subtracting I consignee and I transit from the system average inventory I cs . therefore, the physical holding cost of the consignor for y consignees are obtained as The financial holding cost of the consignor is formulated as (see, for instance, [39]) where,h f mij =c bij I vij . The physical holding cost of y consignees are derived as and the transit holding cost is given as The total cost of the supply chain with y consignees for z items are calculated by adding the equations (4.10), (4.15)-(4.18).
The revenue of the consignor is obtained as (4.20) and the revenue of y consignees for z items are obtained by adding the selling price and investment interest rate of y consignees for z items.
Then the total revenue of the supply chain is calculated by adding the equations (4.20) and (4.21), that is, Hence, the annual profit functionP 1 (m ij , n ij , q ij , l j ) for case 1 can be written as m ij , n ij and l j are positive integers.
In this case, the consignor offers the allowable payment delay to the consignee under the CS policy, i.e., the consignee pays the invoice amount to the consignor by the end of the permissible period τ ij = t ij + α ij t ij , where α ij t ij is delay period offered by the consignor without interest. Sometimes, in reality, the consignee may not pay the invoice amount within the delay period α ij t ij . Therefore, the consignee may pay the invoice amount with interest charges by the end of the period δ ij = t ij + α ij t ij + β ij τ ij , where β ij > 0. The inventory pattern for this case is given by Figure 2, and the system inventory is the same as in the case 1. The opportunity loss of the consignor is written as (see, for instance, [39] The consignee pays the invoice amount by the end of δ ij period, therefore the interest charges for the extra delay by β ij τ ij , which incurs the cost of . The total cost for case 2 is obtained by adding (4.10), (4.15), (4.17), (4.18) and (4.25).
The consignor's revenue is adding obtained by selling cost of the z items to the y consignees and the interest charged for unsettled balances, which is written as Similarly, the revenue of y consignees is obtained by adding sales and investment (i.e.,) Therefore, the total revenue is calculated by adding (4.27) and (4.28).
Hence, the annual profit functionP 2 (m ij , n ij , q ij , l j ) for case 2 is calculated by subtracting the total cost (4.26) from the total revenue (4.29) of the supply chain.
m ij , n ij and l j are positive integers.
Case 3. CS policy with NDIP -DIS under CLT h p mij < h p rij . In this case, the consignor faces the problem of limited storage in the consignee's warehouse. The consignor will not offer any payment delay to the consignee. The inventory system with maximum delayed shipment is given in Figure 3, and the average inventory of the system is calculated as the same as in case 1. The average inventory of consignee is derived by dividing the area The calculation is on the following:  The average inventory of the consignor I consignor is equal to the system average inventory I cs (4.11) minus the consignee's inventory I consignee (4.32) minus transit inventory I transit (4.18).
The total cost of the supply chain is derived as and the revenue of the consignor and consignee seems to be same as the case 1. Hence, the annual profit P 3 (m ij , n ij , q ij , k ij , l j ) can be written as  In this case, the consignor offers a payment delay to the consignee and simultaneously, there was a limited storage space in the consignee's warehouse. The inventory of consignor, consignee, transit is depicted in Figure 4. The revenue of the consignor (i.e., Eq. (4.27)) and consignee (i.e., Eq. (4.28)) in case 2 are taken in this case. Then, the consignor's opportunity loss is given in (i.e., Eq. (4.25)). The total cost of the supply chain is calculated as C 4 (m ij , n ij , q ij , l j ) = z i=1 y j=1 (γ icpi +c pri +c bij )d ij + S vi + n ijÕbij + m ijctij + n ij B(l j ) d ij n ij q ij and the total revenue is same as the case 3. Hence, the annual profitP 4 (m ij , n ij , q ij , k ij , l j ) can be written as m ij , n ij and l j are positive integers.

Defuzzification methodology
Defuzzification is the method of generating a quantifiable result in crisp logic, from fuzzy sets and corresponding membership functions (i.e., the process of reducing the fuzzy set to a crisp set or converting the fuzzy quantity to a crisp quantity). There may be situations in which the output of a fuzzy process must be a single scalar quantity as opposed to a fuzzy set. The left and right λ cuts of S vi , O bij , h f mij , h p mij , h p dij , h p rij , c tij , c pri , c bi , c ci , and c pi are given below, Therefore, by using the signed distance method 4.2, if c = 1, 2 then k ij = 0 and if c = 3, 4 then k ij = n ij − 1, By inserting (5.1)-(5.11) into the equations (5.12)-(5.15) yields the defuzzified profit function, For Case 1.
subject to n ij q ij − (n ij − 1) q ij dij pi ≤ I max , q ij > 0, m ij , n ij and l j are positive integers.
For Case 2.
≤ I max , q ij > 0, m ij , n ij and l j are positive integers.
For Case 3.
≤ I max , q ij > 0, k ij ≤ n ij − 1, m ij , n ij and l j are positive integers. For Case 4.
≤ I max , q ij > 0, k ij ≤ n ij − 1, m ij , n ij and l j are positive integers.

Solution procedure
In this section, we derive the optimal value of q ij and demonstrate the concavity of the integrated profit function with respect to the decision variable q ij . In this model, the number of shipment n ij , the number of payment m ij and the number of delayed deliveries k ij are assumed to be positive integer variables. The given integrated profit function (5.12)-(5.15) seems to be non-linear. To solve this non-linear programming problem, we have focused on some property to obtain the optimal solutions.
Proof. On taking the first and second order partial derivatives of (5.12)-(5.15) with respect to q ij , we obtain For Case 1. and For Case 2. and For Case 3. and For Case 4. and Therefore, for fixed m ij , n ij and l j ∈ [l j,f , l j,f −1 ], d 0 (P c ,0) is concave in q ij . Hence, this completes the proof of Property 6.1.
Result. From the Property 6.1, For Case 1. We obtain the optimal value of q ij (6.9) by equating (6.1) to zero, which maximize the d 0 (P 1 ,0).
For Case 2. We obtain the optimal value of q ij in (6.10) by equating (6.3) to zero, which maximize the d 0 (P 2 ,0).
For Case 3. We obtain the optimal value of q ij in (6.11) by equating (6.5) to zero, which maximize the d 0 P 3 ,0 .
For Case 4. We obtain the optimal value of q ij in (6.12) by equating (6.7) to zero, which maximize the d 0 P 4 ,0 . Proof. On taking the first order derivatives of profit functions (5.12)-(5.15) will lead to ∂d 0 P c (l j ) ,0 ,0 is flat on l j . Therefore, under each case, the profit function d 0 P c (l j ),0 is linear on l j . Hence, this completes the proof of Property 6.2.
For the fixed values of m ij , n ij , and q ij , the maximum d 0 P c (l j ),0 always occurs only at the end points of

Conclusion and future directions
This article has considered a single-consignor multi-consignee for multi-item under controllable lead time in a fuzzy environment. This paper adopts the CS policy, in particular, which is more beneficial for the consignor. This paper compares four different cases to show which cases are the most profitable for the supply chain. Moreover, this paper studies the impacts of controllable lead time for multiple consignees, which is a more critical and practical factor, and this never been studied under CS policies. The numerical results showed that the supply chain players (consignor and consignee) attain the highest profit in case 4 compared to case 3, and case 1 attains higher profit than case 4. However, case 2 was shown to be the most profitable when compared to all other cases. This model can be extended in many ways; basically, this model has some limitations, so it is possible to develop this model by resolving these limitations. Primarily in this model, we assume that the production process is perfect, so it can be extended by turning this model into an imperfect production process. Another extension is possible by relaxing the equal-sized shipment and fixed demand rate in the proposed model (refer, [11,38] and Ganesh Kumar and Uthayakumar [10] for unequal-sized shipments, Sarkar et al. [29] for price and advertisement-dependent demand, Karthick and Uthayakumar [19,20] for fuzzy demand). Exploring the changes that occur in this model by combining concepts such as learning and forgetting can be considered an extension (refer, [13]). The production rate is assumed to be constant in this model, so considering the production rate as a variable for a flexible production process is another extension (refer, [17,27]). The incorporation of the consignee's royalty reduction (refer, [28]) and radio frequency identification (refer, [31]) in the CS policy would be a reasonable extension of this model.