The Inventory Model for Deteriorating Items under Conditions Involving Cash Discount and Trade Credit

In the year 2004, Chang and Teng investigated an inventory model for deteriorating items in which the supplier not only provides a cash discount, but also allows a permissible delay in payments. The main purpose of the present investigation is three-fold, as follows. First, it is found herein that Theorem 1 of Chang and Teng (2004) has notable shortcomings in terms of their determination of the optimal solution of the annual total relevant cost Z(T) by adopting the Taylor-series approximation method. Theorem 1 in this paper does not make use of the Taylor-series approximation method in order to overcome the shortcomings in Chang and Teng (2004) and alternatively derives all the optimal solutions of the annual total relevant cost Z(T). Secondly, this paper systematically revisits the annual total relevant cost Z(T) in Chang and Teng (2004) and presents in detail the mathematically correct ways for the derivations of Z(T). Thirdly, this paper not only shows that Theorem 1 of Chang and Teng (2004) is not necessarily true for finding the optimal solution of the annual total relevant cost Z(T), but it also demonstrates how Theorem 1 in this paper can locate all of the optimal solutions of Z(T). The mathematical analytic investigation presented in this paper is believed to be useful for correct managerial considerations and managerial decisions.

It should be remarked that Sarkar and Sarkar [27] studied a significantly improved inventory model involving what may be referred to probabilistic deterioration. Their solution of the model depended heavily upon Control Theory. On the other hand, Sarkar et al. [35] considered an EOQ model for deteriorating items together with time-dependent increasing demand. They found the component cost as well as the selling price as a continuous time-rate. Sarkar and Sarkar [27] also considered an inventory model involving infinite replenishment rate, stock-dependent demand, time-varying deterioration rates as well as partial backlogging. A production-inventory model involving probabilistic deterioration in two-echelon supply chain was discussed by Sarkar [26]. Derivation of the marketing policy for non-instantaneous deteriorating items involving a generalized-type deterioration and holding-cost rates can indeed be found in the investigation by Shah et al. [18]. Chung et al. [28] considered inventory modelling involving non-instantaneous receipt and exponentially deteriorating items in the case of an integrated three-layer supply chain system under two-level trade-credit. A discussion in the case of several lot-sizing policies for deterioration items under two-level trade credit with partial trade credit to credit-risk retailer and limited storage capacity can be seen in the work by Liao et al. [33]. Wu et al. [36] presented inventory-related policies for perishable products with expiration dates and advance-cash-credit payment schemes. By making use of the Stackelberg and the Nash equilibrium solution, Jaggi et al. [37] explored inventory and credit decisions in the case of deteriorating items with displayed stock-dependent demand involving the two-echelon supply chain. Many other remarkable studies about deteriorating items can be found in the article by Kawale and Sanas [38] (see also [39,40]).
By combining all elements of the trade credit, the cash discount and the deteriorating items, Chang and Teng [41] studied a certain inventory model for deteriorating items in the case when the supplier does not only provide a cash discount, but also allows a permissible delay in payments. We remark in passing that a cash discount can encourage the customer to pay on delivery and it can also reduce the default risk. In the past few years, marketing researchers and practitioners in the area of supply chain management appear to have recognized and understood the phenomenon that the supplier offers a permissible delay in payment to the retailer if the outstanding amount is paid within the permitted fixed settlement period, known as the trade-credit period. During the trade-credit period, the retailer is allowed to accumulate revenues received upon selling items and earning interests. Consequently, without any incentive to make early payments and with the possibility and prospect of earning interest by means of the accumulated revenue which is received during the credit period, the retailer chooses to postpone payment until the last moment of the permissible period which is allowed by the supplier. Thus, clearly, the offer of the trade credit does lead to delayed cash inflow and to thereby increase the risk of cash-flow shortage as well as bad debt. From the suppliers' viewpoint, it is always hoped that they will be able to find a trade-credit policy to increase sales and to decrease the risk of cash-flow shortage and bad debt. In reality, however, especially on the side of the operations management, a supplier is generally willing and ready to provide the retailer with either a cash discount or a permissible delay in payments or both.
The main purpose of the present investigation is three-fold as stated below. 1 To observe that Theorem 1 of Chang and Teng [41] has notable shortcomings in their determination of the optimal solution of the annual total relevant cost Z(T) by adopting the Taylor-series approximation method. Theorem 1 in this paper does not make use of the Taylor-series approximation method in order to overcome the shortcomings in Chang and Teng [41] and thereby derive the optimal solutions of the annual total relevant cost Z(T).

2
To systematically revisit the annual total relevant cost Z(T) in Chang and Teng [41] and to present in detail the mathematically correct ways for the derivations of Z(T).

3
To not only show that Theorem 1 of Chang and Teng [41] is not necessarily true for finding the optimal solution of the annual total relevant cost Z(T), but to also demonstrate how Theorem 1 in this paper can locate all of the optimal solutions of Z(T).
The mathematically correct analytic investigation of the model, which we have presented in this paper, is believed to be useful for correct managerial considerations and right managerial decisions (see also [41]).

The Mathematical Modelling of the Problem
This paper adopts the same assumptions and notations as described in Chang and Teng [41].

Assumptions
(1) The demand for the item is constant with time.
(3) Replenishment is instantaneous. (4) During the time the account is not settled, generated sales revenue is deposited in an interest-bearing account. At the end of this period (that is, M 1 or M 2 ), the customer pays the supplier the total amount in the interest-bearing account, and then starts paying off the amount owed to the supplier whenever the customer has money obtained from sales. (5) Time horizon is infinite.
Notations D = the demand rate per year. h = the unit holding cost per year excluding interest charges. p = the selling price per unit. c = the unit purchasing cost, with c < p. I c = the interest charged per $ in stocks per year by the supplier or a bank. I d = the interest earned per $ per year. S = the ordering cost per order. r = the cash discount rate, 0 < r < 1. θ = the constant deterioration rate, where 0 ≤ θ < 1. T * = the optimal replenishment time of Z(T).
The annual total relevant cost Z(T) consists of the following items: Chang and Teng [41] reveal that (a) Cost of placing order = S T , (c) Cost of carrying inventory = hD if Policy II is adopted. to pay the supplier. From the difference between the total purchase cost c(1 − r)I(0) and the total amount of money in the account, i.e., the following two cases to occur: Equation (3) in Chang and Teng [41] implies that and the interest payable per year = 0 From Equations (2) to (9), the annual total relevant cost Z 5 (T) is given by From Equations (2) to (6) and (10) to (12), we find the annual total relevant cost Z 1 (T) given by Under Case I (B), the customer sells DT units in total at time T , and has cDT to pay the supplier in full at time M 1 . Therefore, Chang and Teng [41] derived the annual total relevant cost Z 2 (T) as follows: Combining Case I (A) and Case I (B), we have For convenience, all the items Z 2 (T), Z 5 (T) and Z 1 (T) are defined on T > 0. Case 1 in Chang and Teng [41] only discussed Equation (11) and ignored Equation (7) of this paper. When Equation (11) of this paper holds true, then Equation (3) in Chang and Teng [41] implies that Case I (B) and Equation (16) reveal that the domain of the function TVC 1 (T) is the set given by (0, M 1 ] [W 1 , ∞), but not the set (0, ∞). The interval (M 1 , W 1 ) is not contained in the domain of the function TVC 1 (T). Therefore, the annual total relevant cost of Cases 1 and 2 in Chang and Teng [41] can be expressed as follows: The functional behavior of TVC 1 (T) on the interval (M 1 , W 1 ) was not discussed in Chang and Teng [41]. This does not make sense, since the domain of TVC 1 (T) should be (0, ∞). The correct derivation of TVC 1 (T) should consider Equations (7) and (11) together. Consequently, it is the shortcoming of the modelling of Chang and Teng [41]. Equations (15a), (15b) and (15c) correct the claims made by Chang and Teng [41].
Case II. The customer adopts Policy I and According to Equations (2) to (6) and (17), the annual total relevant cost TVC 1 (T) can be expressed as follows: Case III. The customer adopts Policy II and W 3 > M 2 Following the same arguments as in Case I, the annual total relevant cost TVC 2 (T) can be expressed as follows: where and Since For convenience, all the items Z 4 (T), Z 6 (T) and Z 3 (T) are defined on T > 0. Let us now use the inequalities given by Equations (24) and (25) as follows: and respectively. Case 3 in Chang and Teng [41] only discussed Equation (25) and ignored Equation (24) above. When Equation (25) in this paper holds true, Equation (3) in Chang and Teng [41] implies that Case II (D) and Equation (26) reveal that the domain of the function TVC 2 (T) is the set (0, is not contained in the domain of the function TVC 2 (T). Therefore, the annual total relevant cost of Cases 3 and 4 in Chang and Teng [41] can be expressed as follows: The functional behavior of TVC 2 (T) on (M 2 , W 3 ) was not discussed in Chang and Teng [41]. This exclusion does not make sense, since the domain of TVC 2 (T) should be (0, ∞). The correct derivation of TVC 2 (T) should, in fact, consider the equations (24) and (25) together. Consequently, it is another shortcoming of the modelling by Chang and Teng [41]. Equations (19a), (19b) and (19c) correct the claims by Chang and Teng [41].
Case IV. The customer adopts Policy II and W 3 ≤ M 2 .
Following the same arguments as in Case II, the annual total relevant cost TVC 2 (T) can be expressed as follows: Upon combining Cases I to IV, the annual total relevant cost Z(T) can be expressed as follows: The objective here is to determine which policy [T * 1 (Policy I) or T * 2 (Policy II)] satisfies the following condition:

The Convexity of Z i (T)
By making use of Equations (10), (13), (14), (20), (21) and (22), we find for the first and the second derivatives of the annual total relevant cost Z 2 (T) that and where We now suppose that Then, clearly, we have the results asserted by Theorem 1 below.

Theorem 1.
Each of the following assertions holds true:
(H) See, for details, Lemma 2 in the paper by Huang and Liao [8].

Theorems for the Optimal Cycle T * of Z(T)
Equations (30), (32), (34), (36), (38) and (40) yield and Case I. Policy I is adopted and M 1 < W 1 . Let Case II. Policy I is adopted and M 1 ≥ W 1 . Let Now, since M 1 ≥ W 1 , Theorem 1 (C) implies that Case III. Policy II is adopted and M 2 < W 3 . Let and Case IV. Policy II is adopted and M 2 ≥ W 3 . Let Thus, since M 2 ≥ W 3 , Theorem 1 (D) implies that Henceforth, in our investigation, we assume that and Theorem 1 (E) to Theorem 1 (H), together, imply that the function Z i (T) is convex when T > 0 for all i = 1, 2, 3, 4, 5, 6. Let T i denote the root of the following equation: From the convexity of the function Z i (T) when T > 0, we conclude that Therefore, clearly, the function Z i (T) is decreasing on (0, T i ] and increasing on [T i , ∞) for i = 1, 2, 3, 4, 5, 6.

Proposition 1.
Suppose that Policy I is adopted and M 1 < W 1 . Then the following assertions hold true:

Proposition 2.
Suppose that Policy I is adopted and M 1 ≥ W 1 . Then the following assertions hold true: Proof.

Proposition 3.
Suppose that Policy II is adopted and M 2 < W 3 . Then the following assertions hold true: Proof. The proof of Proposition 3 is similar to that of Proposition 1. We, therefore, choose to skip the details involved.

Proposition 4.
Suppose that Policy II is adopted and M 2 ≥ W 3 . Then the following assertions hold true: Proof. The proof of Proposition 4 would run parallel to that of Proposition 2. The details involved are, therefore, omitted.
Then each of the following assertions holds true: Therefore, clearly, we have T * = T * 1 (Policy I) or T * 2 (Policy II) associated with the least cost. Chang and Teng [41] do not make the comparison between TVC 1 (T * 1 ) and TVC 2 (T * 2 ). Consequently, in general, the claimed assertion of Theorem 1 (1) in Chang and Teng [41] is not necessarily true. (B) About Theorem 1 (2) in the paper by Chang and Teng [41]: our Theorem 1 (B) (ii) implies that Since Z 2 (M 1 ) > 0, M 1 is not the optimal solution of TVC 1 (T). Therefore, in general, the result claimed in Theorem 1 (2) of Chang and Teng [41] is not true. (C) About Theorem 1 (3) in the paper by Chang and Teng [41]: Let T * CT denote the optimal solution obtained by Theorem 1 in Chang and Teng [41]. In this case, we consider the following example.  We thus observe that M 1 < W 1 , M 2 < W 3 , 3B 1 > A 1 , 3B 1 > A 1 and 3B 3 > A 3 . Furthermore, we get 25 = −1.59779 × 10 −4 < 0, 46 = 0.1698 > 0 and 51 = 0.02795 > 0. Then, by applying Theorem 2 (ii) of this paper, we have T * = T 5 or T 4 . The familiar Intermediate Value Theorem (see, for example, Varberg et al. [42]) can now be used to locate T 5 and T 4 . We thus find that T 5 = 0.08231, T 4 = 0.08207, TVC 1 (T 5 ) = 14950.0759 and TVC 2 (T 4 ) = 15176.1460. Since we have T * = T 5 . Moreover, by applying the Intermediate Value Theorem, we conclude that since 51 > 0, G > 0 and lim Therefore, T 1 does not satisfy Equation (24) in Chang and Teng [41]. Consequently, Theorem 1 (3) in Chang and Teng [41] can be used. We then get T * CT = T 4 . However, the accurate optimum solution of the above Example should be T * = T 5 . Therefore, by contradiction, Theorem 1 (3) in Chang and Teng [41] is not necessarily true.
(D) About Theorem 1 (4) in the paper by Chang and Teng [41]: The proof in this case is similar to that in (B) above. Therefore, Theorem 1 (4) in Chang and Teng [41] is not true. (E) About Theorem 1 (5) in the paper by Chang and Teng [41]: Our reasoning here is the same as that of (A) above. Therefore, Theorem 1 (5) in the work of Chang and Teng [41] is not necessarily true.
By incorporating (A) to (E) above, it is concluded that in general, Theorem 1 in Chang and Teng [41] is not necessarily true.

Concluding Remarks and Observations
In our present investigation, we have successfully divided all our mathematical analytic derivations of the annual total relevant cost Z(T) into the following four cases: When the above Case 2 and Case 4 hold true, the annual total relevant costs in this paper are seen to be consistent with those of Chang and Teng [41]. However, if the above Case 1 and Case 3 hold true, then the annual total relevant costs in the work by Chang and Teng [41] are observed to be incorrect. Furthermore, this paper has also indicates that Theorem 1 in Chang and Teng [41] is based on the assumption that θT is small. However, our present investigation does not include this assumption. On the other hand, in general, Theorem 1 in the work by Chang and Teng [41] is not necessarily true. Theorems 2 to 5 in this paper have been fruitfully used to characterize the optimal solutions and to demonstrate the fact that they can locate all optimal solutions of Z(t). By incorporating the above arguments. we conclude that our present investigation has not only removed all those shortcomings in the paper by Chang and Teng [41], but it has also presented solvable ways for the problem considered by Chang and Teng [41]. Consequently, in this paper, we have corrected and substantially improved the work of Chang and Teng [41]. Therefore, it can significantly reduce the cost of the inventory model.
The mathematically correct analytic investigation of the model, which we have presented in this paper, is believed to be useful for correct managerial considerations and right managerial decisions.
The proposed model, for which we have presented a mathematical analytic investigation in this article, is capable of being extended in several different directions. Among other such possibilities of extension and generalization of our study here, it may be worthwhile to extend the constant demand rate to hold true in the case of a more realistic situation when the time-varying demand rate is a function of the time, the selling price, the advertisement of the product quality, and sundry other considerations. Yet another direction for future research on the subject-matter of our present investigation is the possibility of generalization and extension of the model with a view to allowing for shortages, quantity discounts, inflation rates, and other business-related considerations. Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflicts of interest.