A unified presentation of inventory models under quantity discounts, trade credits and cash discounts in the supply chain management

Abstract

As is well known, trade credit represents one of the most flexible sources of short-term financing available to firms, principally because it arises spontaneously with the firm’s purchases. The decision to offer trade credit and the determination of the firm’s terms of sale are important managerial considerations. In addition, the purchasing firm’s decision to take (or not to take) advantage of a cash discount and the motivations behind such a decision are also important. Our literature review reveals the fact that the research about the inventory model under the conditions of cash discount and trade credit is still a popular topic in the area of operations and inventory management. The main object of this paper is, therefore, to present a combination of all such important factors as (for example) quantity discounts, trade credits and cash discounts in order to establish and investigate a new inventory model when the cash discount for the retailer depends on the ordering quantity and the cash discount for the customer depends on the time when the customer buys an item. We first develop the annual total relevant cost. Then, by using the mathematical analytic tools and techniques dealing with the functional behaviors (such as continuity, discontinuity, increasing, decreasing, convexity, and so on) of the annual total relevant cost, we prove four theorems to determine the optimal replenishment cycle time. Finally, the sensitivity analysis is executed to study the variation of different parameters on the optimal policy. By including citations of a number of closely-related recent works, we also propose to try to incorporate the concepts of quantity discounts into the inventory model considered thus far in order to develop a newer unified inventory model. It is sincerely believed that this proposal should be a rather interesting research topic for future investigations.

Appendices

Appendix 1: Proof of Lemma 2

• (A) If $\triangle _{22}\leqq 0$, then the Eq. (65) implies that

  $$\begin{aligned} G_2 = 2A+pDM_2I_e\left[ (1-r)M_2+2rM_0-2M_1\right] \geqq DhD^2_2>0 \end{aligned}$$

  (74-1)

  Equations (38), (48) and (74-1) demonstrate that Lemma 2(A) holds true.

• (B) If $\triangle _{21}\leqq 0$, the Eq. (72) implies that $\triangle _{22}<\triangle _{23}\leqq 0$. So, Lemma 2(A) demonstrates that Lemma 2(B) holds true.

• (C) If $\triangle _{2W}\leqq 0$, the Eq. (74) implies that $\triangle _{22}<\triangle _{2W}\leqq 0$. So, Lemma 2(A) demonstrates that Lemma 2(C) holds true.

• (D) If $\triangle _{34}\leqq 0$, then the Eq. (68) implies that

  $$\begin{aligned} G_4&=2A+cDM^2_0I_k+pDI_e\left[ (1-r)M^2_2 +2rM_0M_2-2M_1M_2-\left( M^2_0-M^2_1\right) \right] \nonumber \\&\geqq DhM^2_0+cDM^2_0I_k>0. \end{aligned}$$

  (74-2)

  Equations (42), (50) and (74-2) demonstrate that Lemma 2(D) holds true.

• (E) If $\triangle _{4W}\leqq 0,$ then the Eq. (72) implies that $\triangle _{34}<\triangle _{4W}\leqq 0$. Consequently, Lemma 2(D) demonstrates that Lemma 2(E) holds true.

  By incorporating and combining (A)–(E), we have evidently completed the proof of Lemma 2.

Appendix 2: Proof of Theorem 1

[A]: If $\triangle _{12}>0,~\triangle _{22}>0$ and $\triangle _{5W}>0,$ then the Eqs. (57)–(59) imply that $0<\triangle _{12}, 0<\triangle _{22}\leqq \triangle _{23}\leqq \triangle _{34} \leqq \triangle _{4W}$ and $0<\triangle _{5W}$. Therefore, by Lemma 3[A, B, C, D, E](i), we have

• (a1) $TRC_1(T)$ is decreasing on $(0, T^*_1]$ and increasing on $[T^*_1, M_2]$.

• (a2) $TRC_2(T)$ is increasing on $[M_2, M_1)$.

• (a3) $TRC_3(T)$ is increasing on $[M_1, M_0]$.

• (a4) $TRC_4(T)$ is increasing on $\left[ M_0,\frac{W}{D}\right)$.

• (a5) $TRC_5(T)$ is increasing on $\left[ \frac{W}{D}, \infty \right)$.

Thus, clearly, there are the following two cases to occur:

• (i) If $H_{45}\geqq 0,$ by combining (a1)–(a5) and the Eqs. (16)–(19), we get

  $$\begin{aligned} TRC(T^*)=\min \left\{ TRC_1(T^*_1), TRC_5\left( \frac{W}{D}\right) \right\} . \end{aligned}$$

• (ii) If $H_{45}<0,$ by combining (a1)–(a5) and the equations (16)–(19), we get

  $$\begin{aligned} TRC(T^*)=TRC_1(T^*_1). \end{aligned}$$

   [B-R]: In view of Lemmas 1, 2 and 3, the techniques of proofs as in [A] can similarly be used to show that the assertions [B] to [T] hold true.

By incorporating and combining all arguments in [A] to [T], we have thus completed the proof of Theorem 1.