Zu Yue 2023a - Revision history

Mingshiyue at 11:47, 14 June 2024

2024-06-14T11:47:07Z

Mingshiyue at 13:17, 31 August 2023

2023-08-31T13:17:40Z

Mingshiyue at 13:19, 20 August 2023

2023-08-20T13:19:10Z

Mingshiyue at 13:17, 20 August 2023

2023-08-20T13:17:10Z

Mingshiyue at 14:13, 14 August 2023

2023-08-14T14:13:25Z

Mingshiyue at 13:21, 13 August 2023

2023-08-13T13:21:49Z

Mingshiyue at 13:20, 13 August 2023

2023-08-13T13:20:20Z

Mingshiyue at 14:32, 28 June 2023

2023-06-28T14:32:50Z

Mingshiyue at 14:26, 28 June 2023

2023-06-28T14:26:21Z

Mingshiyue at 14:22, 28 June 2023

2023-06-28T14:22:52Z

← Older revision		Revision as of 11:47, 14 June 2024
Line 440:		Line 440:

	<br>		<br>
−	'''Proof of Proposition 5:''' If the supplier extends the promotion period, the retailer will always have an option to place the same special order as before. As a consequence, a longer promotion can only benefit the retailer instead of making it worse off.	+	'''Proof of Proposition 5:''' If the supplier extends the promotion period, the retailer will always have an option to place the same special order as before. As a consequence, a longer promotion can only benefit the retailer instead of making it worse off

@@ Line 33: / Line 33: @@
 The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section 3 describes the model. The retailer's optimal order decision and the corresponding managerial insights are presented in Section 4. In Section 5, we conduct numerical experiments to validate the proposed model. Section 6 concludes this study. All proofs are presented in Appendix.
-==2. Literature review==
+==2. Literature Review==
 The literature is reviewed from two perspectives: price discounts at a future time and price discounts over a short period.
@@ Line 45: / Line 45: @@
 Our study is also related to Arcelus et al. <span id='citeF-35'></span>[[#cite-35|[35]]], who investigated the retailer's special order time and special order quantity under a promotion period of unknown length. This study also examines the retailer's ordering strategy with a promotion period of arbitrary length, but differs from their model in two aspects. First, they allow the retailer to repeatedly place special orders throughout the promotion period, which enables the retailer to fully take advantage of the price discount. In contrast, we restrict the number of special orders to hurry the retailer into placing a larger special order for the purpose of improving the supplier's cash flow. Second, shortages are prohibited in their study, but they are allowed in this study, which endow the retailer with more flexibility in its decision-making.
-==3. Model setup==
+==3. Model Setup==
 Consider a supply chain setting in which a supplier sells a product to a retailer and charges a wholesale price <math display="inline">w</math> for each unit of its product. To promote sales, the supplier offers the retailer a disposable coupon whereby the latter can place a special order in a promotion period <math display="inline">[t_s,t_e]</math> at a reduced wholesale price <math display="inline">\gamma w</math>, <math display="inline">0<\gamma{<1}</math>. The promotion period may include one or more regular replenishment points. If that is the case, the retailer needs to decide whether to continue placing some regular orders after the coupon is available (at time <math display="inline">t_s</math>) to prepare for the special order. Shortages are allowed and fully backordered throughout the time horizon. The length of the time horizon is exogenously given and long enough to include the special order period. In addition to the special order time <math display="inline">t_r</math>, the special order quantity <math display="inline">q_r</math>, the retailer needs to determine the number of regular orders <math display="inline">n</math> placed in <math display="inline">[t_s,t_r]</math>. For convenience, we denote simply by a triple <math display="inline">(q_r,t_r,n)</math> the retailer's ordering strategy with a disposable coupon. To highlight the retailer's inventory mechanism, we adopt a constant demand rate under the framework of the classical EOQ model. To better illustrate our analytical model, we consider a two-echelon supply chain in which Coca-Cola and Costo act as the supplier and the retailer, respectively. The product is cola, which is produced by Coca-Cola and sold to Costco. It is worthy noting that the local demand for cola has tended to be steady <span id='citeF-36'></span>[[#cite-36|[36]]]. A summary of the model notation is listed in the [[#tab-1|Table 1]].
@@ Line 106: / Line 106: @@
-===3.1 Inventory level===
+===3.1 Inventory Level===
 In this subsection, we characterize the retailer's inventory level with respect to its ordering strategy. To this end, we first examine the fixed number of regular EOQ orders placed before the promotion period. Let <math display="inline">m=\lceil t_sD/Q^*\rceil </math>, where <math display="inline">\lceil x\rceil </math> denotes the smallest integer greater than or equal to <math display="inline">x</math>. Thus, <math display="inline">m</math> denotes the number of regular orders placed in <math display="inline">[0,t_s)</math>, which satisfies <math display="inline">(m-1)Q^*/D<t_s\leqslant mQ^*/D</math>.
@@ Line 120: / Line 120: @@
 |}
-===3.2 Total cost===
+===3.2 Total Cost===
 In this subsection, we examine the retailer's total cost with respect to the ordering strategy <math display="inline">(q_r,t_r,n)</math> by adding up its total costs in the regular order interval, the special order interval, and the remaining time horizon.
@@ Line 216: / Line 216: @@
 While a lower discount rate can help the supplier sell its products to more retailers, it may hurt the supplier by cutting its sales revenue. Proposition 5 indicates that the supplier can attract more retailers by properly extending the promotion period in addition to reducing the wholesale price. The intuition is that a longer promotion period endows the retailer more flexibility in ordering decision-making, which benefits the retailer and, thus, renders the supplier better off. This result enlightens the supplier on the promotion strategy.
-==5. Numerical experiments==
+==5. Numerical Experiments==
 In this section, some numerical experiments are performed to illustrate the validity of the model.

@@ Line 33: / Line 33: @@
 The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section 3 describes the model. The retailer's optimal order decision and the corresponding managerial insights are presented in Section 4. In Section 5, we conduct numerical experiments to validate the proposed model. Section 6 concludes this study. All proofs are presented in Appendix.
-==2. Literature revie==
+==2. Literature review==
 The literature is reviewed from two perspectives: price discounts at a future time and price discounts over a short period.

@@ Line 33: / Line 33: @@
 The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section 3 describes the model. The retailer's optimal order decision and the corresponding managerial insights are presented in Section 4. In Section 5, we conduct numerical experiments to validate the proposed model. Section 6 concludes this study. All proofs are presented in Appendix.
-==2. Literature review==
+==2. Literature revie==
 The literature is reviewed from two perspectives: price discounts at a future time and price discounts over a short period.

@@ Line 244: / Line 244: @@
 A graphical illustration of Proposition 3 can be seen in [[#img-4|Figure 4]]. Note that shortages are currently not attractive to the retailer in the regular EOQ ordering strategy <span id='citeF-37'></span>[[#cite-37|[37]]]. There are two noteworthy observations. First, the curve corresponding to “Shortages” is lower than that corresponding to “No shortages”, indicating that although shortages cannot render the retailer better off in its regular orders, they benefits the retailer in the promotion season. Second, the gap between the two curves becomes wider as the discount rate decreases. This implies that making use of shortages in due time can help the retailer cut back on more spending from a lower discount rate.
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:50%;"
 |-style="background:white;"

@@ Line 286: / Line 286: @@
 ==Acknowledgments==
 This work was supported by the National Natural Science Foundation of China (11271175) and the Natural Science Foundation of Shandong Province (ZR2021MA079, ZR2021MA088).
 ==References==

← Older revision		Revision as of 13:20, 13 August 2023
Line 283:		Line 283:

	This paper has some limitations. First, the analysis in our model is constructed on the assumption that the market demand is common knowledge between the retailer and the supplier. The model could be generalized by considering a robust model with uncertain parameters (e.g., unpredictable demands and changeable lead times) <span id='citeF-38'></span><span id='citeF-39'></span>[[#cite-38\|[38,39]]]. Second, for analytical tractability, we assume that the demand rate is constant while normalize the leading time to zero. It could be interesting to consider <math display="inline">(s,S)</math> inventory systems with random leading times and multi-period resupply <span id='citeF-40'></span><span id='citeF-41'></span>[[#cite-40\|[40,41]]]. Third, in this paper, the supplier sells the product only through the retailer. In addition to the resell channel, the supplier can directly sell to end consumers by establishing a direct selling channel. Future research would be conducted to incorporate supplier encroachment <span id='citeF-42'></span><span id='citeF-43'></span>[[#cite-42\|[42,43]]].		This paper has some limitations. First, the analysis in our model is constructed on the assumption that the market demand is common knowledge between the retailer and the supplier. The model could be generalized by considering a robust model with uncertain parameters (e.g., unpredictable demands and changeable lead times) <span id='citeF-38'></span><span id='citeF-39'></span>[[#cite-38\|[38,39]]]. Second, for analytical tractability, we assume that the demand rate is constant while normalize the leading time to zero. It could be interesting to consider <math display="inline">(s,S)</math> inventory systems with random leading times and multi-period resupply <span id='citeF-40'></span><span id='citeF-41'></span>[[#cite-40\|[40,41]]]. Third, in this paper, the supplier sells the product only through the retailer. In addition to the resell channel, the supplier can directly sell to end consumers by establishing a direct selling channel. Future research would be conducted to incorporate supplier encroachment <span id='citeF-42'></span><span id='citeF-43'></span>[[#cite-42\|[42,43]]].
		+
		+	==Acknowledgments==
		+	This work was supported by the National Natural Science Foundation of China (11271175) and the Natural Science Foundation of Shandong Province (ZR2021MA079, ZR2021MA088).
		+

	==References==		==References==

@@ Line 273: / Line 273: @@
 |style="text-align: center;padding:10px;"| [[Image:Draft_Yue_357804329-Figure6.png|700px|Retailer's Total Cost for γ and tₑ (D=1000, A=0.1, h=0.07, c<sub>f</sub>=0.2, cₗ=0.1, w=3, tₛ=0.25, k=7).]]
 |-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 6'''. Retailer's total cost for <math>\gamma </math> and <math>t_e</math> (<math>D=1000</math>, <math>A=0.1</math>, <math>h=0.07</math>, <math>c_f=0.2</math>, <math>c_l=0.1</math>, <math>w=3</math>, <math>t_s=0.25</math>, <math>k=7</math>)
+| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 6'''. Retailer's total cost for  <math>\gamma </math> and <math>t_e</math> (<math>D=1000</math>, <math>A=0.1</math>, <math>h=0.07</math>, <math>c_f=0.2</math>, <math>c_l=0.1</math>, <math>w=3</math>, <math>t_s=0.25</math>, <math>k=7</math>)
 |}

← Older revision		Revision as of 14:22, 28 June 2023
Line 114:		Line 114:
	Given that all items purchased through the last regular order before the promotion period will be sold out at time <math display="inline">((m+n)Q^-S^)/D</math>, we refer to <math display="inline">[0,((m+n)Q^-S^)/D]</math> as the regular order interval. Since all items purchased through the special order will be sold out at time <math display="inline">((m+n)Q^-S^+q_r)/D</math>, we refer to <math display="inline">(((m+n)Q^-S^)/D,((m+n)Q^-S^+q_r)/D]</math> as the the special order interval. The retailer's inventory level is illustrated in [[#img-1\|Figure 1]], where “RI”, “SI”, and “RH” denote the regular order interval, the special order interval, and the remaining time horizon, respectively.		Given that all items purchased through the last regular order before the promotion period will be sold out at time <math display="inline">((m+n)Q^-S^)/D</math>, we refer to <math display="inline">[0,((m+n)Q^-S^)/D]</math> as the regular order interval. Since all items purchased through the special order will be sold out at time <math display="inline">((m+n)Q^-S^+q_r)/D</math>, we refer to <math display="inline">(((m+n)Q^-S^)/D,((m+n)Q^-S^+q_r)/D]</math> as the the special order interval. The retailer's inventory level is illustrated in [[#img-1\|Figure 1]], where “RI”, “SI”, and “RH” denote the regular order interval, the special order interval, and the remaining time horizon, respectively.
	{\| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"		{\| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
−	\|-style="background:white;"	+	\|- style="background:white;"
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
	\|style="text-align: center;padding:10px;"\|[[Image:Draft_Yue_357804329-Figure1.png\|412px\|Retailer's Ordering Strategy with n=0.]]		\|style="text-align: center;padding:10px;"\|[[Image:Draft_Yue_357804329-Figure1.png\|412px\|Retailer's Ordering Strategy with n=0.]]
−	\|
−	\|
−	\|
−	\|
−	\|
−	\|
−	\|
−	\|
−	\|
−	\|
	\|-		\|-
−	\|	+	\| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" \| '''Figure 1'''. Retailer's ordering strategy with n=0
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"\| '''Figure 1'''. Retailer's ordering strategy with ~~<math>~~n=0~~</math>~~	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
−	\|	+
	\|}		\|}