Optimal inventory replenishment strategy for deteriorating items in a demand-declining market with the retailer’s price manipulation

Abstract

Due to rapid technological innovation and global competitiveness, the demand of many fashion-typed products usually decline significantly over time. A retailer facing such a market can employ replenishment strategies to increase its profit. This study, from the perspective of the retailer in a two-echelon supply chain, develops the optimal replenishment strategy for products experiencing deterioration, continuous decrease in market demand and price changes. This model help determine the optimal product life for products. Numerical examples are systematically conducted to verify the performances of the proposed model.

Appendices

Appendix A: Proof of Proposition 1

The analysis of production (Fig. 1) is valid only if the condition $I_{0}^{S}(t_{0}) \ge q_{1}^{*}$ is satisfied. From (20),

$$ I_{0}^{S}(t_{0}) = \frac{p_{0}}{\theta} \bigl( 1 - e^{ - \theta \cdot t_{0}} \bigr) $$

(A.1)

Then

$$ \frac{p_{0}}{\theta} \bigl( 1 - e^{ - \theta \cdot t_{0}} \bigr) \ge q_{1}^{*} $$

(A.2)

Therefore,

$$ p_{0} \ge\frac{q_{1}^{*} \cdot \theta}{1 - e^{ - \theta \cdot t_{0}}} $$

(A.3)

Similarly, from (21), for the initial period of production is $t_{0} = \frac{1}{\theta} \ln[ \frac{p_{0}}{p_{0} - q_{1}^{*}\theta} ]$, the following condition is satisfied: $t_{0} \le\frac{1}{\theta} \ln[ \frac{P_{0}}{P_{0} - q_{1}^{*}\theta} ]$.

Appendix B: Proof of Proposition 2

Assume that a production rate, $p'_{i}$, is larger than p _i , the value of Eq. (21).

$$ p'_{i} > p_{i} = \frac{q_{i + 1} \cdot \theta}{ ( 1 - e^{ - \theta \cdot t_{i}} )}, \quad i = 0,1,2,\dots,n - 1 $$

(B.1)

The supplier’s revenue is the same because the redundant products will not be sold

$$ \mathit{RE}_{i}^{S'} = \mathit{RE}_{i}^{S}, \quad i = 0,1,2,\dots,n - 1 $$

(B.2)

Then the production and storage cost will increase by

$$ \mathit{PC}_{i}^{S'} = U \cdot p_{i}^{\prime} \cdot t_{i} > \mathit{PC}_{i}^{S} = U \cdot p_{i} \cdot t_{i}, \quad i = 0,1,2,\dots,n - 1 $$

(B.3)

(B.4)

(B.5)

(B.6)

Therefore, the supplier’s total profit is decreased to

(B.7)

This ends the proof.

Appendix C: Design of experiments

Run	Basic design										TP S	TP R
	h A	k B	O r C	U D	ν E	ρ F=ABCD	η G=ABCE	ω H=ABDE	A J=ACDE	θ K=BCDE
1	–	–	–	–	–	+	+	+	+	+	62,865	1,750,296
2	–	–	–	–	+	+	–	–	–	–	161,944	163,802
3	–	–	–	+	–	–	+	–	–	–	29,376	102,998
4	–	–	–	+	+	–	–	+	+	+	102,573	1,060,363
5	–	–	+	–	–	–	–	+	–	–	41,103	559,582
6	–	–	+	–	+	–	+	–	+	+	105,557	78,236
7	–	–	+	+	–	+	–	–	+	+	100,131	383,125
8	–	–	+	+	+	+	+	+	–	–	61,594	805,549
9	–	+	–	–	–	–	–	–	+	–	113,713	163,627
10	–	+	–	–	+	–	+	+	–	+	84,735	474,552
11	–	+	–	+	–	+	–	+	–	+	14,451	1,169,658
12	–	+	–	+	+	+	+	–	+	–	178,344	245,895
13	–	+	+	–	–	+	+	–	–	+	128,364	238,772
14	–	+	+	–	+	+	–	+	+	–	141,427	1,737,788
15	–	+	+	+	–	–	+	+	+	–	25,695	835,875
16	–	+	+	+	+	–	–	–	–	+	53,620	51,935
17	+	–	–	–	–	–	–	–	–	+	71,341	105,999
18	+	–	–	–	+	–	+	+	+	–	130,025	874,818
19	+	–	–	+	–	+	–	+	+	–	28,660	2,033,061
20	+	–	–	+	+	+	+	–	–	+	112,553	159,524
21	+	–	+	–	–	+	+	–	+	–	205,224	371,875
22	+	–	+	–	+	+	–	+	–	+	93,536	979,951
23	+	–	+	+	–	–	+	+	–	+	13,192	453,066
24	+	–	+	+	+	–	–	–	+	–	77,834	80,191
25	+	+	–	–	–	+	+	+	–	–	36,729	993,226
26	+	+	–	–	+	+	–	–	+	+	236,794	250,660
27	+	+	–	+	–	–	+	–	+	+	49,100	159,321
28	+	+	–	+	+	–	–	+	–	–	66,838	587,698
29	+	+	+	–	–	–	–	+	+	+	64,928	1,024,148
30	+	+	+	–	+	–	+	–	–	–	72,690	50,371
31	+	+	+	+	–	+	–	–	–	–	60,423	247,587
32	+	+	+	+	+	+	+	+	+	+	97,107	1,464,150