Optimal pricing, ordering, and credit period policies for deteriorating products under order-linked trade credit

Yu-Chung Tsao; Hanifa-Astofa Fauziah; Thuy-Linh Vu; Nur Aini Masruroh

doi:10.3934/jimo.2021152

Article Contents

2022, Volume 18, Issue 6: 4151-4182. Doi: 10.3934/jimo.2021152

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Optimal pricing, ordering, and credit period policies for deteriorating products under order-linked trade credit

1.
Department of Industrial Management, National Taiwan University of Science and Technology, Taipei, Taiwan
2.
Artificial Intelligence for Operations Management Research Center, National Taiwan University of Science and Technology, Taipei, Taiwan
3.
Department of Business Administration, Asia University, Taichung, Taiwan
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
5.
Department of Mechanical and Industrial Engineering, Universitas Gadjah Mada, Yogyakarta, Indonesia

^* Corresponding author: Yu-Chung Tsao
^* Corresponding author: Yu-Chung Tsao

Received: March 2021

Revised: June 2021

Early access: September 2021

Published: September 2022

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Abstract

In the modern global economy, trade credit financing is typical in business transactions for both sellers and buyers. The seller offers a credit period to attract new buyers or stimulate demand, and the buyer takes the opportunity to accumulate revenue. To obtain this benefit, the seller prefers trade credit policies that are dependent on the quantity ordered, referred to as order-linked trade credit. The buyer can obtain the benefits from a fully delayed payment if their order is sufficiently large. Similarly, the seller can sell many products while granting a credit period. Otherwise, the buyer receives only partial trade credit, and the seller can take the opportunity of both cash and credit payments. In this study, an economic order quantity (EOQ) inventory model for deteriorating products, under default risk control-based trade credit, is formulated using a discounted cash flow approach. The seller offers to the buyer order-linked trade credit with price-and credit-period-dependent demand. The optimal selling price, credit period policies, and replenishment cycle time are determined simultaneously, while maximizing the present value of the seller's total profit. Moreover, this research provides numerical examples and sensitivity analysis to illustrate the theoretical results, solution procedure, and gain managerial insights. 200 words.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Conceptual model for deteriorating products under order-linked trade credit

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Figure 2. Inventory level

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Figure 3. Graphical representation of Sub-case 1.1

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Figure 4. Graphical representation of Sub-case 1.2

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Figure 5. Graphical representation of Sub-case 2.1

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Figure 6. Graphical representation of Sub-case 2.2

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Figure 7. The graph of $ PTP^{}_{1.1}(p, M^{}, T^{}) $

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Figure 8. The graph of $ PTP^{}_{1.1}(p, M^{}, T^{}) $

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Figure 9. The graph of $ PTP^{}_{1.1}(p, M^{}, T^{}) $

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Table 1. Summary of literature review

Author	Payment Terms	Time value of money	Pers-pective	Decision Variables	Demand Function	Deterio-ration	Credit-risk customer
Sarkar [22]	FTC	No	Buyer	Cycle Time	Time	Time-varying	No
Taleizadeh[29]	PTC	No	Buyer	Cycle Time	Rate	No	No
Huang [13]	Order Linked of FTC and PTC	No	Buyer	Cycle Time	Rate	No	No
Chen [7]	Order Linked of FTC and PTC	No	Buyer	Cycle Time	Rate	No	No
Ting [31]	Order Linked of FTC and PTC	No	Buyer	Cycle Time	Rate	Constant	No
Shah [24]	Order Linked of FTC and PTC	No	Buyer	Cycle Time; Selling Price	Selling Price	No	No
Taleizadeh[28]	Order Linked of FTC and PTC	No	Buyer	Cycle Time	Rate	Time-varying	No
Tiwari [33]	Order Linked of FTC and PTC	No	Buyer	Cycle Time	Rate	Constant	No
Wang [37]	FTC	No	Seller	Cycle Time; Credit Period	Credit Period	Time-varying	Yes
Shah [23]	FTC	No	Seller	Cycle Time; Credit Period	Credit Period; Time	No	Yes
This Research	Order Linked of FTC and PTC	Yes	Seller	Cycle Time; Selling Price; Credit Period	Selling Price; Credit Period	Constant	Yes
Note: FTC corresponds to full trade credit and PTC corresponds to partial trade credit

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Table 2. Notations

Notation	Description
$ i $	Index of case based on demand, $ i=\lbrack 1, 2\rbrack $
$ j $	Index of case based on time, $ j=\lbrack 1, 2\rbrack $
$ A $	Replenishment cost per order, in dollar
$ c $	Procurement cost per unit, in dollar
$ h $	The inventory holding cost rate per unit per unit time, in dollar
$ I_{e} $	Interest earned per dollar per unit time
$ I_{Loss} $	Interest revenue loss due to offering trade credit per dollar per unit time
$ r $	Annual compound interest rate per dollar per unit time
$ D(p, M) $	Annual demand rate per unit time, as a function of both $ p $and $ M $
$ F(M) $	The rate of default risk
$ \theta $	Deterioration rate, $ 0\le \theta \le 1 $
$ D_{d} $	The specific threshold in which permits the full trade credit
$ \alpha $	The fraction of the delay payments is permitted, $ 0\le \alpha \le 1 $
$ I(t) $	Inventory level at time $ t $
$ Q $	Seller's order quantity
$ OC $	The present value of ordering cost
$ HC $	The present value of holding cost
$ PC $	The present value of procurement cost
$ SR_{i} $	The present value of revenue in case i, $ i=\lbrack 1, 2\rbrack $
$ IE_{i, j} $	The present value of interest earned in case (i, j), $ i, j=\lbrack 1, 2\rbrack $
$ IL_{i, j} $	The present value of interest loss in case (i, j), $ i, j=\lbrack 1, 2\rbrack $
$ T $	Length of the replenishment cycle (decision variable)
$ M $	The credit period policies offered by the seller (decision variable)
$ p $	Selling price offered by the seller per unit (decision variable)
$ PTP(p, M, T) $	The present value of total annual profit, which is the function of $ p $, $ M $ and $ T $

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Table 3. Interest earned and interest revenue loss cases

Case (1):$ D_{d}<D(p, M) $	Case (2):$ D_{d}\ge D(p, M) $
Sub-case1.1: $ 0\le M\le T\le T+M $	Sub-case 2.1: $ 0\le M\le T\le T+M $
Sub-case1.2: $ 0\le T\le M\le T+M $	Sub-case 2.2: $ 0\le T\le M\le T+M $

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Table 4. A comparison of four cases:

Cases	$ p^{} $(＄)	$ M^{} $ (years)	$ T^{} $ (years)	Demand (items)	$ PTP^{}_{i.j}(p^{}, M^{}, T^{}) $ (＄)
Ordered-link trade credit	25.3312	0.081921	0.27852	375	5482.41
No ordered-link trade credit	26.2325		0.25352	344	5398.24
Full trade credit	25.505	0.054	0.27741	368	5478.92
Partial trade credit	25.7267	0.151	0.2979	372	5485.39

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Table 5. Summary of sensitivity analysis

Parameter	$ p^{} $	$ M^{} $	$ T^{} $	$ PTP^{}_{i.j}(p^{}, M^{}, T^{}) $
D$ _{d} $=350	25.3312	0.0819213	0.266878	5482.41
D$ _{d} $=450	25.5224	0.149572	0.297398	5493.47
c=6	23.7634	0.260868	0.360191	7123.23
c=8	24.6262	0.196586	0.320289	6280.3
c=10	25.5224	0.149572	0.297398	5493.47
c=12	26.4370	0.112410	0.28441	4761.02
c=14	27.3616	0.081181	0.277903	4081.91
h=0.6	25.5873	0.177748	0.360788	5518.11
h=0.8	25.5501	0.161851	0.324659	5505.19
h=1	25.5224	0.149572	0.297398	5493.47
h=1.2	25.5012	0.139723	0.275927	5482.67
h=1.4	25.4845	0.131596	0.258471	5472.62
$ \theta $=0.03	25.5515	0.162354	0.325811	5505.41
$ \theta $=0.04	25.5359	0.155573	0.310662	5499.3
$ \theta $=0.05	25.5224	0.149572	0.297398	5493.47
$ \theta $=0.06	25.5106	0.144213	0.285662	5487.86
$ \theta $=0.07	25.5002	0.139388	0.275185	5482.47
b=15	39.6903	0.291871	0.409115	11962.1
b=20	30.7678	0.198115	0.322033	7882.19
b=25	25.5224	0.149572	0.297398	5493.47
b=30	22.0502	0.114364	0.291447	3947.53
b=35	19.5771	0.084373	0.29072	2882.45
l=0.06	27.3895	0.790391	0.771212	5638.77
l=0.08	26.0368	0.328971	0.397466	5537.01
l=0.1	25.5224	0.149572	0.297398	5493.47
l=0.12	25.2051	0.0372622	0.268559	5476.34
l=0.14	25.1003	6.15627$ \times $10$ ^{-27} $	0.266878	5475.06

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Table 6. Summary of sensitivity analysis (cont')

Parameter	$ p $	$ M $	$ T $	$ PTP^_{i.j}(p^, M^, T^) $
$ \alpha $=0.48	26.3120	0.427639	0.398781	5544.5
$ \alpha $=0.64	25.7899	0.243915	0.329357	5510.66
$ \alpha $=0.8	25.5224	0.149572	0.297398	5493.47
$ \alpha $=0.96	25.3619	0.092792	0.281188	5484.03
$ \alpha $=1	25.3312	0.081921	0.27852	5482.41
r=0.024	25.9844	0.307269	0.445592	5546.65
r=0.032	25.7022	0.211592	0.349793	5516.32
r=0.04	25.5224	0.149572	0.297398	5493.47
r=0.048	25.3915	0.103826	0.264839	5475.44
r=0.056	25.2871	0.067011	0.242921	5460.93
I$ _{Loss} $=0.036	25.7848	0.242724	0.314041	5504.49
I$ _{Loss} $=0.048	25.6232	0.185330	0.304028	5497.72
I$ _{Loss} $=0.06	25.5224	0.149572	0.297398	5493.47
I$ _{Loss} $=0.072	25.4539	0.125260	0.292728	5490.55
I$ _{Loss} $=0.084	25.4043	0.107692	0.289277	5488.43
A=12	25.4177	0.120489	0.232408	5523.65
A=16	25.4741	0.136219	0.26707	5507.64
A=20	25.5224	0.149572	0.297398	5493.47
A=24	25.5651	0.161264	0.324661	5480.61
A=28	25.6034	0.171715	0.349611	5468.74
n=60	25.1003	1.57412$ \times $10$ ^{-9} $	0.266878	5475.06
n=80	25.1003	6.67337$ \times $10$ ^{-9} $	0.266878	5475.06
n=100	25.5224	0.149572	0.297398	5493.47
n=120	26.6491	0.452890	0.489132	5575.1
n=140	29.2678	1.039800	101428	5783.31
I$ _{e} $=0.024	25.4698	0.131901	0.258682	5472.69
I$ _{e} $=0.032	25.4934	0.139910	0.276078	5482.71
I$ _{e} $=0.04	25.5224	0.149572	0.297398	5493.47
I$ _{e} $=0.048	25.5589	0.161542	0.324322	5505.12
I$ _{e} $=0.056	25.6065	0.176891	0.359702	5517.93

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