The Inventory Model for Deteriorating Items under Conditions Involving Cash Discount and Trade Credit
Abstract
:1. Introduction, Motivation and Preliminaries
- 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 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 .
- 2
- To systematically revisit the annual total relevant cost in Chang and Teng [41] and to present in detail the mathematically correct ways for the derivations of .
- 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 , but to also demonstrate how Theorem 1 in this paper can locate all of the optimal solutions of .
2. The Mathematical Modelling of the Problem
- (1)
- The demand for the item is constant with time.
- (2)
- Shortages are not allowed.
- (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, or ), 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.
- the demand rate per year.
- the unit holding cost per year excluding interest charges.
- the selling price per unit.
- the unit purchasing cost, with .
- the interest charged per $ in stocks per year by the supplier or a bank.
- the interest earned per $ per year.
- the ordering cost per order.
- the cash discount rate, .
- the constant deterioration rate, where .
- the period of cash discount.
- the period of permissible delay in settling account, with .
- the replenishment time interval.
- , where .
- Policy I: The customer accepts the cash discount and makes the full payment at time .
- Policy II: The customer does not accept the cash discount and makes the full payment at time .
- the annual total relevant cost when the customer adopts Policy (I).
- the annual total relevant cost when the customer adopts Policy (II).
- the optimal replenishment time of .
- the optimal replenishment time of .
- the annual total relevant cost
- the optimal replenishment time of .
- =
- =
- (a)
- The cost of placing order;
- (b)
- The cost of purchasing units;
- (c)
- The cost of carrying inventory (excluding interest changes);
- (d)
- The cost discount earned;
- (e)
- The interest earned from sales revenue during the permissible period or
- (f)
- The cost of interest change for unsold items after the permissible delay or .
- (A)
- Chang and Teng [41] showed that(e)(f) The internet payable per yearThe customer buys units at time 0, and owes to the supplier. At the time , the customer sells units in total, and has plus the interest earned to pay the supplier. From the difference between the total purchase cost and the total amount of money in the account, i.e.,
- (B)
- Under Case I (B), the customer sells units in total at time T, and has to pay the supplier in full at time . Therefore, Chang and Teng [41] derived the annual total relevant cost as follows:
- (C)
- If , then Equation (11) holds true. Following the same arguments as in Case I (A) (ii), we find thatAccording to Equations (2) to (6) and (17), the annual total relevant cost can be expressed as follows:Since is continuous except when if .
3. The Convexity of
- (A)
- if .
- (B)
- (i) is increasing if .(ii) if .
- (C)
- if
- (D)
- if
- (E)
- Both and are convex on .
- (F)
- If , then both and are convex on .
- (G)
- If , then are convex on .
- (H)
- If , then are convex on .
- (A)
- See Lemma 1 in Huang and Liao [8].
- (B)
- LetEquations (43) and (44) yieldEquations (45) and (46) imply that both functions and are increasing when . Therefore, we getWe now set . Consequently, we have
- (i)
- is increasing if .
- (ii)
- if .
- (C)
- LetThenSo, we haveEquation (47) implies that the function is increasing if . Therefore, we haveConsequently, we obtain
- (D)
- The proof is similar to that of Theorem 1 (C).
- (E)
- Equations (31), (37) and Theorem 1 (A) imply that and . Therefore, both functions and are convex on .
- (F)
- If , then . Equations (33) and (39) imply that and . Therefore, both functions and are convex when if .
- (G)
- See, for details, Lemma 2 in the paper by Huang and Liao [8].
- (H)
- See, for details, Lemma 2 in the paper by Huang and Liao [8].
4. Theorems for the Optimal Cycle of
- Case I.
- Policy I is adopted and .LetSince , Theorem 1 (B) implies that
- Case II.
- Policy I is adopted and .LetNow, since , Theorem 1 (C) implies that
- Case III.
- Policy II is adopted and .LetSince , Theorem 1 (B) implies that
- Case IV.
- Policy II is adopted and .LetThus, since , Theorem 1 (D) implies that
- (A)
- If then .
- (B)
- If then .
- (C)
- If then .
- (A)
- If , then . From Equations (73a), (73b) and (73c), we thus find that
- (i)
- is decreasing on and increasing on .
- (ii)
- is increasing on .
- (iii)
- is increasing on .
Since the function is continuous when if , Equations (15a), (15b) and (15c) and the above observations (i) to (iii) imply that . - (B)
- If , from Equations (73a), (73b) and (73c), we find that
- (iv)
- is decreasing on .
- (v)
- is decreasing on and increasing on .
- (vi)
- is increasing on .
Since the function is continuous when if , Equations (15a), (15b) and (15c), together with the above observations (iv) to (vi), imply that . - (C)
- If , then . From Equations (73a), (73b) and (73c), we obtain
- (vii)
- is decreasing on .
- (viii)
- is decreasing on .
- (ix)
- is decreasing on and increasing on .
Since is continuous on if , Equations (15a), (15b) and (15c), and the above observations (vii) to (ix), imply that .
- (A)
- If then .
- (B)
- If then .
- (C)
- If then or is associated with the least cost.
- (A)
- If , then . From Equations (73a), (73b) and (73c), we have
- (i)
- is decreasing on and increasing on .
- (ii)
- is increasing on .
Since , Equations (18a) and (18b), together with the above observations (i) and (ii), imply that . - (B)
- If , from Equations (73a), (73b) and (73c), we have
- (iii)
- is decreasing on .
- (iv)
- is increasing on .
Since , Equations (18a) and (18b) and the above observations (iii) and (iv) imply that . - (C)
- If , then . Thus, from Equations (73a), (73b) and (73c), we get
- (v)
- is decreasing on .
- (vi)
- is decreasing on and increasing on .
Since , Equations (18a) and (18b), together with the above observations (v) and (vi), imply that or is associated with the least cost.
- (A)
- If then .
- (B)
- If then .
- (C)
- If then .
- (A)
- If then .
- (B)
- If then .
- (C)
- If then or is associated with the least cost.
- (i)
- If then or is associated with the least cost.
- (ii)
- If and then or is associated with the least cost.
- (iii)
- If and , or is associated with the least cost.
- (iv)
- If and then or is associated with the least cost.
- (v)
- If and then or is associated with the least cost.
- (vi)
- If and then or is associated with the least cost.
- (vii)
- If and then or is associated with the least cost.
- (i)
- If then or is associated with the least cost.
- (ii)
- If and then or is associated with the least cost.
- (iii)
- If and then or is associated with the least cost.
- (iv)
- If and then then or is associated with the least cost.
- (v)
- If and then or is associated with the least cost.
- (vi)
- If and then or is associated with the least cost.
- (vii)
- If and then or is associated with the least cost.
- (i)
- If then or is associated with the least cost.
- (ii)
- If and then or is associated with the least cost.
- (iii)
- If and then or is associated with the least cost.
- (iv)
- If and then or is associated with the least cost.
- (v)
- If and , then or is associated with the least cost.
- (vi)
- If and then or is associated with the least cost.
- (vii)
- If and then or is associated with the least cost.
- (i)
- If then or is associated with the least cost.
- (ii)
- If and then or is associated with the least cost.
- (iii)
- If and then or is associated with the least cost.
- (iv)
- If and then or is associated with the least cost.
- (v)
- If and then or is associated with the least cost.
- (vi)
- If and then or is associated with the least cost.
- (vii)
- If and then or is associated with the least cost.
5. Discussions Concerning Theorem 1 of Chang and Teng [41]
- (A)
- About Theorem 1 (1) in Chang and Teng [41]:Equation (29) reveals that
- (B)
- About Theorem 1 (2) in the paper by Chang and Teng [41]:IfSince , is not the optimal solution of . 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 denote the optimal solution obtained by Theorem 1 in Chang and Teng [41]. In this case, we consider the following example.
- (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.
6. Concluding Remarks and Observations
- (1)
- (2)
- (3)
- (4)
Author Contributions
Funding
Conflicts of Interest
References
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Chung, K.-J.; Liao, J.-J.; Lin, S.-D.; Chuang, S.-T.; Srivastava, H.M. The Inventory Model for Deteriorating Items under Conditions Involving Cash Discount and Trade Credit. Mathematics 2019, 7, 596. https://doi.org/10.3390/math7070596
Chung K-J, Liao J-J, Lin S-D, Chuang S-T, Srivastava HM. The Inventory Model for Deteriorating Items under Conditions Involving Cash Discount and Trade Credit. Mathematics. 2019; 7(7):596. https://doi.org/10.3390/math7070596
Chicago/Turabian StyleChung, Kun-Jen, Jui-Jung Liao, Shy-Der Lin, Sheng-Tu Chuang, and Hari Mohan Srivastava. 2019. "The Inventory Model for Deteriorating Items under Conditions Involving Cash Discount and Trade Credit" Mathematics 7, no. 7: 596. https://doi.org/10.3390/math7070596