EOQ Model for Deteriorating Items with Stock-Level-Dependent Demand Rate and Order-Quantity-Dependent Trade Credit

Min, Jie; Ou, Jian; Zhong, Yuan-Guang; Liu, Xin-Bao

doi:https://doi.org/10.1155/2014/962128

Mathematical Problems in Engineering

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Abstract Introduction Numerical Example Conclusions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 962128 | https://doi.org/10.1155/2014/962128

EOQ Model for Deteriorating Items with Stock-Level-Dependent Demand Rate and Order-Quantity-Dependent Trade Credit

Jie Min,¹Jian Ou,¹Yuan-Guang Zhong,²and Xin-Bao Liu³

Academic Editor: Zhen-Lai Han

Received24 Jul 2014

Accepted29 Oct 2014

Published16 Dec 2014

Abstract

This paper develops a generalized inventory model for exponentially deteriorating items with current-stock-dependent demand rate and permissible delay in payments. In the model, the payment for the item must be made immediately if the order quantity is less than the predetermined quantity; otherwise, a fixed trade credit period is permitted. The maximization of the average profit per unit of time is taken as the inventory system’s objective. The necessary and sufficient conditions and some properties of the optimal solution to the model are developed. Simple solution procedures are proposed to efficiently determine the optimal ordering policies of the considered problem. Numerical example is also presented to illustrate the solution procedures obtained.

1. Introduction

In classical inventory models, the demand rate of items was often assumed to be either constant or time dependent. However, many marketing researchers and practitioners have recognized that the demand rate of many retail items is usually influenced by the amount of inventory displayed. For example, Whitin [1] stated, “For retail stores the inventory control problem for style goods is further complicated by the fact that inventory and sales are not independent of one another. An increase in inventories may bring about increased sales of some items.” Levin et al. [2] and Sliver and Peterson [3] also observed that sales at the retail level tend to be proportional to the inventory displayed and that large piles of goods displayed in a supermarket will lead customers to buy more. The reason behind this phenomenon is a typical psychology of customers. They may have the feeling of obtaining a wide range for selection when a large amount is stored or displayed. Conversely, they may have doubt about the freshness or quality of the product when a small amount is stored. Based on the observed phenomenon, it is clear that in real life the demand rate of items may be influenced by the stock levels. Many researchers have developed lots of inventory models to cover this phenomenon. Gupta and Vrat [4] were the first to develop an inventory model with the initial-stock-dependent consumption rate. Padmanabhan and Vrat [5] proposed an EOQ model for initial-stock-dependent demand and exponential decay items. Giri et al. [6] presented a model under inflation for initial-stock-dependent consumption rate and exponential decay. In addition, Baker and Urban [7] investigated another kind of inventory model in which the demand of item is a polynomial function of the instantaneous stock level. Mandal and Phaujdar [8] developed an inventory model for deteriorating items with linearly current-stock-dependent demand. Pal et al. [9] extended the model of Baker and Urban [7] to the case of deteriorating items. Padmanabhan and Vrat [10] further developed three models for deteriorating items under current-stock-level-dependent demand rate: without backorder, with complete backorder, and with partial backorder. Chung et al. [11] modified Padmanabhan and Vrat’s [10] models by showing the necessary and sufficient conditions of the existing optimal solution for models with no backorder and complete backorder. Zhou and Yang [12] developed an inventory model for a general inventory-level-dependent demand with limited storage space. Moreover, many further extensive models were developed by researchers such as Balkhi and Benkherouf [13], Dye and Ouyang [14], Zhou and Yang [15], Wu et al. [16], and Alfares [17].

In all the above inventory models with stock-level-dependent demand rate assumed that the retailer must pay for the items as soon as the items are received. However, in the real life, the supplier sometimes will offer the retailer a delay period, that is, a trade credit period, in paying for the amount of purchasing cost. Before the end of the trade credit period, the retailer can sell the goods and accumulate the revenue to earn interest. A higher interest is charged if the payment is not settled by the end of trade credit period. In real world, the supplier often makes use of this policy to promote their commodities. During the recent two decades, the effect of a permissible delay in payments on the optimal inventory systems has received more attention from numerous researchers. Goyal [18] explored a single item EOQ model under permissible delay in payments. Aggarwal and Jaggi [19] extended Goyal’s [18] model to the case with deteriorating items. Jamal et al. [20] further generalized the above inventory models to allow for shortages. Other related articles can be found in Sarker et al. [21], Teng [22], Huang [23], Chang et al. [24, 25], Chung and Liao [26], Teng et al. [27], Liao [28] and Teng and Goyal [29], and their references. More recently, Chen et al. [30] studied an economic order quantity model under conditionally permissible delay in payments, in which the supplier offers the retailer a fully permissible delay periods if the retailer orders more than or equal to a predetermined quantity, otherwise partial payments will be provided. Chang et al. [31] developed an appropriate inventory model for noninstantaneously deteriorating items where suppliers provide a permissible payment delay schedule linked to order quantity. However, all of the abovementioned papers with trade credit financing did not address inventory-level-dependent demand rate.

The present paper establishes an EOQ model for deteriorating items with current-inventory-level-dependent demand rate and permissible delay in payments which is dependent on the retailer’s order quantity. That is, if the order quantity is less than the predetermined quantity, the payment for the item must be made immediately; otherwise, a fixed trade credit period is permitted. We then establish the proper mathematical model and study the necessary and sufficient conditions of the optimal solution to the model. Some properties of the optimal solution to the model are proposed for obtaining the optimal replenishment quantity and replenishment cycle, which make the average profit of the system maximized.

The remainder of the paper is organized as below. The next section introduces some notations and assumptions used in this paper. We then present in Section 3 the formulation of the inventory model with current-stock-level-dependent demand rate and order-quantity-dependent trade credit. Section 4 examines the existence and uniqueness of the optimal solution to the considered inventory system and shows some properties of the optimal ordering policies. In Section 5, numerical example is given to illustrate the solution procedure obtained in this paper. Some conclusions are included in Section 6.

2. Notations and Assumptions

The following notations and assumptions are used in the paper.

Notations:ordering cost one order in dollars,:purchase cost per unit in dollars,:selling price per unit in dollars (),:stock holding cost per year per unit in dollars (excluding interest charges),:deteriorating rate of items (),:the qualified quantity in units at or beyond which the delay in payments is permitted,:the time interval in years in which units are depleted to zero due to both demand and deterioration,:interest which can be earned per $ per year by the retailer,:interest charges per $ in stocks per year by the supplier,:the retailer’s trade credit period offered by supplier in years,:inventory cycle length in years (decision variable),:the retailer’s order quantity in units per cycle,:the retailer’s average profit function per unit time in dollars.

Assumptions(1)The demand rate is a known function of retailer’s instantaneous stock level , which is given by (2)The time horizon of the inventory system is infinite.(3)The lead time is negligible.(4)Shortages are not allowed to occur.(5)When the order quantity is less than the qualified quantity , the payment for the item must be made immediately. Otherwise, the fixed trade credit period is permitted.(6)If the trade credit period is offered, the retailer would settle the account at and pay for the interest charges on items in stock with rate over the interval as , whereas the retailer settles the account at and does not need to pay any interest charge of items in stock during the whole cycle as .(7)The retailer can accumulate revenue and earn interest from the very beginning of inventory cycle until the end of the trade credit period offered by the supplier. That is, the retailer can accumulate revenue and earn interest during the period from to with rate under the condition of trade credit.(8), , which are reasonable assumptions in reality.

3. Mathematical Formulation of the Model

Based on the above assumptions, the considered inventory system goes like this: at the beginning (say, the time ) of each replenishment cycle, items of units are held, and the items are depleted gradually in the interval due to the combined effects of demand and deterioration. At time , the inventory level reaches zero, and the whole process is repeated.

The variation of inventory level, , with respect to time can be described by the following differential equation: with the boundary condition .

The solution to the above differential equation is So the retailer’s order size per cycle is We observe that if the order quantity , then the payment must be made immediately. Otherwise, the retailer will get a certain credit period, . Hence, from (4) one has the fact that the inequality holds if and only if , where is given as below: The elements comprising the profit function per cycle of the retailer are listed below.(a)The ordering cost = .(b)The holding cost (excluding interest charges) = .(c)The purchasing cost = .(d)The sales revenue = .(e)According to the above assumption, there are two cases to occur in interest payable in each order cycle.

Case 1 (). In this case, the delay in payments is not permitted. That is, all products in inventory have to be financed with annual rate . Therefore, interest payable in each order cycle

Case 2 (). In this case, the delay credit period in payments is permitted, which arouses two subcases: Subcase 2.1 where and Subcase 2.2 where .

Subcase 2.1 (). Since the cycle time is shorter than the credit period , there is no interest paid for financing the inventory in stock. So the interest payable in each order cycle = 0.

Subcase 2.2 (). When the credit period is shorter than or equal to the replenishment cycle time , the retailer starts paying the interest for the items in stock after time with rate . Hence, the interest payable in each order cycle (f)Similarly, there are also two cases to occur in interest earned in each order cycle.

Case 1 (). In this case, the delay in payments is not permitted, so the interest earned in each order cycle = 0.

Case 2 (). In this case, the delay in payments is permitted, which causes two subcases as below.

Subcase 2.1 (). Since the cycle time is shorter than the credit period , from time 0 to the retailer sells the goods and continues to accumulate the sales revenue to earn interest , and from time to the retailer can use the sales revenue generated in to earn interest . Therefore, the interest earned from time 0 to per cycle

Subcase 2.2 (). In this case, the delay in payments is permitted, and during time 0 through the retailer sells the goods and continues to accumulate sales revenue and earns the interest with rate . Therefore, the interest earned from time 0 to per cycle From the above, the profit function for the retailer can be expressed as On simplification and summation, we get the profit function as the following three forms:(A)if , then (B)if but , then (C)if and , then where the meaning of notations and in the above expressions is For convenience, we will consider the problem through the two following situations: (1) and (2) .

(1) Suppose That . In the situation of , has three different expressions as follows:

It is easy to verify that, in the case of , continues except at . In fact, if then we could prove that . (Proof of is shown in Appendix A.)

(2) Suppose That . If , then has two different expressions as follows:

Here, and are given by (11) and (13), respectively. continues except at . In fact, we could prove that when . (Proof of is shown in Appendix B.)

In the next section, we will determine the retailer’s optimal cycle time for the above two situations using some algebraic method.

4. The Solution Procedures

In order to decide the optimal solution of function , we need first to study the properties of function () on (), respectively. The first-order necessary condition for in (11) to be maximum is , which implies that For convenience, we set the left-hand expression of (17) as . Taking the derivative of with respect to , we get

From (18) we know that if , then ; that is, is increasing on (). It is obvious that and ; therefore the Intermediate Value Theorem implies that , that is, , has a unique root (say ) on (). Moreover, taking the second derivative of with respect to gives

It is easy to show that the inequality holds for all , so will always be negative on () if ; that is, is convex on (). Therefore, is the unique maximum point of on (). But if , (18) means that is decreasing on (); since , then is negative and is positive for all ; hence is strictly increasing on ().

Then we have the following lemma to describe the property of on ().

Lemma 1. If , is convex on and is the unique maximum point of . Otherwise, is strictly increasing on .

Similarly, the first-order necessary condition for in (12) to be maximized is , which leads to We set the left-hand expression of (20) as . Taking the first derivative of with respect to gives Consequently, from the sign of there are two distinct cases for finding the property of as follows.(1)When , we have ; that is, is strictly increasing on . Since and , has a unique solution (say ) on . Moreover, from (13) we have Since and , we get . Hence, is convex on and is the unique maximum point of .(2)When , there will exist a unique root to the equation . Denote this root by ; then (A)If , that is , then for and for ; that is, is increasing on and decreasing on .(a)When , has a unique root (say ) on due to . Hence, is increasing on and decreasing on . On the other interval of , however, because of and for , has a unique zero point (say ) on . Thus, is decreasing on and increasing on . Hence, is increasing on , decreasing on , and increasing on .(b)When , is negative on because is increasing on , whereas is also negative on since is decreasing on . Therefore, is always nonpositive on and is increasing on .(B)If , that is, , then for any we have ; that is, is strictly decreasing on . Since , is negative on . Hence is increasing on .

Then we have the following lemma to describe the property of on .

Lemma 2. (1) For , is convex on and is the unique maximum point of .
(2) For , is increasing on , decreasing on , and increasing on if , and is increasing on otherwise.
(3) For , is always increasing on .

Likewise, the first-order necessary condition for in (13) to be maximum is , which implies that We set the left-hand expression of (24) as . Taking the first derivative of with respect to gives We consider the following two cases.(1)If , then . Therefore, is increasing on (). It is easy to get +− and . Thus, if then has a unique solution (say ) on (). Hence, will be negative on () and positive on (), which implies is increasing on () and decreasing on (). That is, is unimodal on () and is the unique maximum point of if . In contrast, if then will always be nonnegative on (); hence is always decreasing on ().(2)If , then is strictly decreasing on (). Noting that for , one easily obtains that if there will exist a unique root (say ) to the equation ; hence is positive on () and negative on (), which implies is decreasing on () and increasing on (). But if , is always negative on (); then is increasing on ().

Summarizing the above arguments, the following lemma to describe the property of on () will be obtained.

Lemma 3. (1) For , is increasing on and decreasing on if , and is decreasing on otherwise.
(2) For , is decreasing on and increasing on if , and is increasing on otherwise.

Based upon Lemmas 1–3, we now will decide the optimal solution to from the below two situations.

4.1. Decision Rule of the Optimal Cycle Time When

When , the piecewise profit function has three different expressions, that is, , , and , respectively. From (17), (20), and (24), we have As shown in Appendix C, we will find that . For convenience, we rewrite the notation as , and the following two theorems would be obtained to decide the optimal solution when .

Theorem 4. If and , then one has the following results.(1)When , the following applies.(A)If , , and , then . Hence is .(B)If , , and , then . Hence is , where and satisfies (20).(C)If , , and , then . Hence is or associated with maximum profit.(D)If , , and , then . Hence is or associated with maximum profit.(E)If , , and , then . Hence is or associated with maximum profit.(2)When , the following applies.(A)While , one has the following.(a)If , , and , then . Hence is or associated with maximum profit.(b)If , , and , then . Hence is .(c)If , , and , then . Hence is , , or associated with maximum profit.(d)If , , and , then . Hence is or associated with maximum profit.(e)If , , and , then . Hence is , , or associated with maximum profit.(f)If , , and , then . Hence is or associated with maximum profit.(B)While , one has the following.(a)If , , and , then . Hence is .(b)If , , and , then . Hence is or associated with maximum profit.(3)When , the following applies.(A)If , , and , then . Hence is .(B)If , , and , then . Hence is or associated with maximum profit.(C)If , , and , then . Hence is , , or associated with maximum profit.(D)If , , and , then . Hence is or associated with maximum profit.(4)When , is .

Proof. See Appendix E for detail.

Theorem 5. If and , then one has the following results.(1)When , the results are the same as (1) in Theorem 4.(2)When , the results are the same as (2) in Theorem 4.(3)When , is .

Proof. See Appendix E for detail.

4.2. Decision Rule of the Optimal Cycle Time When

When , the piecewise profit function has only two different expressions, that is, and , respectively. From (24), we have

As shown in Appendix D, we will find that . The following theorem would be given to decide the system’s optimal cycle time when .

Theorem 6. If , then one has the following results.(1)When , the following applies.(A)If and , then . Hence is .(B)If and , then . Hence is or associated with the maximum profit.(C)If and , then . Hence is or associated with the maximum profit.(2)When , is .

Proof. See Appendix F for detail.

5. Numerical Example

In this section, we will provide the following numerical example to illustrate the present model.

Example 1. Given units, units, , , = $50, = $1/unit/year, = $ 5 per unit, = $9 per unit, = 0.13/$/year, = 0.19/$/year, and year, it is easy to get , , and . Consequently, by using Bisection method or Newton-Raphson method we could obtain , , and . Because of , , and , according to Theorem 4(1) the retailer’s optimal cycle time should be ; hence the optimal order quantity and the maximum average profit are and , respectively.

Next, we further study the effects of changes of parameters , , , and on the optimal solutions. The values of other parameters keep the same as in the above example when each of parameters , , , and varies. Table 1 presents the observed results with various parameters , , , and .

The following inferences can be made based on Table 1.(1)The retailer’s optimal order quantity and the optimal average profit increase as the value of increases. It implies that when the market demand is more sensitive to the inventory level, the retailer will increase his/her order quantity to make more profit under permissible delay in payments.(2)The optimal order quantity and the maximum average profit decrease as the value of increases.(3)When increases, the optimal order quantity and the optimal cycle time are increasing but the optimal average profit is decreasing.(4)The retailer’s order quantity is increasing as increases. This verifies the fact that the retailer could indeed increase the sales quantity by adopting the trade credit policy provided by his/her supplier.

6. Conclusions

In this paper, we develop an EOQ model for deteriorating items under permissible delay in payments. The primary difference of this paper as compared to previous related studies is the following four aspects: (1) the demand rate of items is dependent on the retailer’s current stock level, (2) many items deteriorate continuously such as fruits and vegetables, (3) the retailer who purchases the items enjoys a fixed credit period offered by the supplier if the order quantity is greater than or equal to the predetermined quantity, and (4) we here use maximizing profit as the objective to find the optimal replenishment policies, which could better describe the essence of the inventory system than a cost-minimization model. In addition, we have discussed in detail the conditions of the existence and uniqueness of the optimal solutions to the model. Three easy-to-use theorems are developed to find the optimal ordering policies for the considered problem, and these theoretical results are illustrated by numerical example. By studying the effects of , , , and on the optimal order quantity and the optimal average profit , some managerial insights are derived.

The presented model can be further extended to some more practical situations. For example, we could allow for shortages, quantity discounts, time value of money and inflation, price-sensitive and stock-dependent demand, and so forth.

Appendices

A. Proof of

If , we have . From (11) and (12), we get It is evident that and for ; then As a result, we have .

This completes the proof of Appendix A.

B. Proof of

If , we have . From (11) and (13), we get Because of and , we have .

This completes the proof of Appendix B.

C. Proof of

If , from (26) and (27), we have Let ; then Hence, ; that is, .

This completes the proof of Appendix C.

D. Proof of

From (26) and (24), we have For and , then Let ; then it is easy to obtain for , so ; hence .

This completes the proof of Appendix D.

E. Proof of Theorems 4 and 5

Before the proof of Theorems 4 and 5, according to the anterior analysis in Section 4, we can get the following results.(a)If , is increasing on (). Otherwise, is decreasing on ().(b)If , is increasing on (). Further, if and , that is, , is also increasing on (). As a result, if then is always increasing on (), and in this case if and there must exist a unique root to the equation on (), and we denote this root by to differentiate with and . But if , that is, , then for and for . However, when , that is, , we have on (), and in this case if and there must exist a unique root (say ) to the equation on ().(c)If , is increasing on (). Otherwise, is decreasing on ().

Therefore, to solve the optimal order cycle time over (), the possible interval which belongs to needs be considered; hence we should compare the size relations among values of , , and with . Since and , one will derive ≤ and ≤ ≤ . So, there are three cases to occur: (i) ≤ , (ii) < ≤ , and (iii) > . Notice that once , the subfunction is always increasing over (), so the profit function has no optimal solution over (). Hence, we need only to study the two cases as below: (i) ≤ and (ii) > . According to these two cases and from some analysis, the proof of Theorems 4 and 5 would be obtained.

Proof of Theorem 4. If and , then we could obtain that ≤ ≤ ≤ . Hence, we have the following.(1)When , since and is increasing on () there are the following five possible cases to occur.(A)If , , and , then and . These yield that (i) is increasing on (); (ii) is increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we conclude that ; hence is .(B)If , , and , then , , and . These yield that (i) is increasing on (); (ii) is increasing on () and decreasing on (); (iii) is decreasing on (). Combining (i)–(iii) and , we conclude that ; hence is .(C)If , , and , then , , and . These yield that (i) is increasing on () and decreasing (); (ii) is increasing on () and decreasing on (); (iii) is decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is or associated with maximum profit.(D)If , , and , then and . These yield that (i) is increasing on () and decreasing (); (ii) is increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we could conclude that . Hence is or associated with maximum profit.(E)If , , and , then and . These yield that (i) is increasing on () and decreasing on (); (ii) is decreasing on (); (iii) is decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is or associated with maximum profit.(2)When , then for and for .(A)While , we have the following.(a)If , , and , then , , , and . These yield that (i) is increasing on (); (ii) is increasing on (), decreasing on (), and increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is or associated with maximum profit.(b)If , , and , then , , and . These yield that (i) is increasing on (); (ii) is increasing on () and decreasing on (); (iii) is decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is .(c)If , , and , then , , , and . These yield that (i) is increasing on () and decreasing (); (ii) is increasing on (), decreasing on (), and increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is , , or associated with maximum profit.(d)If , , and , then , , and . These yield that (i) is increasing on () and decreasing (); (ii) is increasing on () and decreasing on (); (iii) is decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is or associated with maximum profit.(e)If , , and , then , , and . These yield that (i) is increasing on () and decreasing (); (ii) is decreasing on () and increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is , , or associated with maximum profit.(f)If , , and , then , and . These yield that (i) is increasing on () and decreasing (); (ii) is decreasing on (); (iii) is decreasing on . Combining (i)–(iii) and , we conclude that . Hence is or associated with maximum profit.(B)While , then and .(a)If , , and , then and . These yield that (i) is increasing on (); (ii) is increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is .(b)If , , and , then and . These yield that (i) is increasing on () and decreasing on (); (ii) is increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is or associated with maximum profit.(3)When , since and is decreasing on () there are the following four possible cases to occur.(A)If , , and , then and . These yield that (i) is increasing on (); (ii) is increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is .(B)If , , and , then and . These yield that (i) is increasing on ) and decreasing on (); (ii) is increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is or associated with maximum profit.(C)If , , and , then , and . These yield that (i) is increasing on () and decreasing on (); (ii) is decreasing on () and increasing on (); (iii) is increasing on () and decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is , , or associated with maximum profit.(D)If , , and , then and . These yield that (i) is increasing on () and decreasing on (); (ii) is decreasing on (); (iii) is decreasing on (). Combining (i)–(iii) and , we conclude that . Hence is or associated with maximum profit.(4)When , because on the interval of () the average profit function is always increasing, the optimal solution is evidently.
This completes the proof of Theorem 4.

Proof of Theorem 5. If > , then one has ≤ < ≤ or ≤ ≤ . Since has no optimal solution when , so there are three possible situations to occur as below.(1)When , it is obvious that the results are the same as (1) in Theorem 4.(2)When , the results are the same as (2) in Theorem 4.(3)When , because on the interval of () the average profit function is always increasing, the optimal solution is evidently.
This completes the proof of Theorem 5.

F. Proof of Theorem 6

According to the anterior analysis in Section 4, we could get the below results.(a)If , is increasing on (). Otherwise, is decreasing on ().(b)If , then is increasing on (). Otherwise, is strictly decreasing on ().

Similarly, in order to find the system’s optimal cycle time , the possible interval which belongs to needs be considered if ; hence we should compare the size relations between and . If , then one has ≤ .(1)When , we have the following.(A)If and , then and . These yield that (i) is increasing on (); (ii) is increasing on () and decreasing on (). Combining (i)-(ii) and , we conclude that . Hence is .(B)If and , then and . These yield that (i) is increasing on () and decreasing on (); (ii) (T) is decreasing on (). Combining (i)-(ii) and , we conclude that . Hence is or associated with the maximum profit.(C)If and , then and . These yield that (i) is increasing on and decreasing on (); (ii) is increasing on () and decreasing on . Combining (i)-(ii) and , we can conclude that . Hence is or associated with the maximum profit.(2)When , because on the interval of the average profit function is always increasing, the optimal solution is evidently.

This completes the proof of Theorem 6.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China Project under project Grant nos. 71101002, 71072165, 71201044, 71131003, and 71231004. The authors have benefited substantially from the insightful comments of the editor and anonymous referees, which were instrumental for elevating the quality of this work.

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Copyright © 2014 Jie Min et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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