Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 809149, 16 pages

http://dx.doi.org/10.1155/2014/809149

Research Article

Three-Echelon Inventory Model with Permissible Delay in Payments under Controllable Lead Time and Backorder Consideration

M. F. Yang and Wei-Chung Tseng

Department of Transportation Science, National Taiwan Ocean University, Keelung City 202, Taiwan

Received 25 January 2014; Accepted 19 March 2014; Published 30 April 2014

Academic Editor: Sri Sridharan

Copyright © 2014 M. F. Yang and Wei-Chung Tseng. 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.

Abstract.

This paper proposes a three-echelon inventory model with permissible delay in payments under controllable lead time and backorder consideration to find out the suitable inventory policy to enhance profit of the supply chain. In today’s highly competitive market, the supply chain management has become a critical issue in both practice and academic and supply chain members have to cooperate with each other to bring more benefits. In addition, the inventory policy is a key factor to influence the performance of the supply chain. Therefore, in this paper, we develop a three-echelon inventory model with permissible delay in payments under controllable lead time and backorder consideration. Furthermore, the purpose of this paper is to maximize the joint expect total profit on inventory model and attempt to discuss the inventory policy under different conditions. Finally, with a numerical example provided here to illustrate the solution procedure, we may discover that decision-makers can control lead time and payment time to enhance the performance of the supply chain.

1. Introduction

Inventory policy and supply chain are closely linked to each other; inventory policy can easily influence the performance of supply chain [1]. Jammernegg and Reiner [2] pointed out that the efficient inventory management can enhance performance of supply chain. To enhance the performance of the supply chain, firms play the different role in the supply chain, and they should have a good partnership and share the information with each other. Cachon and Fisher [3] established a platform for firms in the supply chain; it could share information, built the forecast model, and reduced the risk from inventory model. Olson and Xie [4] proposed that vendor and buyer should use the same inventory system to cooperate with each other, and inappropriate inventory policy might cause firms to bear great loss. With the above discussion, the proper inventory policy is a critical part in the supply chain. We would like to develop an integrated inventory model to obtain the optimal solutions and attempt to discuss the inventory policy, and next we have to review some previous literatures to define the research gaps.

Stock is a major part in inventory policy. Excess stocks cause higher cost, and shortage cannot satisfy customer immediately. Frankel [5] believed that how to control stocks was a vital practice in supply chain. Das Roy et al. [6] extended Maddah and Jaber’s [7] model to discuss an optimal shipment strategy for defective items in shortage situation. Sana [8] researched two-warehouse systems in EOQ model with limited capacity of own warehouse. Due to the stochastic demand, they regarded lost sale as penalty in shortage conditions. Nasiri et al. [9] emphasized that firms should keep safety stocks to satisfy customer under instable demands; hence, how to keep adequate stocks is an extremely critical issue in inventory policy. Shortage often causes a great sales loss, normally; firms adopt backorder method to lower sales loss; backorder means the buyer does not receive cargos in time and allows cargos delivery late. Lo et al. [10] indicated that backorder was an important factor in their inventory model; Pan et al. [11] added backorder into their inventory model that considered lead time and safety factor. When shortage occurs, in addition to using backorder method to handle unsatisfied customers, firms can also offer few discounts to save customers. Ouyang et al. [12] discussed the impact of backorder discount in the inventory model under backorder consideration. Pan and Hsiao [13] established an integrated inventory model with controllable lead time and backorder discount considerations. Therefore, backorder consideration should be mentioned in inventory model.

In addition to backorder method, compressed lead time can also alleviate shortage condition. Lead time includes preparative time for send orders, set-up time of facilities, productive time, transportation time, and so on. When the firms face the uncertain demands, they usually compress the lead time. Chandra and Grabis [14] indicated that short lead time could enhance the service level and lower inventory level effectively, but short lead time also caused highly order cost. The previous researches have regarded lead time as a constant, but more and more researches have regarded lead time as a variable or controllable parameter in recent years; Ben-Daya and Raouf [15] added lead time as a decision variable in the inventory model. Xia and Huang [16] established an inventory model with quantity discount and controllable lead time when demand distribution was unknown. And Song et al. [17] studied the effects of controllable lead time in the inventory model; they indicated that shorter lead time can not only satisfy customer’s demand, but also lower inventory level effectively. In 2008, Lo and Yang [18] proposed a notable idea. They showed the relationship between lead time and delay payment time, and their cost functions in the model would be divided into two cases by different payment times. But they considered lead time as a constant. If the lead time is regarded as a variable, it is interesting to know whether we can find out the better inventory policy by controlling lead time and payment time.

Permissible delay in payments is a common commercial strategy in real life; the purpose of this strategy is to enhance elasticity of capital. In other words, firms can obtain additional interest income from unpaid payments, but equally firms may pay the additional opportunity cost in some conditions. Lo and Yang [18] indicated that firms should let payment time be longer than lead time to obtain additional interest income. Chen and Kang [19] established an integrated inventory model with permissible delay in payments. Zhu et al. [20] considered permissible delay in payments in their inventory model with limited storage capacity and shortage. Huang et al. [21] proposed an inventory policy between buyer and vendor and studied the relationship between permissible delay in payments and order-processing time reduction. Sana and Chaudhuri [22] proposed a deterministic EOQ model with delays in payments and price-discount offers; they considered various types of deterministic demands for retailer to operate their model. They all have added delay payment method in their inventory models. To fit the real life, inventory model should take delay payment into account.

There are many issues that can be discussed in inventory model, and we reviewed the literatures about backorder, controllable lead time, and permissible delay in payment. Shortage is an extremely common condition in real life and always accompanies sales loss. Hence, in the second paragraph, we simply defined backorder. To lower the sales loss, the backorder method is widely used in years. After reviewing Ouyang et al. [12] and Pan and Hsiao’s [13] research, we would like to regard backorder as a major condition in our inventory model.

In the third paragraph, we knew that compressed lead time was another method to alleviate shortage condition. Recently researches have indicated that the controllable lead time is a popular issue in inventory model. Lead time consists of many components, and these components such as productive process and transportation process can be compressed by renewing facilities or using fast vehicles. Short lead time can respond to demand quickly and enhance the service level. With high service level, firms can gain more revenue from their customers but compressing lead time will cause additional cost to firms. With inspiration from Lo and Yang’s [18] research, it is interesting to know whether we can find out better inventory policy by controlling lead time and payment time.

In the fourth paragraph, we discussed the permissible delay in payment, a common commercial strategy in business; the strategy aims to enhance the elasticity of capital. Vendor always benefit from interest income by delaying payment time, but buyer would pay an additional opportunity cost. Generally, supply chain can benefit from delay strategy; hence, how to maximize the total profit by adjusting payment time is a potential issue to discuss.

To simulate supply chain completely, we would like to discuss the above-mentioned ideas in three-echelon inventory model. There are several researches that have studied inventory problems in three-echelon models. Roy et al. [23] built a three-echelon supply chain model involving newsboy problem to deal with the inventory control problems. They also claimed that uncertain demand may cause shortage condition in their integrated inventory model. Sana [24] investigated an EOQ/EPQ model in three-echelon supply chain management; he focused on the volume flexibility and replenishment lot size problem in a collaborating system. Pal et al. [25] established a three-echelon inventory model; they considered multi-items and multisuppliers to make model more realistic. Sana et al. [26] proposed a three-echelon supply chain model with multiple suppliers, manufacturers, and retailers for multiple items. They aimed to compare the expected profit between the collaborating system and Stakelberg game structure. Three-echelon inventory model has clearly shown how supply chain works in the above researches.

With the above discussion, we would like to combine the concepts of Pan and Hsiao [13] and Lo and Yang’s [18] researches to develop a three-echelon integrated inventory model with delay payment, controllable lead time, and backorder consideration. In this paper, we can determine our proposed inventory model’s optimal solutions and sensitivity analysis of decision variable. Finally, with appropriate backorder condition, we can attempt to determine better inventory policy by controlling lead time and payment time.

2. Notations and Assumptions

Tersine [27] believed that lead time was the time between the placement of an order and delivery of cargos. And lead time can be divided into preparative time for send orders, set-up time of facilities, productive time of products, transportation time, and so on. Lead time always includes transportation and productive process, but supplier just supplies the raw materials to the manufacturer. Hence we may suppose that manufacturer’s lead time only includes transportation process, and retailer’s lead time includes both. Lead time can be reduced by paying additional cost, and this additional cost was called “crashing cost” [12, 20, 21]. According to Pan and Hsiao’s [13] research, retailer’s lead time consisted of mutually independent components; every component has a normal duration and a minimum duration , , given for to compute the crashing cost. To establish the inventory model, the following notations and assumptions are used.

2.1. Notations

 : the length of lead time of retailer, a decision variable : economic order quantity of the ith model, a decision variable () : backorder discount per unit during the short-age period, a decision variable : average demand per year : production rate of manufacturer : demand rate of retailer : manufacturer’s permissible delay period : retailer’s permissible delay period : the safety factor of the inventory : the probability of shortage during an order period : the proportion of permissible backorder during the shortage period : the upper bound of the backorder ratio : marginal profit per unit : the random demand during the lead time : the average of the daily demand during the lead time : the standard deviation of the daily demand during the lead time : supplier’s purchasing cost per unit : supplier’s ordering cost per order : supplier’s inspecting cost per unit : supplier’s repair cost per unit : the probability of imperfect products from supplier : supplier’s holding cost per unit per production run : supplier’s capital opportunity cost per dollar per year : supplier’s interest earned per dollar per year : manufacturer’s purchasing cost per unit : manufacturer’s ordering cost per order : manufacturer’s inspecting cost per unit : manufacturer’s repair cost per unit : the probability of imperfect products from manufacturer : manufacturer’s transportation cost per order : manufacturer’s holding cost per unit per production run : the length of lead time of manufacturer : manufacturer’s capital opportunity cost per dollar per year : manufacturer’s interest earned per dollar per year : the number of shipments per production run from manufacturer to retailer : retailer’s purchasing cost per unit : retailer’s selling price per unit : retailer’s ordering cost per order : retailer’s transportation cost per order : retailer’s holding cost per unit per production run : retailer’s capital opportunity cost per dollar per year : retailer’s interest earned per dollar per year : the number of shipments per production run from retailer to customer : the crashing cost at the jth component in ( ) : the jth component in normal duration () : the jth component in minimum duration  : probability density function : probability distribution function : the total crashing cost per production run T: supplier’s total annual profit T: manufacturer’s total annual profit T: retailer’s total annual profit JTP: the joint total annual profit EJTP: the expected joint total annual profit.

2.2. Assumptions

(1)This is a single supplier, a single manufacturer, and a single retailer for a single product.(2)Shortages are not allowed at supplier and manufacturer.(3)Lead time of manufacturer includes transportation process, and lead time of retailer includes transportation and productive process.(4)The lead time of retailer consists of n mutually independent components; the jth components have a normal duration and a minimum duration , .(5)Let be the length of lead time with components crashed to their minimum values, and can be expressed as , for ; hence, with a given lead time , the crashing cost per production run is given by for .(6)The relationship between the supplier’s purchasing cost , manufacturer’s purchasing cost , retailer’s purchasing cost , and retailer’s selling price is .(7)We regard imperfect production as basic condition in our proposed model, and inspecting costs are equal to and and repair costs are equal to and [18].(8)The imperfect products will completely and immediately fix [18].(9)The supplier offers the manufacturer a permissible delay period . During this permissible delay period, the manufacturer sells the goods and uses the sale revenue to earn interest at a rate of . At the end of this time period, the manufacturer pays the purchasing cost to the supplier and the rest of the goods in the warehouse bring a capital opportunity cost at a rate of .(10)The time horizon is infinite.(11)Lead time is deterministic and lead time demand has finite mean and variance [13].(12)The reorder point , where is a safety point [13].(13)Inventory is continuously reviewed and replenishments are made whenever the inventory level falls to the reorder point .

3. Building Model

In this section, we discuss and develop the supplier’s model, manufacture’s model, and retailer’s model in following section and combine the above models to an integrated inventory model. To build the proposed model, we extended Lo and Yang’s [18] research to compute relevant opportunity cost and interest income and referred to Pan and Hsiao’s [13] research to compute relevant shortage and crashing cost. As for the holding cost, it could be computed by using integration in Appendix A’s Figures 1, 2, and 3 [28].

Figure 1: Supplier and manufacture’s inventory level.

Figure 2: .

Figure 3: .

3.1. The Supplier’s Total Annual Profit

In each production run, the supplier’s costs include ordering cost, inspecting cost, repair cost, holding cost, and opportunity cost. And the supplier’s revenues include sales revenue and interest income. With permissible delay in payments, we knew that, if payment time was longer than lead time, then it would bring additional interest income to the manufacturer; in addition, the supplier must pay equally opportunity cost. If payment time was shorter than lead time, then it would bring additional opportunity cost and fewer interest income to manufacturer; also, the supplier could earn interest income and pay fewer opportunity cost. Owing to the fact that the supplier’s profit function has that two following cases, based on length of lead time and payment time , the different parts between two possible cases are as follows.

Case 1. If , then

(i)transfer opportunity cost =

Case 2. If , then

(i)transfer opportunity cost = ;(ii)transfer interest income = .

Thus, and are given by

3.2. The Manufacturer’s Total Annual Profit

In each production run, the manufacturer’s costs include ordering cost, inspecting cost, repair cost, holding cost, transportation cost, and opportunity cost. And the manufacturer’s revenues include sales revenue and interest income. We have already known the relationship between length of lead time and payment time from Section 3.1. Owing to the fact that the manufacturer’s profit function has the two following cases, based on length of lead time and payment time , the different parts between two possible cases are as follows.

Case 1. If , then

(i)interest income = ;(ii)transfer opportunity cost = .

Case 2. If , then

(i)transfer opportunity cost = .(ii)opportunity cost = .(iii)transfer interest income = .(iv)interest income = .

Thus, and are given by

3.3. The Retailer’s Total Annual Profit

In each production run, the retailer’s costs include ordering cost, shortage cost, holding cost, transportation cost, crashing cost, and opportunity cost. And the retailer’s revenues include sales revenue and interest income. We have already known the relationship between length of lead time and payment time from Section 3.1. Owing to the fact that the retailer’s profit function has the two following cases, based on length of lead time and payment time , the different parts between two possible cases are as follows.

Case 1. If , then

(i)interest income = .

Case 2. If , then

(i)opportunity cost = ;(ii)interest income = .

Thus, and are given by

3.4. The Expected Joint Total Annual Profit

Hence, the expected joint total annual profit function, EJTP, can be expressed as

where

4. Solution Procedure

4.1. Determination of the Optimal Backorder Discount for Any Given and

We would like to find the minimum value of the joint expected total profit . For any given and, to take first and second partial derivation of , with respect to , we have

Because (7) < 0, therefore, for fixed and, is a concave function in . So as there exists a unique value of which maximize, can be obtained by solving the equation/ = 0 in (6) and is given by

4.2. Determination of the Optimal Order Quantity for Any Given and

For any given and, to take first and second partial derivation of , with respect to , we have

Because (10) , therefore, for fixed and, is a concave function in . So as there exists a unique value of which maximize, can be obtained by solving the equation/ = 0 in (9) and are given by

Algorithm. Summarizing the above arguments, we establish the following algorithm to obtain optimal values of ().

Step 1. Arrange the value of such that .

Step 2. Determine by solving (6) = 0.

Step 3. With the different duration, substitute and into (11) to find .

Step 4. Substitute into (5) to find .

Step 5. For , repeat Steps 1 to 4 and compute ().

Step 6. For and , repeat Steps 1 to 5 and obtain ().

Step 7. Observe () by varying and .

Step 8. Determine optimal ().

5. Numerical Example

To illustrate the preceding model, we consider an inventory system with the following data and lead time data in Table 1: (average demand per year) = 1000 unit/year, (production rate of manufacturer) = 10, (demand rate of retailer) = 8, (probability of shortage) = 0.3, (marginal profit per unit) = $6 per unit, (standard deviation) = 7 unit, (supplier’s purchasing cost per unit) = $10 per unit, (supplier’s ordering cost per order) = $100 per order, (supplier’s inspecting cost per unit) = $0.6 per unit, (supplier’s repair cost per unit) = $0.2 per unit, (supplier’s holding cost per unit per production run) = $2 per unit, (supplier’s capital opportunity cost per dollar per year) = $0.04 per year, (supplier’s interest earned per dollar per year) = $0.03 per year, (probability of imperfect items from supplier) = 0.1, (manufacturer’s purchasing cost per unit) = $25 per unit, (manufacturer’s ordering cost per order) = $120 per order, (manufacturer’s inspecting cost per unit) = $0.6 per unit, (manufacturer’s repair cost per unit) = $0.25 per unit, (probability of imperfect items from manufacturer) = 0.1, (manufacturer’s transportation cost per unit ) = $40 per order, (manufacturer’s holding cost per unit per production run) = $2.5 per unit, (lead time of manufacturer) = 7 days, (manufacturer’s capital opportunity cost per dollar per year) = $0.04 per year, (manufacturer’s interest earned per dollar per year) = $0.03 per year, (retailer’s purchasing cost per unit) = $30 per unit, (retailer’s selling price per unit) = $50 per unit, (retailer’s ordering cost per order) = $150 per order, (retailer’s transportation cost per unit) = $50 per order, (retailer’s holding cost per unit per production run) = $3 per unit, (retailer’s capital opportunity cost per dollar per year) = $0.04 per year, and (retailer’s interest earned per dollar per year) = $0.03 per year.

Table 1: Lead time data of numerical example.

Lead time component,
Normal duration, (days)
Minimum duration, (days)
Crashing cost, ($/day)

1
20
14
1.2

2
16
10
5

5.1. The Sensitivity Analysis of Decision Variable

First, we need to know how and influence the () and individual profit, and, according to function (4), we developed four different models, and we could reach the optimal () by applying the algorithm; in addition, we take the () as an example here; the results are shown in Table 2.

Table 2: The profit of and individual.

(, , )

(, , )

*1K
15
55.4568
*16366.8507
1K
1
75.5802
14194.8814

1K
20
60.0695
16068.2148
1K
2
75.4669
14278.6926

1K
25
64.8780
15662.1435
1K
3
75.4287
*14306.6807

1K
30
69.7979
15139.5899
1K
4
75.4666
14278.6926

1K
35
74.7770
14507.1236
1K
5
75.5802
14194.8814

Supplier

Supplier

1K
15~35
10.0861
3446.6683
1K
1~5
10.0861
3446.6683

Manufacturer

Manufacturer

1K
15~35
6.3162
−13561.9435
1K
1~5
6.3162
−13561.9435

Retailer

Retailer

1K
15
15.0492
16870.0034
1K
1
104.3418
20571.7856

1K
20
48.3484
19208.4841
1K
2
104.2893
20572.1597

1K
25
68.7905
19972.7063
1K
3
104.2134
*20572.2606

1K
30
85.9062
20359.0393
1K
4
104.1140
20572.0941

1K
35
101.2845
*20550.3897
1K
5
103.9911
20571.6672

Optimal solution of .

*1K = 1000.

The results are shown in Table 2; we could observe how lead time of retailer and backorder discount influenced the () and individual profit. The total profit of () decreased with longer lead time of retailer , and the profit of retailer increased with longer lead time . We might indicate that performance of () could be enhanced by compressing the lead time but it also caused lower profit for the retailer. The profit of () and retailer could reach the optimal results when ; it is suggested that backorder discount should be offered appropriately to the buyer. Owing to the fact that the expected joint total annual profit was greater than individual’s profit (supplier + manufacturer + retailer) and shorter lead time was more benefit for (). It seems that compressing the lead time was an available policy in our proposed inventory models. And we continue discussing the more detailed inventory policy in the following sections.

5.2. The Permissible Delay Period and EJTP

Let us observe the value of profit with by varying the permissible delay period , and, with applying the proposed procedure, the solution results are contained in Figure 4 (see Appendix B).

Figure 4: The value of profit by change in .

To compare which condition is more beneficial to our proposed inventory model, we have more detailed results in Table 3.

Table 3: The value of profit in different condition by varying .

75.43~85.19
69.34~78.18
64.2~72.26

8~107
8~107
8~107

(, , )
14306.68~15847.64
14837.16~16500.41
14666.7~16527.05

104.71~124.4
83.92~94.93
71.26~78.76

8~107
8~107
8~107

(, , )
18067.25~19600.13
17393.2~18804.44
16317.84~17750.81

76.1~74.8
69.88~68.7
63.62~64.69

7~0
7~0
7~0

(, , )
14144.93~14200.08
14691.76~14711.56
14512.15~14553.64

87.56~92.06
75.83~78.7
69.91~72.07

7~0
7~0
7~0

(, , )
*20313.71~20678.1
18369.36~18776.31
16028.2~16488.16

Optimal solution of .

According to Section 3.1, we knew that, if payment time was longer than lead time, then it would bring additional interest income to the buyer. On the other hand, if payment time was shorter than lead time, it would cause buyer to bear additional opportunity cost. From the above discussion, we had four possible cases in , for , and considered the different duration of , for . Hence, we had twelve possible values of profit. According to the data from Table 3, we could indicate that the optimal profit occurred in with normal duration condition, and the worst profit occurred in with normal duration condition.

The Hessian matrix H at the optimal solution , , and is which is negative definite because all the eigenvalues are negative. Hence, the required optimal solution is , , and and .

5.3. The Permissible Delay Period and EJTP

In this section, we observe the value of profit by varying the permissible delay period . The detailed results have been shown in Table 4.

Table 4: The value of profit in different condition by varying .

75.43~85.19
69.34~78.18
64.21~72.26

37~136
31~130
25~124

(, , )
14306.68~15847.64
14837.2~16500.4
14666.7~16527.05

72.43~104.71
67.02~83.92
62.45~71.26

36~0
30~0
24~0

(, , )
13749.61~18067.25
14326.88~17393.2
14198.61~16317.84

76.1 ~86.16
69.87~78.94
64.69~72.95

37~136
30~129
24~123

(, , )
14144.93~15664.35
14691.76~16336.02
14553.64~16399.41

59.79~87.55
59.89~75.83
60.69~69.91

36~0
30~0
24~0

(, , )
*16859.1~20313.7
15392.02~18369.36
13701.0~16028.2

Optimal solution of .

This time, we varied the permissible delay period . We noticed that the optimal profit still occurred in with normal duration condition. But the worst profit was different from previous result; it occurred in with minimum duration condition.

The Hessian matrix H at the optimal solution , , and is which is negative definite because all the eigenvalues are negative. Hence, the required optimal solution is , , and and .

No matter which payment time we varied, the optimal profit still occurred in the same condition. Hence, through controllable lead time and payment time, we could find the optimal policy in our proposed integrated inventory model.

6. Conclusions and Future Works

In this paper, we discussed the issue of a three-echelon inventory model with permissible delay in payments under controllable lead time and backorder consideration to find out the suitable inventory policy to enhance profit of the supply chain. Owing to information revolution, the community can get anything they want easily. It causes that any goods in the market always have a stable price due to the inspection by community. Hence, the enterprises are hardly raising the price on their product. If they want to enhance the profit, to control the relevant cost and reduce it efficiently is a good approach to enhance profit for them.

In the manufacture part, the firms could delay the payment time to obtain additional interest income, and the proper payment time could be a great benefit for the firms. In the customer part, length of lead time usually influenced the price of products. Fast lead time could respond to customer’s demand quickly, and firms could request customers for high price. Inevitably, firms must pay the additional cost for reducing lead time. In the stock part, firms usually adopted backorder approach to minimize sales loss and offered some proper discount to save customers.

From the above discussion, we proposed our inventory model and analyzed its profits, in a three-echelon inventory model case. First, we observed that integrated profit is greater than individual’s profit. Also we concluded that the compressed lead time and appropriate backorder discount are most benefit factors for entire supply chain in Section 5.1. Second, we fixed the backorder ratio and observed the optimal profit by varying payment time in Sections 5.2 and 5.3. With the different duration and four cases in , we had twelve possible values of profit, and the optimal profit occurred in with normal duration. Finally, we discover that the profit of supply chain will be different by compressing lead time and varying payment time.

In the supply chain, with the real data as input parameters, our proposed model can help decision-makers enhance the performance of the supply chain by controlling lead time and payment time. As far as the authors’ knowledge goes, no such type of three-echelon inventory model involving controllable lead time has yet been discussed in relevant literatures. These are the new major contributions of the present paper.

Finally, it is our hope that this work will encourage future work in this area and related areas. And we will improve our further research in more real-world complexities, such as the concept of present value, multiple vendor and buyer, and apply the fuzzy approach into the proposed models, and we will cooperate real case to get the actual data that can illustrate real numerical examples to our future research.

Appendices

A. Inventory Level

See Figures 1, 2, and 3.

B. Profit Variation

See Figure 4.

C. Proof

The following test can be applied at a nondegenerate critical point . If the Hessian is positive definite at , then attains a local minimum at . If the Hessian is negative definite at , then attains a local maximum at . If the Hessian has both positive and negative eigenvalues, then is a saddle point for (this is true even if is degenerate). Otherwise the test is inconclusive. If all the principal minor of H, the Hessian matrix H is positive definite at ; it also means there will be a local minimum at ; if all the principal minor of H, the Hessian matrix H is negative definite at ; it also means there will be a local maximum at .

The Hessian matrix H of can be shown as

where

Next, we can evaluate the principal minor of H at point ; the first principal minor of H is

The second principal minor of H is

And the third principal minor of H is

where

With , we have (C.6) > 0 and, substituting it into (C.5), we can obtain . Therefore, from (C.3) to (C.5), it follows that the Hessian matrix H is negative definite at .

Conflict of Interests

None of the authors has any conflict of interests regarding this study.

Acknowledgments

The authors would like to thank the editors and referees for their helpful comments and suggestions.

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