Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 1490712, 10 pages
http://dx.doi.org/10.1155/2016/1490712

Research Article

Optimal Replenishment Policy for Weibull-Distributed Deteriorating Items with Trapezoidal Demand Rate and Partial Backlogging

Lianxia Zhao

School of Management, Shanghai University, Shanghai 200444, China

Received 2 December 2015; Accepted 27 March 2016

Academic Editor: Alireza Amirteimoori

Copyright © 2016 Lianxia Zhao. 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.

An inventory model for Weibull-distributed deteriorating items is considered so as to minimize the total cost per unit time in this paper. The model starts with shortage, allowed partial backlogging, and trapezoidal demand rate. By analyzing the model, an efficient solution procedure is proposed to determine the optimal replenishment and the optimal order quantity and the average total costs are also obtained. Finally, numerical examples are provided to illustrate the theoretical results and a sensitivity analysis of the major parameters with respect to the stability of optimal solution is also carried out.

1. Introduction

Deteriorating is a general phenomenon for many products, in that fruits or vegetables are spoiled directly while alcohol undergoes physical depletion over time, and electronic products deteriorate through a gradual loss of potential utility as time passed. Based on the above situation, the effect of deteriorating for items can not be disregarded in many inventory systems. An inventory model with a constant decaying rate for items is proposed by Ghare and Schrader in 1963 [1], while Covert and Philip [2] extended Ghare and Schrader’s model to an EOQ model with a two-parameter Weibull-distributed deteriorating rate. Since then, there are many researchers who focused on such topic; Wee et al. [3] considered an inventory model for deteriorating items with quantity discount, pricing, and partial backlogging, and more related articles can be referred to like Yang et al. [4], Shah et al. [5], Yang et al. [6], and so forth.

However, it is impossible for deteriorating items that the demand rate increases continuously during their full life cycle. Based on such reasons, Hill [7] proposed an inventory model with ramp-type demand rate, which extended to allow shortage proposed by Mandal and Pal [8]. Further, Wu [9] extended Mandal and Pal’s model to have Weibull-distributed deterioration and time-dependent backlogging. Skouri et al. [10] considered a model with general ramp-type demand rate, partial backlogging, and Weibull deterioration rate. Tan and Weng [11] considered a discrete inventory model for deteriorating items with partial backlogging. Pal et al. [12] considered a production inventory model for deteriorating items with ramp-type demand rate under the effect of inflation and shortages under fuzziness. There are many other related literatures about such inventory model, such as Ahmed et al. [13], Agrawal et al. [14], Singh and Pattnayak [15], Wu et al. [16], and Rabbani et al. [17].

While, for some short life cycle products, the demand rate may increase up to a certain level in the early stage of marketing, then reach a stabilized period, and finally decrease to zero and reach the end of their life cycle, Cheng and Wang [18] proposed an inventory model for deteriorating items with trapezoidal type demand rate. Further, Cheng et al. [19] extended the model to be of time-dependent deteriorating items with trapezoidal type demand rate and partial backlogging. Uthayakumar and Rameswari [20] studied a model for defective items with trapezoidal type demand rate to determine the optimal product reliability. Lin [21] and Lin et al. [22] considered the inventory model which was proposed by Cheng and Wang [18]. Zhao [23] studied an inventory model for Weibull-distributed deterioration items with trapezoidal type demand rate and partial backlogging in which the model begins with no-shortage. While, in real situation, customers will respond differently in case of shortages, either to wait until replenishment or to look for alternative products, in this paper, we study an inventory model beginning with shortage, Weibull-distributed deterioration items, trapezoidal type demand rate, and time-dependent partial backlogging. By analyzing the studied model, we propose an optimal replenishment strategy for it and also carry out a sensitivity analysis of the main parameters.

The remainder of the paper is organized as follows. Notations and assumptions are described in Section 2. Section 3 constructs and analyzes the inventory model starting with shortage. Some numerical examples to illustrate the solution procedure are provided; then sensitivity analysis of the major parameters is also carried out in Section 4. Finally, a conclusion is presented.

2. Notations and Assumptions

The fundamental notations and assumptions used in this paper are given as follows.

(i) is the inventory level at time , , where is the length of inventory cycle.(ii) denotes the replenishment time, and denotes the optimal .(iii) is the maximum inventory level for each ordering cycle.(iv) denotes the optimal ordering quantity.(v) denotes the fixed cost per order.(vi) is the cost of each deteriorated item, is the unit inventory holding cost per unit time, is the unit shortage cost per unit time, and is the unit lost sales cost.(vii), , denotes the average total cost per unit time under different conditions, respectively.(viii) denotes the average total cost per unit time.(ix)The demand rate, , which is positive and consecutive, is assumed to be a trapezoidal type function of time; that is,

 where is time point changing from the increasing demand function to the constant demand , and is time point changing from the constant demand to the decreasing demand function .(x)The replenishment rate is infinite; that is, replenishment is instantaneous.(xi)The time to deterioration of the item is distributed as Weibull ; that is, the deterioration rate is , where is the scale parameter, , and  is the shape parameter, (xii)Shortage is allowed and let be fraction of shortages backlogged, where is the waiting time up to the next replenishment and is a tiny positive constant; that is, .

3. Inventory Model Starting with Shortage

The system of model begins with shortage and until the shortage achieves its maximum; then replenishment occurs at time . All of the shortage demand during is partial backlogged (Figure 1). At , the total number of backlogged items will be satisfied by the replenishment and the inventory level up to , while the deteriorating occurs during the time of . According to the notations and assumptions mentioned above, the model is described as shown in Figure 1:

with boundary condition ,  ,  .

Figure 1: Graphical representation of inventory level over the cycle starting with shortage.

Based on the values of , , and , three possible cases are presented as follows.

Case 1 (). If the replenishment time , from (2), we have

Solving the differential equations (3) with , we have

The total replenishment quantity can be computed as

The total shortage quantity during the interval is

The total number of perished items during the interval , say , is

The total number of inventories carried during the interval , say , is

The total of lost sales during the interval , say , is

Therefore, the average total cost per unit time under the condition can be given by

Taking the first and the second order derivative of with respect to , respectively, we have

Then, we have

and let

Therefore, (i) if and , there exists a unique solution satisfying , and obtains its minimum at ; (ii) if and , it means that is a nonincreasing function and obtains its minimum value at point ; (iii) if the sign of is indefinite, then the curve of is -shape, and may obtain its minimum at more than one point; without loss of generality, set , where is one of the solutions of .
Thus, the optimal value of the order level, , is

and the optimal order quantity is

Case 2 (). If the replenishment time , then from (2), we have

Solving the differential equation (17) with , we have

From (18), the total replenishment quantity can be computed:

The total shortage quantity during the interval is

The total number of perished items in the interval is

The total number of inventories carried during the interval is

The total of lost sales during the interval is

Therefore, the average total cost per unit time under the condition can be given by

Taking the first and the second order derivative of with respect to , respectively, we have

Let

Therefore, (i) if , , and , there exists a unique solution satisfying , and obtains its minimum at ; (ii) if ,  , and , it means that is a nonincreasing function and obtains its minimum at point ; (iii) if , , and , it means that is a nondecreasing function and obtains its minimum at point ; (iv) if the sign of is indefinite, then the curve of is -shape, and may obtain its minimum value at more than one point; without loss of generality, set , where is one of the solutions of .
Thus, the optimal value of the order level, , is

and the optimal order quantity is

Case 3 (). If the replenishment time , from (2), we have

Solving (30) with , we have

From the last equation of (31), the total replenishment quantity can be computed:

The total shortage quantity during the interval is

The total number of perished items in the interval is

The total number of inventories carried during the interval is

The total of lost sales during the interval is

Therefore, the average total cost per unit time under the condition can be given by

Taking the first and the second order derivative of with respect to , respectively, we have

We know that

Therefore, (i) if and , there exists a unique solution satisfying , and obtains its minimum at ; (ii) if and , it means that is a nondecreasing function and obtains its minimum value at point ; (iii) if the sign of is indefinite, then the curve of is -shape, and may obtain its minimum value at more than one point; without loss of generality, set , where is one of the solutions of .
Therefore, the optimal value of the order level, , is

and the optimal order quantity is

From the above analysis, we obtain that the total average cost of the system over the interval is

where ,  , and are obtained from (10), (24), and (37), respectively.

Basing on the analyzed results above, we provide the results which ensure the existence of a unique , say , to minimize the total average cost for the model system starting with shortages. The procedure is as follows:

Step 0. Input the parameters, ,  ,  ,  ,  ,  ,  ,  ,  ,  , and ,  .

Step 1. Calculate and .

Step 1.1. If and , solve a unique from , and the total average cost and the order quantity can be obtained by (10) and (16), respectively.

Step 1.2. If and , then the optimal average cost and the optimal order quantity can be obtained by (10) and (16) at , respectively.

Step 1.3. If the sign of is indefinite, then solve equation . If the solution is not unique, by comparing their average costs to obtain the optimal replenishment point, compute the order quantity by (16).

Step 2. Calculate and .

Step 2.1. If ,  , and , then seek a unique from equation , and the optimal total average cost and the optimal order quantity can be obtained by (24) and (29) at , respectively.

Step 2.2. If ,  , and , then obtains its minimum at , and compute the order quantity by (29).

Step 2.3. If ,  , and , then obtains its minimum at , and compute the order quantity by (29).

Step 2.4. If the sign of is indefinite, then solve equation . If the solution is not unique, by comparing their average costs to obtain the optimal replenishment point, compute the optimal order quantity by (29).

Step 3. Calculate .

Step 3.1. If and , seek the unique from equation , and the total average cost and the order quantity can be obtained by (37) and (42), respectively.

Step 3.2. If and , then obtains its minimum at , and compute the order quantity by (42).

Step 3.3. If the sign of is indefinite, then solve equation . If the solution is not unique, by comparing their average costs to obtain the optimal replenishment point, compute the order quantity by (42).

Step 4. Find and accordingly select the optimum , and then obtain corresponding optimal order quantity .

4. Numerical Examples and Sensitivity Analysis

In order to demonstrate the procedure to obtain the optimal solution of the model, we present four examples for the model. Examples are based on piece-wise demand rate, such as and .

Example 1. The parameter values of an inventory system are given as follows: weeks, weeks, weeks, , , , units, units, units, = $500, = $2, = $3, = $12, = $8.
Since , , for all , for all , and for all , then we get local minimum value in and in . By solving equation , we have . From (10), we obtain . Therefore, the optimal ordering quantity is and the total average cost is .

Example 2. The parameter values of an inventory system are given as follows: weeks, weeks, weeks, , , , units, units, units, = $500, = $5, = $10, = $12, = $8.
Since , , for all , for all , and for all , then we get local minimum value in and in . By solving equation , we have . From (24), we obtain . Therefore, the optimal ordering quantity is and the total average cost is .

Example 3. The parameter values of an inventory system are given as follows: weeks, weeks, weeks, , , , units, units, units, = $500, = $5, = $10, = $12, = $8.
Since , , for all , for all , and for all , then we get local minimum value in and in . By solving equation , we have . From (37), we obtain . Therefore, the optimal ordering quantity is and the total average cost is .

Example 4. The parameter values of an inventory system are given as follows: weeks, weeks, weeks, , , , units, units, units, = $500, = $5, = $10, = $12, = $8.
Since , , for all , for all , and for all , then we get local minimum value in and in . By solving equation , we have . From (37), we obtain . Therefore, the optimal ordering quantity is obtained from (42), and the minimum cost .

In order to clearly indicate the effects of parameters such as ,  , and on , , and   , respectively, we study the sensitivity of different parameters under the studied inventory model. The sensitivity analysis is performed on the base of Example 1, and the results are shown in Tables 1–3, respectively.

Table 1: The sensitivity of for the model in Example 1.

0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08

3.4366
3.4905
3.5463
3.6042
3.6642
3.7263
3.7907
3.8574
3.9265

515.0829
515.0829
511.4210
507.5965
503.6066
499.4433
495.0985
490.5641
485.8321

648.1034
648.1034
644.8369
641.4184
637.8392
634.0900
630.1614
626.0436
621.7267

775.2147
771.8437
768.4699
764.9983
761.4252
757.7473
753.9611
750.063
746.0498

Table 2: The sensitivity of for the models in Example 1.

0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008

3.0141
3.1325
3.2569
3.3870
3.5229
3.6642
3.8105
3.9614
4.1171

415.8882
432.0629
448.9624
466.5543
484.7907
503.6066
522.9206
542.6345
575.5923

522.8982
543.9279
565.9745
589.0106
612.9883
637.8392
663.4729
689.7782
723.0000

638.5314
661.4087
685.1585
709.7639
735.1984
761.4252
788.3972
816.0570
845.0147

Table 3: The sensitivity of for the models in Example 1.

1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2

3.1114
3.1995
3.2513
3.3176
3.40332
3.5148
3.6613
3.8551
4.1128

427.5053
440.1612
447.1691
456.2029
467.9084
483.1709
503.6066
529.5334
564.1669

538.4988
554.7959
563.9494
575.7494
591.0495
611.0248
637.8392
672.0262
717.9964

644.4596
670.3559
680.1679
692.9359
709.5918
731.4743
761.4252
797.6508
845.0147

From Table 1, we observe that inventory level , order quantity , and the total average cost    gradually decrease as the shortage parameter increases for the model, and the replenishment time increases as increases for the model starting with shortage. Also, we observe that the values of , , , and are slightly sensitive to the changes of .

From Table 2, we find that when the deterioration parameter increases, then , , and increase for both models, and the replenishment time increases for the model starting with shortage. We see that the values of , , , and all are temperately sensitive to the changes of for the model.

From Table 3, we find that , , and increase as the deterioration parameter increases for both models. Also, we find that the replenishment time increases as increases for the model starting with shortage. We see that the values of , , , and all are moderately sensitive to the changes of for the model.

5. Conclusion

This paper studies an inventory model for Weibull-distributed deterioration with trapezoidal type demand rate and partial backlogging. First, an optimal replenishment strategy for deteriorating items with the model starting with shortage is proposed. Then, numerical examples are provided to illustrate the theoretical results. From the analysis of Tables 1–3, it can be found that the replenishment time point , order quantity , and the total average cost are moderately sensitive to the changes of and and lowly sensitive to the changes of , respectively. However, if the order cycle is a decision variable, such inventory model will be of more practical significance, which provides us an interesting topic in future.

Competing Interests

The author declares that there are no competing interests.

Acknowledgments

The research is supported partly by National Natural Science Foundation of China (Grant no. 71502100).

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