This paper studies an inventory model for Weibulldistributed deterioration items with trapezoidal type demand rate, in which shortages are allowed and partially backlogging depends on the waiting time for the next replenishment. The inventory models starting with no shortage is are to be discussed, and an optimal inventory replenishment policy of the model is proposed. Finally, numerical examples are provided to illustrate the theoretical results, and a sensitivity analysis of the major parameters with respect to the optimal solution is also carried out.
The effect of deteriorating for items cannot be disregarded in many inventory systems and it is a general phenomenon in real life. Deterioration is defined as any process that decreases the usefulness or the value of the original item, such as decay or physical depletion. For example, fruits, vegetables, or foodstuffs are subject to spoilage directly while being kept in store, and electronic products, radioactive substances, and photographic film deteriorate through a gradual loss of potential or utility with the passage of time.
Due to the variability in economic circumstances, the basic assumptions of the EOQ model should be constantly modified according to the studied inventory model. In recent years, many researchers have studied kinds of EOQ models for deteriorating items. Ghare and Schrader [
practically, the demand rate of deterioration items is impossible to increase continuously all the time. Hill [
In the above mentioned research, one of assumptions was considered: the ramp type demand rate, partial backlogging, and Weibulldistributed deterioration rate. However, for fashionable commodities, hightech products, and other short life cycle products, the demand rate should increase with the time up to certain point at first stage then reach a stabilized period and finally the demand rate decrease to zero and the products retreat from market in their product life cycle, that is, the demand rate with continuous trapezoidal function of time. On the other hand, in many real situations, customers encountering shortages will respond differently. Some customers are willing to wait until the next replenishment, while others may be impatient and go elsewhere as waiting time increases; that is, the willingness for a customer to wait for backlogging is diminishing with the length of the waiting time. In this paper, we consider an inventory model with Weibulldistributed deterioration items, trapezoidal type demand rate, and timedependent partial backlogging. By analyzing the inventory model, a useful inventory replenishment policy is proposed. Finally, numerical examples are provided to illustrate the theoretical results, and a sensitivity analysis of the optimal solution with respect to major parameters is also carried out.
The rest of the paper is organized as follows. Section
The fundamental notations and assumptions used in inventory model and considered in this paper are given as below.
The demand rate,
The replenishment rate is infinite; that is, replenishment is instantaneous.
The deterioration rate of the item is defined as Weibull
Shortages are allowed and they adopt the notation used in Abad [
The time horizon of the inventory model is finite.
In this section, we consider an inventory model starting with no shortage. The behavior of the model during a given cycle is depicted in Figure
Graphical representation of inventory level over the cycle.
In the following, we consider three possible cases based on the values of
Due to the deteriorating and trapezoidal type demand rate, the inventory level gradually diminishes during the time interval
The differential equations governing the inventory model can be expressed as follows:
The differential equations governing the inventory model can be expressed as follows:
In the following, we will provide the results which ensure the existence of a unique
If
If
The optimal value of the order level,
If
The optimal value of the order level,
If
If
The optimal value of the order level,
The above analysis shows that the three average cost functions
In such considered inventory model starting with no shortage, if
In order to demonstrate the above procedure which can be applied to obtain the optimal solution of the model, this paper presents several examples for the model, respectively. Examples are based on piecewise demand rate, such as
The parameter values are given as follows:
The model starting with no shortage; by solving the equation
The parameter values are given as follows:
The model starting with no shortage, by solving the equation
The parameter values are given as follows:
The model starting with no shortage, solving the equation
In order to clearly indicate the effects of parameters such as
By studying the results of Table
By studying the results of Table
By studying the results of Table
By studying the results of Table
By studying the results of Table
By studying the results of Table
By studying the results of Table
The sensitivity of

0  0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08 


8.9664  8.9187  8.8689  8.8167  8.7622  8.7049  8.6449  8.5817  8.5152 

472.9889  469.7083  466.2823  462.7001  458.9498  455.0167  450.8910  446.5519  441.9829 

595.6569  592.5872  589.3822  586.0308  582.5217  578.8412  574.9771  570.9112  566.6265 

805.6323  802.8699  800.0139  797.0588  793.9986  790.8268  787.5365  784.1200  780.5690 
The sensitivity of

0  0.001  0.002  0.003  0.004  0.005  0.006  0.007 


9.4545  9.3115  9.1702  9.0312  8.8950  8.7622  8.6328  8.5072 

428.3286  435.4002  442.0048  448.1317  453.7785  458.9498  463.6555  467.9088 

523.8866  536.6685  548.9558  560.7091  571.9025  582.5217  592.5618  602.0259 

704.6045  722.8867  741.0107  758.9297  776.6035  793.9986  811.0872  827.8474 
The sensitivity of

1.4  1.6  1.8  2.0  2.2  2.4  2.6  2.8 


9.2876  9.1810  9.0142  8.7622  8.4047  7.9422  7.4073  6.8496 

442.9058  447.5894  453.1802  458.9076  462.8345  461.6422  453.4686  438.8904 

545.1340  554.1055  566.4485  582.4772  601.2274  774.6099  766.4364  751.8582 

745.7703  764.3043  791.7624  790.7114  829.3653  588.7417  473.1282  391.1266 
The sensitivity of

0  0.4  1  1.6  2  2.4  2.6  3  3.6 


8.8225  8.8103  8.7921  8.7741  8.7622  8.7503  8.7443  8.7326  8.7150 

463.0980  462.2595  461.0101  459.7707  458.9498  458.1333  457.7266  456.9164  455.7090 

584.1909  583.8529  583.3499  582.8514  582.5217  582.1940  582.5217  581.7061  581.2226 

783.5429  785.6518  788.7984  791.9251  793.9986  796.0633  797.0925  799.1443  802.2061 
The sensitivity of

0  0.4  0.8  1.2  1.8  2.4  3  3.4  3.8 


11.8343  11.267  10.7698  10.326  9.7377  9.2216  8.7622  8.4819  8.2196 

673.7217  633.0852  597.9589  566.9228  526.1187  490.5382  458.9498  439.6922  421.6606 

679.6039  659.4483  642.7261  628.4787  610.4744  595.4175  582.5217  574.8648  567.8284 

67.2679  193.9216  308.8975  413.9636  555.8150  681.6692  793.9986  862.3253  925.9516 
The sensitivity of

10.4  10.6  10.8  11  11.2  11.6  12  12.4  12.8 


8.4424  8.4859  8.5284  8.5698  8.6102  8.6879  8.7622  8.8329  8.9005 

436.9771  439.9713  442.889  445.7332  448.5071  453.854  458.9498  463.8114  468.4587 

573.7973  574.9747  576.1255  577.2506  578.3509  580.4809  582.5217  584.4787  586.3587 

763.8057  767.9203  771.9299  775.8387  779.6505  786.9976  793.9986  800.6779  807.0579 
The sensitivity of

0  2  4  6  8  10  12  14  16 


8.6989  8.7150  8.7309  8.7466  8.7622  8.7775  8.7928  8.8078  8.82272 

454.6069  455.7103  456.8017  457.8815  458.9498  460.0048  461.0527  462.0877  463.1120 

580.7818  581.2231  581.6601  582.0929  582.5217  582.9455  583.367  583.7837  584.1966 

788.1626  789.6445  791.1109  792.5621  793.9986  795.4203  796.8276  798.2207  799.5999 
An inventory model starting without shortage for Weibulldistributed deterioration with trapezoidal type demand rate and partial backlogging is considered in this paper. The optimal replenishment policy for the inventory model is proposed, and numerical examples are provided to illustrate the theoretical results. A sensitivity analysis of the optimal solution with respect to major parameters is also carried out. From Table
The author declares that there is no conflict of interests regarding the publication of this paper.
The author is grateful to the anonymous referees who provided valuable comments and suggestions to significantly improve the quality of the paper. This work was supported partly by Humanities and Social Science Fund of the Ministry of Education of China (no. 11YJCZH019).