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Within the last years, inventory management has received wide attention in such a way the recent researches in this field have focused on the study of inventory system in the presence of supply chain problems such as supply disruption. Various factors can disrupt a supply chain system, including an equipment breakdown, a strike, bad weather, natural disasters, political instability, traffic interruptions, terrorism, and so on [

The representative literature of inventory management with respect to supply disruptions includes [

In this study, we consider a continuous-review inventory system in the presence of supply disruptions, with the relaxation of some assumptions that are made in the previous researches. For instance, we consider backordering in the stockout situations, which means that customers can choose to backorder unfulfilled products or not. The assumption relaxation makes the problem more realistic. In addition, customer differentiation which is included in the proposed model has not been considered in the previous continuous-review inventory systems. Due to the complexity of the problem, we have used simulation modeling to develop the mentioned inventory system. Furthermore, we investigate the impacts of supply disruptions and customer differentiation on the inventory system.

In Section

This paper considers a continuous-review inventory system with single product of a retailer, where supply may be disrupted. The supplier is not always available. The retailer sells products to the customers and replenishes the stock from its single supplier. When a supply disruption occurs, the supplier cannot fulfil the orders from the retailer. Only when the disruption issue is resolved can the orders be processed. We define the time period during which the supplier is available (i.e., under normal conditions) as its ON period, and the time period during which the supplier is not available (i.e., under disruption conditions) as its OFF period. ON and OFF periods represent the frequency and duration of supply disruptions, respectively. In other words, ON and OFF periods reflect the disruption severity of an unreliable supplier. The longer the ON periods, the less frequent the disruptions and the slighter the disruptions. On the contrary, the longer length of OFF periods, the longer the disruption duration and the more severe the disruptions. The standard

The retailer adopts a continuous-review inventory policy

A standard

Figure

A

In this paper, we study partial backordering in the stockout situations. When the retailer is out of stock, a customer may choose to backorder the products he/she needs or to abandon the purchase order and leave for other sellers (i.e., lost sale). The retailer incurs backorder cost or lost sale cost accordingly. In general, unit backorder cost per time unit is less than unit lost sale cost, since the retailer may obtain profits from selling backorders.

In addition, we allow the number of outstanding orders to be more than one. We also incorporate customer differentiation in the discussed inventory system. Customers are segmented based on their backorder probabilities in the stockout situations and are differentiated into two classes. One class has higher backorder probability, while the other class has lower backorder probability. For convenience, these two classes are denoted by classes I and II, respectively. To acknowledge class I for their higher backorder probability, the retailer provides them with high priority to receive backorders.

All the above considerations, combined with the complex nature of a continuous-review inventory system, make it very difficult to study this inventory management problem by using an analytical method. In this paper, we will utilize simulation techniques [

The calculation formulas of the annual ordering cost, annual holding cost, annual backorder cost, and annual lost sale cost from each customer class are as follows:

The structure of the model is made up of two subsystems: customer demand subsystem and inventory replenishment subsystem, as shown in Figures

Customer demand subsystem.

Inventory replenishment subsystem.

Figure

The second situation is that the net inventory level is positive, but there is no enough stock for the customer’s demand. Under this situation, the customer takes all available products and decides whether to backorder the unfulfilled products or not. The third situation is that the net inventory level is nonpositive, and there are no products available at all. Under this situation, the customer can either backorder the unfulfilled products or leave without ordering. In the second and third situations, we differentiate customers since different customer classes have different backorder probabilities.

If a customer chooses to backorder, the net inventory level and inventory position of the retailer decrease by the demand size of the customer. Besides, the backorder quantity of the corresponding customer class is equal to minus net inventory level if it is the second situation or increases by the demand size of the customer if it is the third situation. Now consider the other case. If the customer chooses not to backorder, part of the sale is lost when it is the second situation or the entire sale is lost when it is the third situation. The resulting lost sale cost is calculated accordingly, as shown in Figure

Figure _{I} + BoQ_{II}, where OQ and _{I} ≤ OQ < BoQ_{I} + BoQ_{II}. The third situation is that the arriving products even cannot satisfy the backorders of customer class I (i.e., OQ < BoQ_{I}) [

In this section, we present the input data of the model. The customer demand of the retailer is assumed to follow an empirical distribution, where customer interarrival time follows an exponential distribution with a mean of _{0} = IP_{0} = 10. Such settings prevent the initial inventory status from being unrealistically “empty and idle.” Later we will warmup the simulation model to remove the influences that the initial settings bring about. The inventory policy parameters

Empirical distribution of customer’s demand size.

Demand size | 1 | 2 | 3 | 4 | 5 |

Probability | 0.1 | 0.25 | 0.3 | 0.25 | 0.1 |

In the beginning of the simulation, the model is empty without any inventory. Therefore, the data obtained from that may not be appropriate criteria for analysis. To avoid this matter, a period of time is taken into account for the model as the warmup period. In this study we have used the Welch method [

Determining warmup period.

In this section, we simulate the inventory system according to the model described in Sections

To investigate the impact of supply disruptions on the inventory system, we conduct the following experiment. We reasonably assume that 10% of customers belong to class I, and that when a stockout occurs, 80% of customer class I and 10% of customer class II choose to backorder, that is,

The design for the first experiment.

Parameter | Values (units: days) |
---|---|

20; 60; 120 | |

1; 5; 10 |

On the other hand, when looking into the impact of customer differentiation on the inventory system, we consider two cases of supply disruptions for the sake of completeness:

The design for the second experiment.

Parameter | Values |
---|---|

5%; 10%; 20%; 40% | |

60%; 90% | |

5%; 20% |

As we showed in Section

The experimental results are illustrated in Figures

The impact of

The impact of

The impact of customer differentiation on ATC under the disruption scenario of

The impact of customer differentiation on ATC under the disruption scenario of

Figure

Figures

Figure

Figure

Now, we want to compare the results of the experiments to the one developed by Li and Chen [

Compare current model with previous study under different scenarios of

Scenario number | Disruption | Average annual total cost | |||||

Current model | Previous model | ||||||

1 | 120 | 1 | 10891 | 13900 | |||

2 | 60 | 1 | 11127 | 13900 | |||

3 | 20 | 1 | 11813 | 13950 | |||

4 | 120 | 5 | 13608 | 14150 | |||

5 | 10 | 80 | 10 | 60 | 5 | 13731 | 14400 |

6 | 120 | 10 | 14372 | 14800 | |||

7 | 20 | 5 | 14479 | 15750 | |||

8 | 60 | 10 | 14973 | 15800 | |||

9 | 20 | 10 | 17826 | 18700 |

Compare current model with previous study under different scenarios of

As the results show, the continuous-review model dominates the other model in all scenarios with different values of

Compare current model with previous study under impact of customer differentiation on the inventory system.

Scenario number | Average annual total cost | ||||||

Current model | Previous model | ||||||

1 | 40 | 90 | 5 | 10545 | 13670 | ||

2 | 40 | 90 | 20 | 11041 | 13050 | ||

3 | 20 | 90 | 5 | 11228 | 13670 | ||

4 | 20 | 90 | 20 | 11249 | 13270 | ||

5 | 10 | 90 | 20 | 11348 | 13450 | ||

6 | 5 | 90 | 20 | 11386 | 13600 | ||

7 | 10 | 90 | 5 | 11574 | 13500 | ||

8 | 60 | 1 | 40 | 60 | 20 | 11746 | 13370 |

9 | 5 | 90 | 5 | 11818 | 13700 | ||

10 | 40 | 60 | 5 | 11871 | 14050 | ||

11 | 20 | 60 | 20 | 11988 | 13800 | ||

12 | 10 | 60 | 20 | 12127 | 14050 | ||

13 | 20 | 60 | 5 | 12198 | 14150 | ||

14 | 5 | 60 | 20 | 12215 | 14200 | ||

15 | 10 | 60 | 5 | 12317 | 14250 | ||

16 | 5 | 60 | 5 | 12356 | 14300 |

Compare current model with previous study under impact of customer differentiation on the inventory system.

As the results show, the average annual costs (ATC) of the continuous-review model is lower that the periodic one in all scenarios (

In the paper, a continuous-review inventory model

We have also shown that the ATC of a continuous-review model is significantly lower than a periodic one which is taken from a latest research. For future studies, we recommend to consider the price factor in the model as well. The normal fluctuation on product prices may have a great effect on the inventory system costs.

Annual total cost.

Stochastic demand size of a customer

The mean interarrival time of customers

Replenishment lead time

Time horizon as one year, that is, 365 days

_{0}:

The initial net inventory level of the retailer

The net inventory level of the retailer at time point

_{0}:

The initial inventory position of the retailer

The inventory position of the retailer at time point

The mean duration of ON periods

The mean duration of OFF periods

Setup cost for each order placement

Unit product price

Unit holding cost per time unit

Unit backorder cost per time unit from customer class

Unit lost sale cost from customer class

The backorder quantity from customer class

The number of lost sales from customer class

The backorder probability of customer class

The proportion of customer class I

Annual ordering cost

Annual holding cost

Annual backorder cost from customer class

Annual total backorder cost,

Annual lost sale cost from customer class

Annual total lost sale cost,

The optimal reorder point

The optimal reorder quantity

The minimum annual total cost.