Optimal Emergency Ordering Policy for Inventory with a Random-Ending-Time Supply Disruption

This paper considers the emergency ordering strategy for the classical economic ordering quantity inventory system with a supply disruption. For situations where the ending time of supply disruption is stochastic and the purchase price increases over time during that period, we develop an emergency ordering optimization model based on maximizing retailer’s profits. Through modeling analysis in various situations, the closed-form solution of the model is obtained, and the optimal emergency ordering strategy is provided for retailers. Numerical experiments verify the effectiveness of the model and the influence of related parameters on the optimal ordering strategy.


Introduction
With the development of global market economy, enterprises will face various risks such as capital chain rupture, demand disruption, supply disruption, and legal risk. Among these, supply disruption is a common risk which may bring larger loss to enterprises [1,2]. It is reported that Renesas Electronics Co., for example, is likely to suspend production for at least three months after a re on March 19, 2021, at its semiconductor plant [3]. As a supplier of automotive chips, the Renesas Electronics Co. has a global market share of about 30% in microcontrol unit chips for vehicles. is means that the original car chip shortage will be intensi ed after the re chip shortage due to the new energy vehicle production capacity increase. Recently, the global supply chain was disrupted during the COVID-19 pandemic, and by the time it subsided, it had su ered a shock and remained vulnerable [4]. Due to the large loss caused by the supply disruption to each member in the supply chain especially the downstream enterprises, the supply disruption received much attentions of researchers' [5][6][7][8][9][10][11][12][13][14].
In order to reduce the risk of supply interruption, Hendricks et al. [15] put forward some risk reduction suggestions, such as improving prediction accuracy, synchronous planning and implementation, shortening delivery cycle, working with partners, and related investments. Hopp et al. [16] elaborated on a measure that focuses on the risks that may occur throughout the supply chain structure and planned for potential disruptions from the recovery process. Skipper and Hanna [17] examined the relationship between some attributes and exibility of the emergency response planning process and investigated a strategic approach to emergency response planning to reduce the risk of the supply disruption. Xu et al. [18] introduced the CVaR measure to hedge against the risks for the loss-averse newsvendor. Huang et al. [19] considered centralized and decentralized dual-channel supply chains in the case of demand disruptions to guarantee the robustness of optimal output in the event of demand disruptions. Sawik and Tadeusz [20] studied the supply chain with the risk of the optimal selection and protection under supply disruption, established a mixed integer programming method by combining risk neutral with risk avoidance or average risk supply, and obtained a risk conditional value to control the loss caused by supply disruption risk.
To reduce the losses caused by the supply disruption, as a remedial measure, the emergency order is taken into consideration. For an inventory system with random demand and with stochastic product supply disruption, Risa and Croix [6] o ered retailers the best rush order strategy. When the occurrence time of supply interruption is subject to a certain probability distribution and the ending time of supply interruption is determined, Xu et al. [21] studied the optimal order quantity in the loss-averse newsvendor model with backordering. Huang et al. [22] established an emergency order optimization model based on minimizing the inventory cost. Xu et al. [23] studied the optimal option purchase of a loss-averse retailer under emergent replenishment. Xia et al. [24] considered the supply interruption management problem of inventory model with a loss function and provided the retailer with the optimal emergency replenishment strategy. is paper considers the retailer's emergency ordering strategy during supply interruption such that the recovery time of supply interruption is stochastic and the purchasing price rises with the time during the period. For this problem, if the retailer places a larger rush order, the retailer would be su ered certain loss of pro ts if the supply disruption is ended at an earlier time; if the retailer makes a small emergency procurement, then the retailer would make a second emergency order with a higher purchase price when the supply disruption is ended at a later time. erefore, the retailer should determine the best time to order urgent orders and the number of urgent orders to maximize his pro ts during supply disruption period. e problem considered in this paper happens in reality. For instance, on March 23, 2021, an accident occurred when a super large gold class container ship named "Ever Given" passed through the Suez Canal which resulted in the total paralysis of the transportation of the Suez Canal. As the canal is the main channel between Asia, Europe, and Africa, the accident led to the supply interruption in many parts of the world. As for the time of channel restoration, it ranges from one or two days to one month from outside news. Six days later, the "Ever Given" was still in a single channel, blocked obliquely. Fortunately, on March 29, the shallowing operation of the container ship was successful, and the blockage of the Suez Canal was solved [25]. e rest of this paper is arranged as follows. Section 2 gives relevant symbols and assumptions on the problems considered in this paper. Section 3 establishes an optimization model based on retailer's pro t maximization. Section 4 solves the optimization model and gives the retailer's optimal emergency order strategy. Section 5 conducts some numerical analyses and veri es the in uence of relevant parameters on the model. e last section draws some conclusions.

Notation and Assumptions
For the concerned problem, we assume that the planning horizon of the mechanism is in nite and the demand is stable. As for the supply interruption, we assume that the end time t e is random, but the cut o time for the end of the event is deterministic.
at is, the event end time obeys a certain probability distribution within [0, T]. Further, we assume that shortage is not allowed during the supply disruption and the retailer's purchasing price increases with the time during the period. For the inventory system, when the supply disruption happens, the remaining inventory Q 0 will be depleted after certain period. e time is denoted by t 0 . Due to the supply disruption, the retailer may place one or two emergency orders during the supply disruption period, see Figures 1 and 2. Certainly, when the event is ended, then the ordering policy would turn into the classical economic ordering quantity policy when the emergency order is depleted. In order to maximize the retailer's expected pro ts, the retailer should determine the optimal timing and volume of emergency replenishment. To do this, we need the following notation and associated assumptions in Table 1.
For this model, we have the following assumptions: (1) e demand is stable throughout the planning period, and no shortage is allowed; (2) e leading time of each emergency order is zero; (3) At most two emergency orders are made during the supply disruption period; (4) e ending time of the supply interruption follows the uniform distribution with probability density function ϕ(t e ) 2(t e − t 0 )/(T − t 0 ) 2 for t e ∈ [t 0 , T]; (5) e retailer's purchasing price of the items linearly increases with time during the event period, i.e., the purchasing price is C(t) c 0 + τt where c 0 > c, τ > 0 and t ∈ [0, t e ].

Mathematical Formulation
For a related inventory system, due to the demand rate of t ∈ [0, t e ]λ, the retailer's remaining inventory Q 0 at the 2 Mathematical Problems in Engineering beginning will be exhausted at t 0 Q 0 /λ. As a remedy to the supply disruption, the retailer would make emergency orders during the period. Hence, we break the discussion into two cases.
Case 1. e retailer makes one emergency order. In this case, since the shortage is not allowed and the deadline time of supply disruption is T, and the retailer should place a rush order at or before t 0 with quantity Q d λT − Q 0 ; see Figure 1. Because the end time of this event is stochastic in [t 0 , T], then the retailer's inventory pro t in [0, T] is as follows: where the rst two terms are gross pro ts, the third item is the holding cost of the remaining inventory Q 0 , and the remaining three items are the inventory cost of emergency replenishment [26]. For this emergency ordering policy, as the recovery time of the events obeys uniform distribution, that is, the probability density function ϕ(t e ) 2(t e − t 0 )/(T − t 0 ) 2 for t e ∈ [t 0 , T], so the retailer's expected inventory pro ts in [0, T] are as follows: Case 2. e retailer makes two emergency orders. For this strategy, if the supply interruption ending time t e ends after t d2 , i.e., t d2 < t e , and the retailer would make the rst emergency replenishment at t d1 with size Q d1 and make the second emergency order at t d2 with size Q d2 , see Figure 2. According to the assumptions imposed on the model, it holds us, the retailer's inventory pro t in [0, T] for this case is as follows: where the rst three terms are the gross pro ts, the fourth item is the holding cost of remaining inventory Q 0 , and remaining items are inventory cost of emergency replenishment.
For the emergency ordering policy, if the supply disruption is ended after t d1 but before t d2 , i.e., t d1 < t e < t d2 , then one emergency order su ces and the inventory system returns to the classical EOQ model when the emergency order is depleted; see Figure 3. us, the retailer's inventory pro t in [0, T] for this case is as follows: For this emergency ordering policy, as the recovery time of the events obeys uniform distribution, that is, the probability density function ϕ(t e ) 2(t e − t 0 )/(T − t 0 ) 2 for t e ∈ [t 0 , T], so the retailer's expected inventory pro t in [0, T] is as follows: According to the above discussion, we can obtain the following optimization model of the concerned problem: In the next section, we will find the closed-form solution for the model and provide the retailer with the optimal emergency ordering policy.

Model Solution
For optimization problem (1), from the formation of the objective function, we know that the problem can be divided into the following two optimization problems: which, respectively, correspond to the ordering policy with one emergency order and that with two emergency orders.
For the ordering policy with one emergency order, for optimization model (2), we have the following conclusions.

Theorem 1.
For the emergency ordering policy with one emergency order, the optimal emergency ordering time is Proof. From the assumptions on the model and the discussion in Section 3, if one emergency order is made between the event period, then the retailer's expected inventory profits are as follows: which is a linear function with respect to t d . us, the maximum of the profit is reached at the end points of interval [0, t 0 ]. at is, if (h − τ)(λT − Q 0 ) > 0, the objective function is monotonically increasing with respect to t d in [0, t 0 ], and the maximum of function is reached at t 0 ; otherwise, the maximum of function For the ordering policy with two emergency orders, for optimization model (3), we have the following conclusions. □ Theorem 2. For the emergency ordering policy with two emergency orders, the first emergency order is either placed at t d1 � 0 with quantity and the second emergency order is placed at t d2 � t 0 + Q * d1 /λ with quantity Q d2 � λT − Q 0 − Q * d1 , or the first emergency order is placed at t d1 � t 0 with quantity.
Moreover, the second emergency order is placed at To show the conclusion, we need the Shengjin formula for to univariate cubic equation [27].
Proof. From the assumptions on the model and the discussion above, then the retailer's expected inventory profits in [0, T] are as follows: en, the optimization problem (3) can be written as follows: To solve the problem, we first consider the following optimization problem with respect to t d1 And then solve the optimization problem with respect to Q d1 max F 2 t d1 , Q d1 , For problem (4), since the objective is a linear function with respect to t d1 , we discuss it in two cases.
Case I. If zF 2 (t d1 , Q d1 )/zt d1 > 0, then the objective function F 2 (t d1 , Q d1 ) is monotonically increasing with respect to t d1 in [0, t 0 ], so the maximum of the objective function in [0, t 0 ] is reached at t 0 . Case II. If zF 2 (t d1 , Q d1 )/zt d1 < 0, then the objective function F 2 (t d1 , Q d1 ) is monotonically decreasing in [0, t 0 ], and the maximum of the objective function is reached at 0. Now, consider optimization problem (5). Computing the derivative of the objective function yields the following: which is a cubic function of Q d1 . In order to solve the root of the univariate cubic equation zF 2 (t d1 , Q d1 )/zQ d1 � 0, that is, the derivative zero of the objective function, we discuss the following cases.
). Further, if Q d1 < 0, then the objective function F 2 (t d1 , Q d1 ) decreases with the decrease of Q d1 in (− ∞, Q d1 ] and increases with the increase of Q d1 in [Q d1 , +∞). Hence, the maximum of the objective function is reached at λT − Q 0 ; if Q d1 > 0, then F 2 (t d1 , Q d1 ) decreases with the decrease of Q d1 in [0, Q d1 ) and increases with the increase of Q d1 in [Q d1 , +∞), the maximum of the objective function is be In this case, the equation zF 2 (t d1 , Q d1 )/zQ d1 � 0 has two real roots Q 11 � − s/r + B/A, Q 12 � Q 13 � − B/2A. If r > 0 and Q 11 < 0, the objective function F 2 (t d1 , Q d1 ) decreases with the decrease of Q d1 in (− ∞, Q 11 ] and increases with the increase of Q d1 in [Q 11 , +∞). Hence, the maximum of the objective function can be reached at λT − Q 0 ; if r > 0 and Q 11 > 0, then F 2 (t d1 , Q d1 ) decreases with the decrease of Q d1 in [0, Q 11 ] and increases with the increase of Q d1 in [Q 11 , +∞). Hence, the maximum of F 2 (t d1 , Q d1 ) can be obtained at 0 or λT − Q 0 . Case 4. Δ � B 2 − 4AC < 0. In this case, the equation zF 2 (t d1 , Q d1 )/zQ d1 � 0 has three real roots Combining the discussions mentioned above, we can obtain the optimization problem of optimization problem (3) if the first emergency order is placed at t d1 � 0,

Mathematical Problems in Engineering 7
Or if the retailer's first emergency replenishment time is t d1 � t 0 , For the optimal solution to problem (3), if Q d1 � 0 or Q d1 � λT − Q 0 , then the retailer would place one emergency order; this does not satisfy the assumption of the model. Further, substituting t d1 � 0 or t d1 � t 0 into the expression of Q d1 yields the optimal order quantity of emergency order. e following optimal solutions are obtained under the condition of satisfying the constraints.

Numerical Experiments
To verify the theoretical results given in the previous section, the following numerical experiments are carried out. Example 1. Consider the inventory system with c � 8, c 0 � 10, b � 48, Q 0 � 40, τ � 3, λ � 6, T � 20, h � 2, K � 20, and the retailer makes one emergency order.
By algorithm, when the retailer makes one emergency replenishment, the retailer's best emergency ordering time at the initial moment with size 80, then the retailer's expected inventory profits is 2220. If the retailer's best emergency ordering time is 6.67 with size 80, then the retailer's expected inventory profits is 1687. erefore, the retailer's best rush order strategy is to place 80 rush orders at the initial moment. e detailed numerical results are listed in Table 2, in which we use strategies I and II to represent replenishment strategies at two different times.
Example 2. According to the inventory system considered in Example 1, we continue to consider the case where the retailer makes two emergency orders, leaving the other parameters unchanged.
By algorithm, when the retailer makes two emergency orders, and if the retailer makes the first emergency replenishment at the initial time with quantity 31, the second emergency replenishment time is 5.17 with quantity 49, then the retailer's expected inventory profits under the given strategy is 3211. If the first replenishment is at the emergency time of 6.67 with quantity 31, and make the second replenishment at 11.8 with quantity 49, then the retailer's expected inventory profits is 3773. erefore, the retailer's best replenishment strategy in this case is that the first emergency replenishment time is t 0 � 6.67, the emergency replenishment quantity is 31, the second emergency replenishment time is 11.84, and the emergency replenishment quantity is 49. e detailed numerical results of this example are shown in Table 3, in which we use strategies I and II to represent replenishment strategies at two different times.
By comparing Examples 1 with 2, we nd that when the retailer chooses to make two emergency orders, their inventory pro ts are greater.
We conduct sensitivity analysis on the in uence of relevant parameters on the model. First, the e ect of the demand rate λ on retailers' expected inventory pro ts is          Mathematical Problems in Engineering analyzed. For the inventory system of Example 1 and Example 2, we let λ increase from 1 to 10, and other parameters remain unchanged. e numerical results are shown in Figures 4-7 from which we can see that compared with just making one emergency order, as the demand rate λ increases, the retailer chooses to make two emergency orders and place the rst order at the initial time has a great impact on retailers' expected inventory pro ts. en, we analyze the e ect of the holding cost h on retailer's expected inventory pro ts. For the emergency replenishment system given in Example 1 and Example 2, increase h from 1.5 to 5.5, leaving the other parameters unchanged. As can be seen from Figures 8-11, from which we can see that with the increase of the holding cost h has a great impact on the expected inventory pro t of retailers who choose to make two emergency orders and place the rst emergency order at t 0 .
According to the abovementioned numerical analysis, it can be seen that the changes of the demand rate λ and holding cost h have a great impact on the expected inventory pro ts of the retailer. Among them, the increase of the demand rate λ will have a positive impact on the retailer's pro ts, and the increase of holding cost h will have a negative impact on the retailer's pro ts, which shows that holding cost h and demand rate λ are the key factors for the retailer to choose emergency replenishment strategy. When supply interruption occurs, managers need to pay more attention to the e ect of these factors on the whole inventory system.

Conclusions
is paper mainly studies the emergency ordering problems with the supply disruption whose ending time is random. Based on inventory pro ts maximization, we established an inventory model, and the optimal emergency procurement strategy of retailers was given through model analysis. Finally, the e ectiveness of the optimal strategy was veri ed through numerical experiments. [28].
Certainly, this paper considers the emergency ordering problem of the retailer in the case of supply disruption. As a next step, we can expand on the fact that supply disruptions occur at random times, and taking into account the retailer's risk appetite, which could make the overall study more realistic.
Data Availability e data used to support the ndings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no con icts of interest regarding the publication of this paper.