Computers & Operations Research 39 (2012) 1909–1918
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Computers & Operations Research journal homepage: www.elsevier.com/locate/caor
An inter-temporal resource emergency management model Junchang Qin a,c, Yiting Xing b, Song Wang c, Kanliang Wang d,n, Sohail S. Chaudhry e,1 a
Department of Economics and Information, Shenzhen Institute of Information Technology, Shenzhen 518055, China School of Economics and Management, Chang’an University, Xi’an 710064, China School of Management, Xi’an Jiaotong University, Xi’an 710049, China d School of Business, Renmin University of China, Beijing 100872, China e Department of Management and Operations/International Business, Villanova School of Business, Villanova University, Villanova, PA 19085, USA b c
a r t i c l e i n f o
abstract
Available online 20 July 2011
One of the most important ways for promoting the service level during emergency management in public sectors is through enhancing management of strategic resources. From the perspective of vertical integration of operational process in emergency management, emergency resources can be classified as those resources used during the response period as well as those resources that are consumed during the recovery period. An inter-temporal integrated single-period resource model for solving optimal order quantity is proposed that meets the characteristics during the demand for the recovery resources depending on the stock shortage of the response resources. In the light of the classification of the dependent relationship of the two kinds of resources, namely deterministic or stochastic, the research investigates the analytical properties of the model, based on which, a genetic algorithm-based simulation approach is proposed. Finally, a real case with numerical example is provided to assess and validate our model, as well as managerial insights are obtained through key parameters sensitivity analysis. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Emergency resources Inter-temporal analysis Inventory model Genetic algorithms
1. Introduction During the past few years, millions of the inhabitants of our planet have experienced several major naturally occurring catastrophic events around the world. These naturally occurring events include the 2005 and 2010 Earthquakes in Pakistanadministered Kashmir and Haiti, 2004 and 2011 Tsunami/Earthquakes in South Asia and Japan, the 2005 Hurricane Katrina and the recent 2011 Mississippi River Floods in the USA, to name a few. In such instances, it is imperative that disaster responding units that include the agencies belonging to the governments of the affected countries, Non-Governmental Organizations, the International Red Cross/Crescent and others responding units provide apt and timely emergency response services to their affected citizens facing the calamity. As it is evident from these recent natural calamities, on several occasions, the response from the various responding units was of poor service or late reaction or both. In some situations, there was deficiency of the necessary resources that were required during the response period as well as during the search and recovery phases. Therefore, the natural question is how the governmental agencies and other the n
Corresponding author. Tel.: þ86 10 62514650. E-mail addresses:
[email protected] (K. Wang),
[email protected] (S.S. Chaudhry). 1 Tel.: þ1 610 519 4369. 0305-0548/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2011.07.008
responding units can improve their operations performance when faced against such natural disasters [26]. As we enter the new century, the continual of emergency incidents, such as the various acts of nature as well as humancaused accidents, have brought forward new challenges to the research and practice of emergency service decisions [3,5,8,10,13,21,24,25,34,38]. The Federal Emergency Management Agency (FEMA) of USA proposed a four-stage emergency management operations model that includes mitigation, preparedness, response and recovery. This model has been in operation and widely applied in practice (http://www.fema.gov/government/ index.shtm). In this paper, we consider a joint purchase of the emergency resources for emergency management that is based on the framework of this emergency operations model in the response and recovery stages. Hence, given the demand information for resources used during the response stage as well as the dependent relationship between the shortage quantity of response resources and the quantity demanded of recovery resources, the key issue is to determine the order quantity of these two types of resources so as to minimize total losses in the course of emergency response and recovery stages. The following situation is the impetus behind our research. Over the years, millions of people in China have been displaced or otherwise affected by floods, which are typically caused by torrential rains. In fact, during June 2011, there were over 5 million people affected by floods in the eastern and southern provinces of
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China. Therefore, our research addresses a specific decision-making process of emergency resources inventory in a flood incident. In order to mitigate the loss caused by the flood, the decision makers have to decide on the order quantity of the emergency resources in advance based on the probability associated with flood occurrence. These necessary emergency materials can be classified into two categories in terms of the characteristics of the operational course based on the response-stage and the recovery-stage emergency management pattern. They are the resources used during the response period (i.e., response resources) and resources that are utilized during the recovery period (i.e., recovery resources). Response resources (will be called resource 1 throughout the rest of the paper) such as sandbags and protection network refer to such resources. When there is a danger of floods or flooding occurs, then resource 1 can be used to prevent or limit the flood damage. Recovery resources (will be called resource 2 throughout the rest of the paper) such as food and medicine refer to these types of resources. After the occurrence of the calamity, the use of resource 2 can be used during the recovery phase in the disaster area. When disaster occurs, if the stocked quantity of resource 1 is less than the quantity demanded, a demand for resource 2 will be generated; hence inventory of resource 2 is needed. Obviously, there is a dependent relationship between the two kinds of resources. However, because of the complexity and severity of the calamity, the dependent relationship is non-linear and stochastic. On the contrarily, if resource 1 is overstocked, we may assume that the rescue is sufficient and that the inventory of resource 2 is unnecessary. Furthermore, the loss caused by the emergency resources shortage is also affected by non-linear factors. An analogous classification of emergency resources exists in many other types of emergency incidents. For example, in the emergency management of drought, the emergency resources can be classified into two categories: pumping equipment and irrigation equipment that are used to prevent the spread of devastation from a drought and the relief for food safety and human healthcare. Thus, it can be seen that, the inter-temporal dependent relationship between the inventory quantity of response resources and that of recovery resources should be considered in the emergency management, in order to reduce the cost of emergency operation. In China, two different governmental agencies are generally in charge of the rescue and recovery operations. This implies that the orderings of the response resources and recovery resources are conducted independent of each other. Hence, in this case, it is often observed that there is a shortage or excess of both the two types of resources. Therefore, cooperative buying of the two kinds of resources according to their related demands is necessary for the cost reduction in the process of emergency management. Although emergency management literature is abundant, quantitative analysis on emergency inventory is rare. Ray [31] proposed a single-commodity, multi-modal network flow model over a multi-period planning horizon on a capacitated network to optimally schedule the distribution of food-aid in West Africa. Knott [20] introduced a bulk food transportation problem, developed a linear programming model in which two objectives were considered, namely, minimize transportation cost and maximize the amount of food delivered. To address the routing and scheduling of disaster relief operations, Haghani and Oh [14] proposed a linear programming model. The single objective was to minimize the total cost, which involved vehicle flow, commodity flow, supply or demand carryover and transfer cost. Beamon and Kotleba [2] developed and tested three different inventory management strategies as applied to the complex emergency in south Sudan. The study showed the advantages of utilizing quantitative methods to manage inventory in a relief setting. Balcik et al. [1] pointed out that the significance
of coordination mechanisms in humanitarian relief chains and the challenges as well as opportunities were reviewed and described for related problems. Apparently, previous researchers discussed emergency resources management mainly from the perspective of scheduling single response or recovery resource. As early as in 1985, McLoughlin [27] indicated the importance of integration in emergency management. However, most studies on integration of the four stages of operational mode only provide a qualitative analysis. To the best of our knowledge, none of the studies used the inter-temporal integration perspective in the area of emergency inventory management. Therefore, in this paper, we attempt to integrate the emergency resources purchase of both, the response stage and the recovery stage to minimize the total cost. In general, certain emergency incidents occur seasonally (such as, floods that mostly occur in the summer, and drought that mainly happens during the spring and winter periods in China). Consequently, the demand for food and medicine in the rescue operations can be considered seasonal as well. These characteristics correspond to that of single-period inventory problem. Combined with the inventory dependent relationship of resource 1 and resource 2, we can assume that the inventory management problem of inter-temporal emergency resources can be seen as a type of single-period inventory decision problem under the condition of multi-echelon inventory. Next, we will briefly review the previous research associated with this model. Whitin [36] introduced a single-period inventory problem of components and finished products in a two-stage production process, under the condition that the demands for finished products are stochastic. Brayan et al. [4] and Johnson and Montgomery [17] studied the analogous problems in the context of multi-stage production process. Using an analytical method, Gerchak and Zhang [11] proposed a solution process to the problem that the original inventory of two kinds of products was non-zero. When the cost structure of assemble-in-advance was different from that of assemble-to-order, Hariga [15] presented a new mathematical model and showed the complete conditions under which the solutions exist. Stalinski [33] solved the inventory problem of spare parts and finished products in a multi-stage production process according to different objectives that included minimizing the expected cost and/or maximizing the profit level. Recently, Eynan and Rosenblatt [9] studied two polices of assemble-in-advance and assemble-to-order in different contexts, and provided some practical insights on inventory management policy. The above studies enrich the research findings of multi-stage inventory problem from different perspectives. Based on these studies, we can summarize some common characteristics in their research assumptions as the demand for final product was externally stochastic, the dependent relationship between finished products and spare parts was linear and deterministic, and the costs associated with overstocking or stock shortage are linearly related to the quantity of overstocking or stock shortage. In addition, as to the single-period inventory problem with ˚ and Jornsten ¨ [29] proposed a greedytransshipments, Nonas based approximation strategy to the problem with n locations, and the necessary and sufficient conditions of the optimal solution are given under certain cost constraints. Zou et al. [39] set up a single-period inventory model with two locations in a competitive context, and analyzed the conditions and values to transship based on the solution. Olsson [30] considered the inventory problem within the same stage and with single transshipment direction under continuous physical inventory strategy, and designed a simple and quick solution approach. By analyzing the above studies, the differences between the model in this paper and the inventory models with transshipments are clearly shown, mainly in two aspects: first, the demands among all
J. Qin et al. / Computers & Operations Research 39 (2012) 1909–1918
transit depots in transshipment models are independent, and the transshipment is only to provide the risk pool effect to reduce the inventory carrying cost, so the key decision point in these models is the tradeoff between inventory carrying cost and transshipment cost, whereas in our model the demand of resource 2 is dependent on the shortage quantity of resource 1 in a stochastic way; second, transshipment models have determinate and linear costs relationships which are quite different from the stochastic and non-linear relationships in our model. It is clear that, there are obvious differences between the inventory problem of emergency resources and the problem studied in the above researches. On one hand, due to the complexity of emergency process, the demand for recovery resources is not only dependent on the shortage quantity of response resources, but also affected by external stochastic factors, whereas in the previous studies, the demand for final products was solely affected by external stochastic factors. On the other hand, the structure of multi-stage inventory cost in previous research was linear; yet in the emergency operation process, the demand function of recovery resources and the loss caused by the shortage of response resources are both non-linear. The remainder of the paper is organized as follows. In Section 2, a mathematical model is proposed according to the specific characteristics of our research problem. In terms of the two different dependent relationships between resource 1 and resource 2—determinate or stochastic, the analytical properties of the model are analyzed in Section 3. Section 4 provides the analytical-based simulation algorithm with a numerical example, which is followed by a short conclusion in Section 5.
2. Mathematical model The specific problem studied in this research is to determine the optimal order quantity of emergency resources in the context of the integration of emergency management operational process, given the dependent relationship between the demand quantity of resource 2 and the shortage quantity of resource 1, with the objective to minimize the expected loss related to all emergency resources. In order to reflect the dependent relationship in demand function of resource 2, we introduce a method using a deterministic scalar multiplied with a stochastic variable. Karlin and Carr [18] first proposed this method and this model have been widely adopted in many classical operation research studies. The following notations and assumptions are used in the development of our model.
1911
l1 ðd1 Þ loss function of resource 1 when it is deficient; x2 ¼ Dðg, eÞ demand for resource 2, and Dð Þ is a non-negative function of d1; vi ¼ hi si loss value of one unit of resource i, when resource i overstocked; d2 ¼ maxð0,x2 q2 Þ shortage quantity of resource 2; loss function of resource 2 when it is deficient. l2 ðd2 Þ Assumption 1. f1 ðx1 Þ ¼ lelx1 , demand for resource 1 that follows the negative exponential distribution, according with the general law that severe emergency incidents causing big harm will occur with a small probability; hereinto, l 40 is constant. Assumption 2. e is a stochastic variable defined in the interval ½A,B and follows uniform distribution, with mean value of m ¼ A þ B=2; besides, e is independent with x1 . Assumption 3. Dðg, eÞ ¼ gðd1 Þe, indicating that the demand for resource 2 is constituted by the product of the deterministic part, gðd1 Þ and the stochastic part, e; it is obvious that the mean value and variance of x2 vary with the value of gðd1 Þ, but the ratio of standard deviation to mean value, namely coefficient of variation, is constant, i.e., they are positively correlated. Assumption 4. If d1 4 0, then gðd1 Þ ¼ að1ebd1 Þ; hereinto, a 4 0, b 40 are constant, furthermore, a denotes the maximal demand quantity for resource 2 caused by the calamity; this expression indicates that the deterministic part of the demand for resource 2 is exponentially related to the shortage quantity of resource 1, but less than a. If d1 ¼ 0, then gðd1 Þ ¼ 0, indicating that the deterministic part of the demand for resource 2 will be zero when there is no shortage of resource 1. Assumption 5. If d1 4 0, then l1 ðd1 Þ ¼ að1ebd1 Þ; hereinto, a 4 0,b4 0 are constant, furthermore, a denotes the maximal loss value that can be caused by the calamity; this expression indicates that the loss generated by the shortage of resource 1 is exponentially related to its shortage quantity, but less than a. If d1 ¼ 0, then l1 ðd1 Þ ¼ 0. Assumption 6. l2 ðd2 Þ ¼ kd2 ; hereinto, k 40 is constant. It is obvious that when resource 2 is overstocked, l2 ð0Þ ¼ 0. Assumption 7. l1 ð1Þ 4 c1 , l2 ð1Þ 4 c2 , indicating that when the order quantity is less than the actual demand quantity, loss of resource shortage will be generated.
2.1. Notations and assumptions The following are the main notations and assumptions used in this paper: the resource index, 1, 2; order cost of one unit of resource i, ci 40; disposal price of one unit of resource i when overstocked; hi inventory holding cost of one unit of resource i, hi 4 0; x1 demand for resource 1, x1 Z 0; f1 ðx1 Þ probability density function of x1 ; cumulative probability density function of x1 ; F1 ðx1 Þ order quantities of resourcei, qi Z0; qi d1 ¼ maxð0,x1 q1 Þ shortage quantity of resource 1; the deterministic part of the demand for resource 2, a gðd1 Þ non-negative function of d1; e the stochastic part of the demand for resource 2, a stochastic variable defined in the interval ½A,B, and B ZA 4 0, let R ¼ BA; i ci si
Assumption 8. The initial inventory is zero, and so is the ordering lead time. The inventory holding cost occurs only when the resources are overstocked. Assumption 9. The decision maker has only one opportunity to place an order. Moreover, ci 4 vi , and when the order quantity is more than the actual demand quantity, loss of resource overstocked will be generated. Other assumptions comply with the general newsboy model. 2.2. Mathematical model In order to simplify the analysis, we consider again the example of inventory decision on emergency resources in a flood incident. When the order quantities of resource 1 and resource 2 are equal to their actual demand quantities, the two kinds of resources are utilized completely. Thus, there are no shortages or overstocked resources, so only the order costs needs to be calculated for the model. The loss associated with the overstocked resource will be
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generated when the order quantities of the two kinds of resources exceed their actual demand quantities. On the other hand, loss associated with resource shortage will occur if the order quantities are less than the actual demand quantities. In both these instances, the model will incorporate the loss associated with the overstocked resource and/or the shortage of resource as well as the order cost. The loss function value can be calculated under three different conditions according to the following expressions. If q1 Z x1 , then the loss value will be E1 ¼ v1 ðq1 x1 Þ þv2 q2 þ
2 X
ci qi
ð1Þ
i¼1
ð5Þ
If q1 o x1 and q2 Z x2 , then the loss value will be E2 ¼ l1 ðd1 Þ þ v2 ðq2 x2 Þ þ
2 X
ci qi
ð2Þ
Based on Assumptions 1–6, the expression (5) can be transformed to (6):
ð3Þ
8 ð1 þ ðl=bÞÞ > þ v1 q1 þ v2 q2 vl1 þ c1 q1 < elq1 vl1 þ a llþab þ lkþlab 1 qa2 Eðq1 ,q2 Þ ¼ l=b lq l=b > : þ e l þ1bv2 ð 1 qa2 Þð1q2 ÞabÞþ kelq1 ðaq2 Þ 1 qa2 þ c2 q2
i¼1
If q1 o x1 and q2 o x2 , then the loss value will be E3 ¼ l1 ðd1 Þ þ l2 ðd2 Þ þ
2 X
ci qi
i¼1
According to (1)–(3), when ðq1 ,q2 Þ are given, the total expected loss value will be R1 8 R q1 ½v1 ðq1 x1 Þ þ v2 q2 f1 ðx1 Þdx1 þ q1 l1 ðd1 Þf1 ðx1 Þdx1 > > < 0 2 X R 1 R q2 R1 Eðq1 ,q2 Þ ¼ > ci qi > : þ q1 ½ 0 v2 ðq2 x2 Þdx2 þ q2 l2 ðd2 Þdx2 f1 ðx1 Þdx1 þ i¼1
ð4Þ
We can derive from Eq. (4) that, the ðq1 ,q2 Þ, when the total expected loss value can reach its minimum, satisfy the optimal order quantities of resource 1 and resource 2. In general, there are two types of method to solve this problem, namely, analytical analysis [19] and simulation [28]. The process of analytical analysis is as follows: first, conduct the convexity analysis of the objective function; second, solve the partial differential equations derived from the function; and finally, transform the solutions of the equations and obtain the optimal solutions of the objective function. It is difficult to obtain the analytical solutions since solving Eq. (4) is not easy through the process of analytical analysis. Although the traditional Monte Carlo simulation method is relatively easy to implement, it is also quite time-consuming and inefficient. Therefore, it is critical to improve the efficiency of the algorithm by utilizing the analytical properties of the model, confining properly the search scope of the solution space, and decreasing the running time of the simulation process. In the next section, we analyze the analytical properties of the model according to the classification of the dependent relationship between the two types of resources, deterministic or stochastic, which, provides a basis for the development of an efficient algorithm for the proposed problem. n
to compare with the decision results when mean value of recovery resource demand are employed. Therefore, we first discuss the model for the case when B ¼A. If B ¼ A, namely, Dðg, eÞ ¼ gðd1 Þ, then the dependent relationship between the demand quantity of resource 2 and the shortage quantity of resource 1 will be deterministic as follows: 8 2 X > R R > > q1 ½v1 ðq1 x1 Þ þ v2 q2 f1 ðx1 Þdx1 þ 1 l1 ðd1 Þf1 ðx1 Þdx1 þ > ci qi > 0 q1 > < i¼1 R q ðlnð1ðq2 =aÞÞ=bÞ Eðq1 ,q2 Þ ¼ > þ q11 v2 ðq2 gðd1 ÞÞf1 ðx1 Þdx1 > > > R1 > > : þ q ðlnð1ðq =aÞÞ=bÞ kðgðd1 Þq2 Þf1 ðx1 Þdx1 1 2
n
3. Model analysis In this section, the analytical properties of the model under two conditions are examined based on the dependent relationship between the two types of resources. 3.1. If B ¼ A In practice, decision makers often employ the mean value of stochastic variable by replacing the variable itself. Although, this approach can simplify the model resolution, however, it can cause a decrease in performance outcome. In our model, in order to show the importance to consider stochastic recovery resources demand when response resources are in shortage, it is necessary
ð6Þ
One can observe from Eq. (6) that using the traditional mathematical analytical method to solve the problem is difficult. Therefore, simulation method is adopted to solve the model. Although, the traditional simulation method fits quite nicely with the various conditions and is relatively easy to implement, however, it does not restrict the search scope of the solution space. This leads to a significantly larger computational burden in running the simulation program to derive the solution. Thus, based on the mathematical analysis of the model, we propose an analyticalbased simulation algorithm, which makes use of the analytical properties of the model, and reduces the search scope of the solution space by considerable amount and thereby, improve the efficiency of the simulation program. Next, we present the analysis of the analytical properties analysis of the model. We will limit the sampling space of q1 by two extreme situations. The optimal order quantity of q1 is obviously the sampling upper bound of q1 when the inventory of resource 2 is zero. The optimal order quantity of q1 is the sampling lower bound of q1 when the inventory of resource 2 are always enough to satisfy the demand. Hence, Proposition 1. If the order quantity of resource 2 is zero, then the optimal order quantity of resource 1 will be qn1 ¼
lnðMd Þ
ð7Þ
l
where 2
Md ¼
ðv1 þða þ kaÞðlðl =l þ bÞÞÞ : lv1 þ c1
Proof. When q2 ¼ 0 holds, the function of expected loss value will be Z q1 Eðq1 ,0Þ ¼ v1 ðq1 x1 Þf1 ðx1 Þdx1 0 Z 1 þ ðl1 ðx1 q1 Þ þ kgðx1 q1 ÞÞf1 ðx1 Þdx1 þ c1 q1 & ð8Þ q1
According to Assumptions 1–6, expression (8) can be simplified, as shown in (9): v1 lq1 1 1 1 1 ðe þ lq1 1Þþ alelq1 þ c1 q 1 þ kalelq1 l l lþb l lþb ð9Þ Differentiate (9) with respect to q1 , we can get: 1 1 1 1 2 2 kal elq1 v1 ðlelq1 Þal elq1 þ c1 l lþb l lþb
ð10Þ
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Moreover, differentiate (10) with respect to q1 , we can get: 1 1 1 1 3 3 þkal elq1 v1 lelq1 þ al elq1 ð11Þ l l þb l lþb It is clear that (11) is positive by assumptions. Therefore, the optimal order quantity of q1 can be obtained from (8). Furthermore, setting Eq. (10) equal to zero, then the optimal order quantity of q1 is determined as qn1 ¼ lnðMd Þ=l. Accordingly, qn1 will increase as 1=l, a, a, b, or b increases; and qn1 will decrease as c1 =v1 increases. Furthermore, if, in practice, the supply of resource 2 used in the recovery period of the calamity salvage is scarce and the assumption of Proposition 1 holds, expression (7) can assist to determine the optimal order quantity of resource 1. Proposition 2. If the order quantity of resource 2 can always fulfill the demand, then the optimal order quantity of resource 1 can be determined by (12): qn1 ¼
lnðNd Þ
ð12Þ
l
where 2
Nd ¼
ðv1 þða þ c2 aÞðlðl =l þ bÞÞÞ : lv1 þc1
Proof. According to (4), the expected loss value of resource 1 is Z
q1
v1 ðq1 x1 Þf1 ðx1 Þdx1 þ
0
Z
ðl1 ðx1 q1 Þ þ c2 gðx1 q1 ÞÞf1 ðx1 Þdx1 þ c1 q1
3
1
l
ð13Þ
1 1 1 3 þc2 al elq1 l þb l lþb
ð15Þ
elq1
lðv2 þkÞ q2 l=b 1 bðaq2 Þ a
ð20Þ
Obviously, Eq. (20) is positive, indicating that the expression (18) is convex. Therefore, by letting Eq. (19) equal to zero, we have b=l ! v2 þc2 elq1 q2 ¼ a 1 ð21Þ v2 þ k It can be easily seen from Eq. (21) that, if q1 is big enough, q2 may be negative. Therefore, when resource 1 is sufficient, there will be no need for resource 2. Finally, when given q1 , the optimal order quantity of resource 2 will be b=l !! v2 þ c2 elq1 n q2 ¼ max 0, a 1 v2 þk Particularly, if q1 ¼ 0, i.e., the order quantity of resource 1 is zero, then the optimal order quantity of resource 2 can be expressed as ! v2 þc2 b=l ð22Þ q2 ¼ a 1 v2 þk
Proposition 3. If q1 , the order quantity of resource 1, is given, then the optimal order quantity of resource 2 will be: b=l !! v2 þ c2 elq1 n q2 ¼ max 0, a 1 ð16Þ v2 þ k Proof. When q1 is given, x2 ¼ að1ebðx1 q1 Þ Þ becomes a function of stochastic variable x1 . Thus, the probability density function of x2 can be written as
l elðq1 ðlnð1ðx2 =aÞÞ=bÞÞ bðax2 Þ
&
3.2. If B 4 A If B 4A, namely, Dðg, eÞ ¼ gðd1 Þe, and the demand quantity of resource 2 is determined by the conduct of the deterministic part, gðd1 Þ, and the stochastic part, e. Let u ¼ q2 =gðd1 Þ, then we can derive Eq. (23) from Eq. (4):
Eðq1 ,q2 Þ ¼
8 2 X > Rq R1 > > < 1 ½v1 ðq1 x1 Þ þv2 q2 f1 ðx1 Þdx1 þ l1 ðd1 Þf1 ðx1 Þdx1 þ ci qi 0
q1
i¼1
> R R R > > : þ q1 ð 0q2 =gðd1 Þ vR2 ðq2 gðd1 ÞuÞduþ q1=gðd 1
It is obvious that Eq. (15) is positive. Let Eq. (14) equal to zero, then we can obtain the optimal order quantity of q1 : qn1 ¼ lnðNd Þ=l. If, in practice, the supply of resource 2 used in the recovery period of the calamity salvage is sufficient and the assumption of Proposition 2 holds approximately, the optimal order quantity of resource 1 can be determined by expression (12).
f2 ðx2 Þ ¼
ð19Þ
a
&
Similar to the proof process of Proposition 1, we get the first order and the second order derivative of (13) with respect to q1 , as shown in Eqs. (14) and (15) respectively: 1 1 1 1 2 2 v1 ðlelq1 Þal elq1 c2 al elq1 þ c1 ð14Þ l lþb l lþb v1 lelq1 þ al elq1
q2 l=b v2 elq1 þ c2 elq1 ðk þv2 Þ 1
1
q1
respectively:
ð17Þ
2
1Þ
k R ðgðd1 Þuq2 ÞduÞf1 ðx1 Þdx1
ð23Þ
Proposition 4. If the order quantity of resource 2 is zero, then the optimal order quantity of resource 1 can be expressed as in Eq. (24), where qn1 ¼
lnðMs Þ
ð24Þ
l
where 2
Ms ¼
ðv1 þ ða þ kmaÞðlðl =l þ bÞÞÞ lv1 þc1
Proof. If the order quantity of resource 2 is zero, the loss caused by the shortage of resource 1 is composed of two parts: l1 ðd1 Þ and kDðg, eÞ. Therefore, we derive Eq. (25) from Eq. (4): Z q1 Z 1 v1 ðq1 x1 Þf1 ðx1 Þdx1 þ l1 ðd1 Þf1 ðx1 Þdx1 Eðq1 ,0Þ ¼ 0 q1 Z 1 þ kDðg, eÞf1 ðx1 Þdx1 þc1 q1 & ð25Þ q1
According to (4) and (17), the expected loss value of resource 2 can be expressed as Z q2 Z 1 v2 ðq2 x2 Þf2 ðx2 Þdx2 þ kðx2 q2 Þf2 ðx2 Þdx2 þ c2 q2 ð18Þ 0
q2
By obtaining the first order and the second order derivative of Eq. (18) with respect to q2 , we attain Eqs. (19) and (20),
The expression of the expected loss value can be rewritten in a more simple form as v1 lq1 1 1 ðe þ lq1 1Þ þ alelq1 l l l þb kðA þ BÞ 1 lq1 1 þ ale þ c1 q1 ð26Þ 2 l lþb
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Apparently, the only difference between Eqs. (26) and (9) is the constant ðA þ BÞ=2, which is the mean value of e. Similar to the proof process of Proposition 1, we have: qn1 ¼ lnðMs Þ=l. Proposition 5. If the order quantity of resource 2 can always fulfill the demand, then the optimal order quantity of resource 1 can be determined by Eq. (27): n
q1 ¼
lnðNs Þ
ð27Þ
l
where 2
Ns ¼
ðv1 þða þ c2 maÞðlðl =l þ bÞÞÞ lv1 þ c1
Proof. According to expression (4) and the assumptions of the model, the expected loss value of resource 1 is Z q1 Z 1 v1 ðq1 x1 Þf1 ðx1 Þdx1 þ ðl1 ðx1 q1 Þ þc2 Dðg, eÞÞf1 ðx1 Þdx1 þ c1 q1 & 0
q1
ð28Þ
When Eq. (28) is simplified with some algebraic manipulations, we will find that the only difference between the simplified expression of (28) and (13) is the constant m. Therefore, we obtain qn1 ¼ lnðNs Þ=l. Proposition 6. If q1 , the order quantity of resource 1, is given, then the optimal order quantity of resource 2 will be: b=l !! mv2 þ c2 elq1 n ð29Þ q2 ¼ max 0, a 1 mðv2 þ kÞ Proof. When q1 is given, x2 ¼ að1ebðx1 q1 Þ Þ becomes a function of stochastic variables x1 , e. According to Assumption 2 and Eq. (17), the probability density function of x2 will be 1 l f2 ðx2 Þ ¼ elðq1 ðlnð1ðx2 =aÞÞ=bÞÞ R bðax2 Þ
&
ð30Þ
Thus, we can derive the expected loss value of resource 2 from Eqs. (4) and (30) as Z q2 Z 1 v2 ðq2 x2 Þf2 ðx2 Þdx2 þ kðx2 q2 Þf2 ðx2 Þdx2 þ c2 q2 ð31Þ 0
q2
Next, we obtain the first order and the second order derivative of Eq. (31) with respect to q2 , as shown in Eqs. (32) and (33), respectively: q l=b mv2 elq1 þc2 melq1 ðk þ v2 Þ 1 2 ð32Þ
a
lðv2 þ kÞ q2 l=b elq1 m 1 bðaq2 Þ a
ð33Þ
It is obvious that Eq. (33) is positive, indicating that Eq. (31) is convex. Thus, by setting Eq. (32) equal to zero, we have: b=l ! mv2 þ c2 elq1 q2 ¼ a 1 ð34Þ mðv2 þkÞ It can be easily seen from (34) that, if q1 is large enough, q2 may become negative. Therefore, when resource 1 is sufficient, there will be no need for resource 2. Consequently, when given q1 , the optimal order quantity of resource 2 will be b=l !! mv2 þ c2 elq1 qn2 ¼ max 0, a 1 mðv2 þ kÞ
In particular, if q1 ¼ 0, i.e., the order quantity of resource 1 is zero, then the optimal order quantity of resource 2 can be expressed as ! mv2 þ c2 b=l qn2 ¼ a 1 ð35Þ mðv2 þ kÞ We can now summarize the findings that are based on the above analysis in which we considered both the deterministic and stochastic dependent relationship between resource 1 and resource 2. When the dependent relationship is deterministic, Propositions 1 and 2 restrict the upper bound and lower bound of qn1 ; Proposition 3 introduces the optimal value of q2 when q1 is given. Likewise, when the dependent relationship is stochastic, Propositions 4 and 5 define the upper bound and lower bound of qn1 : Proposition 6 introduces the optimal value of q2 when q1 is given. Besides, note that if the supply of resource 2 is scarce in the practice of the emergency management, and the assumption of Propositions 1 and 4 holds, expressions (7) and (24) can assist to determine the optimal order quantity of resource 1. Equally, if the supply of resource 2 is sufficient and the assumption of Propositions 2 and 5 holds approximately, we can use expressions (12) and (27) to attain the optimal order quantity of resource 1.
4. Genetic algorithm based solution methodology and a numerical example In this section, we briefly describe our solution methodology based on genetic algorithms that is embedded within the simulation procedure. In addition, we an actual example is utilized to provide an assessment and validation of the model proposed in this research. 4.1. Design of genetic algorithm-based simulation approach The total expected loss value can be approximated using Monte Carlo simulation as it difficult to solve the proposed mathematical model using the closed-form approach. In general, the optimal solution c be obtained using the enumeration method over domain of independent variables, however, this is a time consuming process. Although, we can acquire the range of the independent variables and restrict the size of the sample space according to the model analysis as described in Section 3, the enumeration method over the sample space is necessitated with a simple simulation approach, and the computing efficiency might not be evidently improved. Consider the use of genetic algorithm (GA), which is a parallel iterative method and has a certain evolutionary learning abilities, which repeats evaluation, selection, crossover, and mutation process after initialization until the stopping condition is satisfied [6,12,16,22,23,35,37]. Therefore, a GA based on stochastic search is developed and substituted for the enumeration method to reduce program runtime. The particular steps of the GA based solution approach are as follows: Step 1. Input parameters, and specify the value of qu1 and ql1 . Here, they represent the upper bound and lower bound of the order quantity of resource 1, and are determined by expressions (7) and (12) if B ¼ A, and determined by expressions (24) and (27) if B 4 A. Step 2. Initialize the GA parameters, such as the size of population, the number of generations, the probability of crossover and the probability of mutation. Specify the bounds of individual in terms of qu1 and ql1 . Step 3. Determine the fitness of each individual in pop; which is the procedure of computing the fitness function as depicted in Fig. 1. And population evaluation. Step 4. If the termination conditions are satisfied, end the algorithm and output the results.
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Fig. 1. Flow chart of algorithm for the computation of the fitness function.
Step 5. Implementing genetic operators such as crossover and mutation over pop. Moreover, the new individuals generated are then saved into popnew. Step 6. According to the union set between popnew and pop, select and update the individuals by roulette wheel selection operator based on individual maintaining strategy and then, save the individuals selected into pop. The determination of the value domain of parameters can be referred to Davis [7]. The termination conditions rely on the maximum generations or the convergence of algorithm. In Fig. 1, when q1 is given, the order quantity of q2 will be decided by Eqs. (17) and (29), respectively. Ds is the step-length of q1 . n is the sample size of the demand for resource 1 when q1 is given. According to Sheng et al. [32], the standard deviation of Zðq1 ,q2 Þ is inversely proportional to the square root of n, i.e., the error of Zðq1 ,q2 Þ decreases as n increases. 4.2. Numerical example In this section, we present numerical tests for a real case that assesses and validates our model. This example is based on the flood emergency in a local district of Shaanxi Province in China. The local emergency management office is in charge of the response and recovery activities associated with the flood
emergency. For this study, we are unable to provide the original data as they are confidential due to the local government regulations. However, we were able to use and implement the existing data for our problem. Therefore, we first fit the original data to the model and obtained the various parameters. Then, we conducted the linear scaling and normalization process associated with the data and parameters. As a result, based on the data relating to floods over the past ten years, the following parameters were generated for our experimental work with a degree of confidence for these values of over 98%:
l ¼ 0:001;
A ¼ 0:6;
B ¼ 1:7;
b ¼ 0:02;
b ¼ 0:01
In Table 1, Loss-D means that the expected loss value was computed using q1 and q2 that were obtained without considering randomness of e, namely ‘‘A ¼ B ¼ 1:15’’ is used in place of A ¼ 0:6 and B ¼ 1:7. The Loss-S was the value of the expected loss using q1 and q2 obtained under randomness of e. It can be noticed that all the Loss-S values are less than the Loss-D values for the five year period. In addition, it is evident that larger error was induced when the mean of e approximates to it. This indicates the importance to consider the stochastic part of the dependent relationship. Fig. 2 shows the changes of quantity ordered for q1 and q2 during the five year period, which implies that the intertemporal dependent relationship between the inventory quantity
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of response resources and that of recovery resources is very complex in emergency management. The value of parameters in Tables 2–6 equals the average values over the past five year period. From these tables, we can observe the following:
As the value of k increases, the order quantity of response resources has no apparent change while the order quantity of recovery resources increases. This shows that, the shortage cost of recovery resources does not have significant impact on
Table 1 lists the other parameters for past five years in the example and their output results. Number of years
c1
v1
c2
v2
a
a
k
Loss-D
Loss-S
The The The The The
26 30 33 38 37
17 19 19 22 20
15 18 22 24 23
4 6 10 10 9
9800 12,000 13,000 15,000 15,000
1000 1200 1300 1350 1400
45 55 59 65 70
59,258 57,377 71,691 79,686 81,048
49,923 55,357 69,561 79,101 79,565
first second third fourth fifth
their order quantity, but can have significant impact on the order quantity of response resources. The augment of the demand quantity of response resources increases its order quantity generally, and while the one demanded is less, the order quantity of the recovery resources may be zero. Nevertheless, from Table 3, we can see that its quantity can be decreased if the amount of recovery resources is properly adjusted. The variant of a has little impact on the order quantity of recovery resources. However, the variant of response resources behaves in the same way. This apparently shows that the degree of the disaster in response period does not affect the order quantity of recovery resources. Only by increasing the order quantity of response resources can reduce the rescue cost. As a increases, the total amount of emergency resources increased. Whereas when a is large enough, the order quantity of recovery resources may become zero and the one of the response resources is significantly increased. This shows that when the demand of recovery resources is very large, we need to increase the order quantity of response resource to reduce the whole rescue cost, rather than the recovery resources. The uncertainty of e increases the loss of emergency resources. The order quantity of response resources should be increased to reduce the impact of e. That is, when the uncertainty of recovery resource demand increases, we can reduce resource cost in the entire operation by increasing the order quantity of response resources.
Above observations suggest that the performance of inventory management can be improved if the relation between response and recovery resources is rational trade-off according to real situation in emergency management. 5. Conclusions From the perspective of the vertical integration of operational process in emergency management, the emergency resources can be classified into response resources and recovery resources. Based on the deterministic or stochastic dependent relationship of the two
Fig. 2. Quantity ordered of q1 and q2 during five years. Table 2 Numerical results as k varies. k
45
51
54
57
60
64
68
72
76
80
q1 q2 ls
525.48 1178.6 62,523
536.50 1185.9 62,736
531.38 1191.5 63,727
525.49 1204.8 64,534
525.50 1256.6 64,644
532.46 1297.6 65,187
547.43 1298.2 66,104
533.13 1318.9 66,357
525.49 1339.3 67,156
542.01 1369.4 67,414
Table 3 Numerical results as l varies.
l
0.0050
0.0046
0.0041
0.0037
0.0032
0.0028
0.0023
0.0019
0.0014
0.0010
q1 q2 ls
342.5495 0 2.3545
388.3488 0 2.3641
434.0659 0 2.7612
450.1790 0 3.0390
485.1030 3.007 3.2893
494.0575 10.987 3.4675
618.3603 30.987 4.0072
717.2042 108.98 4.6452
923.6478 156.09 5.7625
525.4895 1189.3 6.4746
Table 4 Numerical results as a varies. a
10,000
12,222
14,444
16,667
18,889
21,111
23,333
25,556
27,778
30,000
q1 q2 ls
472.97 1298.5 61,211
512.13 1298 63,438
549.82 1297 65,296
586.14 1296 68,247
627.70 1295 70,631
655.05 1292 72,958
687.80 1292 75,012
725.57 1290 77,024
762.08 1287 78,050
809.00 1281 80,070
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Table 5 Numerical results as a varies.
a
1000
1222
1444
1667
1889
2111
2333
2556
2778
3000
q1 q1 ls
167.9 999.9 36,218
269.6 1221.9 46,276
362 1443.7 56,419
455.2 1665 66,009
524.4 1885.7 74,887
596.7 2105.1 84,458
1432.9 0 87,787
1384.4 0 91,678
1395.1 0 95,102
1618.5 0 95,228
Table 6 Numerical results as e varies. A B
0.5 1
0.6 1.8
0.9 2
1.25 2.3
1.4 2.6
1.45 2.9
1.5 3.1
1.7 3.5
1.8 3.7
2.0 4.0
q1 q2 ls
302.92 1299.5 41,680
311.10 1299.5 52,030
327.45 1299.5 59,110
335.63 1291.4 75,240
507.32 1298.1 83,880
417.38 1299 91,570
662.65 1293.9 94,680
654.48 1295.3 103,180
654.48 1294.3 106,990
850.70 1274.1 114,530
kinds of resources, we proposed an inter-temporal integrated singleperiod inventory model for solving order quantity. The model characterized both the deterministic dependent relationship and stochastic dependent relationship between the shortage quantity of response resources and the demand quantity of recovery resources. Contrary to the traditional multi-echelon inventory model, there is a non-linear relationship between the shortage quantity of response resources and the demand quantity of recovery resources in our model. This form is characterized by the exponential function in our model. Furthermore, the demand quantity of recovery resources is not only determined by the shortage quantity of response resources, but also external stochastic factors. Due to the inherent difficulty in obtaining analytical solutions to the model, we introduced a genetic algorithm-based simulation approach based on the analysis of the mathematical properties of the model. A real example demonstrated the significance of vertically integrated emergency management. In addition, the example demonstrated the importance of the consideration of the stochastic part of the dependent relationship between the two types of resources. Some managerial insights were presented based on parameters sensitivity analysis. Specifically, the main contributions of the paper are as follows. From the theoretical aspects of single-period inventory management, the proposed model is an extension of those models based on multi-stage structure. Our analytic results showed that better decision can be achieved as long as the stochastic non-linear relations for demand information between the disparate resources are considered. Meanwhile, in view of genetic algorithm-based simulation, the algorithmic framework can be used for situations that face similar problems. From the practical perspective, our findings illustrated that the necessary emergency resources should be integrated and managed by changing the status quo in situations where different emergency resources are managed by different departments. In addition, the proposed model could be used to develop decision tools for practitioners in emergency management. Based on the assumptions of the relaxation model, future researches may consider proposing models that are more practical in specific types of the calamity. In addition, given characteristics of emergency logistics, research on integrated operational model that involve optimizing inventory, transportation, and facility location would be a valuable direction in the future.
Acknowledgment This research was partially supported by the National Natural Science Foundation of China under Grants 70890080/70890081.
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