A robust optimization model for blood supply chain in emergency situations

Article history: Received April 4 2016 Received in Revised Format April 27 2016 Accepted May 12 2016 Available online May 14 2016 In this paper, a multi-period model for blood supply chain in emergency situation is presented to optimize decisions related to locate blood facilities and distribute blood products after natural disasters. In disastrous situations, uncertainty is an inseparable part of humanitarian logistics and blood supply chain as well. This paper proposes a robust network to capture the uncertain nature of blood supply chain during and after disasters. This study considers donor points, blood facilities, processing and testing labs, and hospitals as the components of blood supply chain. In addition, this paper makes location and allocation decisions for multiple post disaster periods through real data. The study compares the performances of “p-robust optimization” approach and “robust optimization” approach and the results are discussed. © 2016 Growing Science Ltd. All rights reserved


Introduction
Natural disasters like earthquake, flood, and famine cause many problems around the world annually.Indian Ocean earthquake and tsunami on December16, 2004, Yellow River flood in China, on July 10, 1931, Bam earthquake in Iran, on December 26, 2003 and prevalence of Ebola virus in Africa in 2014, are only a few examples of natural disasters.It is obvious that these disasters have an intense impact on the affected areas and create a huge volume of demands there.So, without a precise schematization, rescue operations are not efficient.One of the most useful applications in this respect is mathematical modeling approach that has helped affected countries' governments during natural disasters (Sheu, 2007).First of all, in disaster management, mathematical modeling approach was used for marine disasters in 1980s.After those achievements, researchers have gradually started using a mathematical approach for other emergency situations as a powerful method in disaster management nowadays (Beamon & Kotleba 2006).
Recent disasters have shown that blood supply chain and its effective operation services are affected by outer disruption (Jabbarzadeh et al., 2014).For example, for the case of Bam earthquake, because of improper blood supply chain only 23% of donated blood units were distributed to the affected areas (Abolghasemi et al., 2008).Similarly, Sichuan earthquake in China disrupted blood supply chain, in 2008 (Sha & Huang, 2012).Likewise, during Japan earthquake and tsunami in 2011, also called Great Sendai Earthquake, the blood management system of this country faced many problems (Nollet et al., 2013).The above-mentioned instances demonstrate complexity of blood supply chains, so we need an ingenious design of blood supply chain during natural disasters, because shortage of blood in disasters always increases mortality rate (Pierskalla, 2005).Statistics show blood demand during disasters has unstable rate and dynamic behavior and demand for blood in necessary during the first hours of incidents.On one hand, this dynamic nature of blood demand absolutely increases complexity.On the other hand, because blood products have a short expiration date, and donation rate has a huge doze at the very early hours, special constrains on blood products should be considered and consequently, it results in more complexity (Delen et al., 2011;Tabatabaie et al., 2010).
Therefore, by considering aforesaid uncertain and dynamic nature of blood demand, this study develops a dynamic optimization model by using robust stochastic approach for determining the number and the location of blood facilities, and also specifying inventory levels in hospitals at the end of each period.Blood donors, blood facilities, processing and testing labs, and hospitals have been considered as the components of blood supply chain in this paper.Our objective function seeks to minimize the total cost in this network such as transportation cost, inventory cost and fix cost.While our model has considered real situations, it will help decision makers implement location and allocation decisions during disasters.This paper is organized as follows: The following section briefly reviews related literature.Section 3 presents the robust network model for blood supply during emergency situations.Also, this section defines basic assumptions of the proposed model.Finally, the p-robust model is proposed in the last part of this section.The computational experiments are proposed in section 4, also this section involves sensitivity analysis about proposed models and compares "robust" and "p-robust" models performance.And the last section presents concluding and remarks some directions for future researches in respect.

Literature Review
Despite the fact that there are a lot of studies about dynamic supply chain management and its related problems, blood supply chain has not been explored profoundly and there are numerous research gaps in this problem.Or and Pierskalla (1979) studied partial blood banking for the first time.A literature review paper by focusing on dynamic network analysis was performed by Beliën and Forcé (2012) which relegates blood supply chain's problems and exposes research gaps on the strategic facility location decisions.Also, a review of tactical and operational models focusing on blood gathering and allocated inventory to each hospital was proposed by Pierskalla (2005).This study also reviewed models for allocating donor areas and transfusion centers to community blood centers, specifying the number of community blood centers in a region, locating these centers, and matching supply and demand.Daskin, Coullard and Shen (2002) expanded an integrated approach to determine the location of distribution centers and the amount of allocated inventory to each center.A nonlinear integer programming model for locating the problem of blood supply chain was presented by Shen et al. (2003).This model also considered inventory decisions in a single-period.Cetin and Sarul (2009) developed a model for determining the number and location of blood banks by minimizing total cost and total distance traveled.
In practical blood supply chain area, Şahin et al. (2007), Sha and Huang (2012) and Nagurney et al. (2012) presented location-allocation models with real case study.Şahin et al. (2007) developed a hierarchical location-allocation model in single-period state for Turkish Red Crescent Society.Sha and Huang (2012) presented a deterministic and multi-period model to determine location-allocation decisions of blood facilities.Their case study was about blood supply chain in Beijing earthquake.Nagurney et al. (2012) developed a blood supply chain network for allocating decisions and determining optimal capacity of blood centers.Arvan et al. (2015) presented a bi-objective, multi-product for blood supply chain by using є-constraint method, but their single-period model did not capture uncertainty in blood demand.Eventually, Jabbarzadeh et al. (2014) proposed a dynamic blood supply chain network in emergency situations.Their robust network analyzed existence of potential earthquakes in Tehran, Iran as a real case.This proposed network considered blood donor, blood facilities, and blood centers without processing and testing labs.
According to our study there are many different performance measures that researchers have used.Wastage, backorders, availability, transportation cost and shortage are the most prevalent classes of performance measures.Table 1 shows these categories.In addition, this table demonstrates that different studies have focused on transportation and delivery costs.

Model Formulation
Our blood supply chain network and basic assumptions are presented in this section.According to Fig. 1, donor points, blood facilities, processing and testing labs, and hospitals are components of this fourlayer network.Fig. 1 shows the schematic form of blood supply chain network.Hospitals receive blood products in each period and help injuries during natural disasters.Processing and testing labs receive blood from blood facilities and record, test and process these blood samples and transport them to hospitals.In laboratories the donated bloods will be completely examined and the demand for them will be considered.Blood facilities are responsible for gathering blood from donors, in addition this layer should transport collected bloods to testing labs.Permanent facilities and mobile facilities are considered as two kinds of blood facilities in this model.Permanent facilities cannot move and have larger capacities than temporary facilities.The objective function of the proposed model is to minimize the total cost of blood supply chain under each scenario.By solving the model the following decisions are specified at each period by using a set of scenarios: 3.1 Robust model Mulvey et al. (1995) introduced a robust optimization due to the optimal design of supply chain in the real world and uncertain environments.By expressing the value of vital input data in a set of scenarios, robust optimization tries to approach the preferred risk aversion.This approach results in a series of solutions that are less sensitive to the model data from a scenario set.Two sets of variables act in this approach: control and design variables.The first ones are subject to adjustment once a specific realization of the data is obtained, while design variables are determined before realization of the uncertain parameters and cannot be adjusted once random parameters are observed.Constraints can be divided into two types as well: structural and control constraints.Structural constraints are typical linear programming constraints which are free of uncertain parameters, while the coefficients of control constraints are subject to uncertainty.Now we present our robust model.
Our decisions in this paper are made in two stages.Stage 1 specifies the location of permanent facilities for long periods of time before occurrence of a specific scenario.After that, stage 2 determines the mobile facilities' location and above decisions such as allocation and inventory decisions according to a specific scenario.

Notations
Following indicates, parameters, and decision variables are used for our robust model: Fixed costs of locating a permanent blood facilities at point j q f  Fixed costs of locating a lab at point j The robust model aims to minimize total costs of blood supply chain under each scenario.Total costs (TOTC) consist of fixed cost (FC), moving cost of mobile facilities, operational cost (OC), transportation cost (TC), and inventory cost (IC).These costs have been shown as follows: The mathematical model can be formulated as follows: , , , , , , , , , , , , , , , , , , Eq. ( 1) shows the objective function that minimizes total costs.As it has been stated above, this objective function consists of fixed cost, moving cost, operational cost, transportation cost, and inventory cost.Eq.
(2) prevents locating more than one facility at each point.Eq. (3) shows that a mobile facility cannot move from a point where no facility has been located in its previous period.Eq. ( 4) enforces donors cannot be assigned to unopened facilities.Eq. ( 5), Eq. ( 10), and Eq. ( 14) clarify coverage radius restriction.Eq. ( 6) determines inventory level and also unsatisfied demand for each product at hospitals.Eq. ( 7), Eq. ( 8), and Eq. ( 15) ensure blood and its products can be transported according to correct assignment.Eq. ( 9) shows the capacity of each donor.Eq. ( 11) clarifies maximum capacity of mobile and permanent facilities.Eq. ( 12) asserts a lab can be assign to a hospital if this lab is located.Eq. ( 13) balances input bloods and output products.Eq. ( 16) expresses each demand product of each hospital, at least partially, should be satisfied.Eq. ( 17) limits transportation time of blood supply.Eq. ( 18) illustrates maximum capacity of each hospital for each product.Eq. ( 19) explains capacity of each lab to hold donation bloods.Eq. ( 20) is an auxiliary equation based on what Yu and li (2000) have proposed.Eq. ( 21) defines binary and positive decision variables.

p-Robust model
The proposed model in the previous part determines location and allocation decisions for preparedness phase in disaster management.Location decisions consist of specifying mobile and permanent facilities and processing labs.Allocation decisions involve assignment of blood facilities to donor points, processing labs to blood facilities, and hospitals to processing labs.Here we complete this model to be more practical in real world.As it is stated in the previous section many studies such as (Jabbarzadeh et al. 2014) assumed facilities, labs, and hospitals remain unaffected during disasters, however, it is obvious these sites may be located on the faults and consequently may be affected during an earthquake.So we used Mont-Carlo simulation to generate scenarios and p-robust method to solve these problems for respond phase in disaster management.We assume two different events can occur after an earthquake: blood facilities or processing labs disruption.The method of generating scenarios for affected sites has been shown in Fig. 2.
To introduce the robustness measure we use in this paper, let E be a set of scenarios.Let (Pe) be a deterministic (i.e., single-scenario) minimization problem, indexed by the scenario index e.(That is, for each scenario e ∈ E, there is a different problem (Pe)).The structure of these problems is identical; only the data is different.For each e, let z*e be the optimal objective value for (Ps); we assume z*e >0 for each e.The notion of p-robustness was first introduced in the context of facility layout (Kouvelis et al., 1992) and used subsequently in the context of an international sourcing problem (Gutierrez and Kouvelis 1995) and a network design problem (Gutiérrez et al., 1996).
Let p ≥ 0 be a constant.Let X be a feasible solution to (Ps) for all e ∈ E, and let z*e (X) be the objective value of problem (Ps) under solution x. x is called p-robust if for all e ∈ E, The left-hand side of the Equation above is the relative regret for scenario e; the absolute regret is given by z*e (X) -z*e (Snyder & Daskin 2006).
.( 2) .(21) The constraints of the above model are the same as the robust model's constraints, however, based on new definition on some variable, these constraints consider each scenario e E  .
Model 2 is solved for each scenario and the optimum value of the objective functions named Z*e.
According to p-robust method, the effect of each scenario must be involved in the optimum structure of the blood supply network.So Model 3 is used to build the network.
Eq. ( 33) is the p-robust model's objective function which considers all scenarios e E  .Eq. ( 36) enforces, for each scenario, the costs cannot be more than 100(p +1) % of its optimal costs Z*e (value of p is related to the necessity of its scenario).Other constraints are the same as model 1 and 2.

Computational Result and Discussion
Because of the strategic and geographical location of Iran, and owing to the fact that 90 percent of Iran is located on faults, earthquakes have always been the most devastating disaster in the country among other natural disasters.Tehran, as a strategic city in Iran, has always been exposed to such disasters.
Regarding earthquakes, Tehran is considered a dangerous region (8 to 10 Mercalli scales).The fault in the north of Tehran is the biggest fault of the city located in the south foothill of Alborz ranges and in the north of Tehran.This fault starts in Lashkarak and Sohanak, continues in Farahzad and Hesarak, and continues towards the west.This fault encompasses Niavaran, Tajrish, Zaferanieh, Elahieh, and Farmanieh on its way.The necessity of paying attention to crisis management is an obvious issue regarding the dangerous and risky situation of Tehran (Sabzehchian et al. 2006).Fig. 3 shows 22 districts in Tehran which also shows donors' locations in this large city.By using the population of each district and the average blood donation rate of 22.05 unit per 1000 population, donation capacity of each district (mi) can be estimated (Torghabeh et al., 2006).Centers of districts have been considered as potential sites for permanent blood facilities.The information about districts' location and their donation capacity is derived from Jabbarzadeh et al. (2014).Potential locations of processing labs are shown in Fig. 3.These potential sites are in districts of 2, 4, 9 and 14.According to Jabbarzadeh et al. (2014) the fixed cost of permanent facilities in Tehran is about $1518.23; in addition, the unit of operational cost of blood products is about $ 0.07 and finally, the capacity of permanent and mobile facilities are 2500 and 700.The cost of moving in of the temporary facilities in the first period is about $ 322.98 and the moving cost of the second period is derived from (Jabbarzadeh et al., 2014).According to Daskin et al. (2002) the unit of inventory cost of blood is about $1. Unit of blood transportation cost between facilities and labs and hospitals is $2.35.Coverage radius for blood facilities, labs, and hospitals are 9, 15, and 21 kilo meters.
The fixed cost for processing labs is $1990.In addition, we assume the average velocity for transporter vehicles is 60 km/h.The maximum capacity for each blood product in each lab is 550.The time that blood remains in each facility is 10 hours and the time that blood products detain in labs is 32 hours.The time window for blood supply is 70 hours.
According to Tabatabaie et al. (2010) and Jabbarzadeh et al. (2014) and generating numbers, we define earthquake scenarios and estimate the demand for blood products for each hospital in two periods.These demands have been shown in Table 2.We assume during an earthquake that, the first period demand for blood products is more than the second one, also we suppose all scenarios have equal possibilities.
Latitude and longitude of hospitals are shown in Table 3. Distance between the two points can be calculated by the following equation.

Table 4
Selected mobile facilities under each scenario at each period Table 5 shows allocation of donors to mobile and temporary facilities under each scenario at each period.This table demonstrates one donor can be assigned to more than one facility.In addition, it has been concluded the number of located facilities in the first period are more than the second one.Table 6 shows gathered blood from donor points at first period in each mobile and permanent facilities.Table 7 describes quantity of bloods that is transported from blood facilities to processing labs at first period under each scenario.Finally, Table 8 and Table 9 show the quantity of blood products transported from lab4 and lab9 to each hospital at each period under each scenario.Unsatisfied demand can be calculated from Table 8.

Table 9
Quantity of blood products transported from lab4 and lab9 to each hospital at each period under each scenario Fig. 5. Tradeoff between minimizing total costs' objective and maximizing satisfied demands In this section, we explored a trade-off between minimizing total costs' objective and maximizing satisfied demand by changing in ω parameter.A decision maker who risks the shortage of blood likes a higher value of ω because of a lower cost.However, another decision maker who doesn't risk the shortage of blood prefers a lower value of ω.Fig. 5 helps decision makers find the best decision by choosing their favorite ω.With the increase in value of ω, total costs will increase and unsatisfied demands will decrease.For example at ω=50 total cost is $64431.At ω=100 total cost is increased to $75176 because reduction in unsatisfied demands.
This section proposes a sensitivity analysis of the vital parameters for deterministic models.The first parameter for this purpose is the demand of blood product p at hospital k in period t, that is shown by Dpk t .Based on Arvan et al. (2015) blood demand depends on diverse factors such as population, age, gender, unpredictable events and so on.Table 10 shows a uniform distribution used in sensitivity analysis of demand for each product.Fig. 6 demonstrates the variation of the objective function in the deterministic model by changing the amount of blood demand product p at hospital k in period t.This figure shows a sudden increase in the mentioned objective function.By increasing the blood demand the model is unable to satisfy all demands.Consequently, shortage cost promotes noticeably and results in this sudden increase.The maximum donation capacity of each donor is the second parameter that is used for sensitivity analysis.Donation capacity is a random variable because it depends on different elements such as population and this means it could have a noxious impact on the blood supply during disasters.So, being attentive to this parameter's changes would be advantageous for decision makers.To analyze these changes the uniform distributions of donation capacity have been proposed in Table 11.Fig. 7 displays the changes of objective function of deterministic model by variation in the donation capacity.Abrupt increase in this figure can be explained by the same reason that was mentioned for blood demand.Based on the proposed flowchart in Fig. 2, three permanent facilities will be down during earthquake, these facilities for special scenario are: F2, F9, and F15.Values of the objective functions for four scenarios are seen in Table 12.As stated before, these values go into the p-robust model as Ze* parameter.
According to these scenarios and Table 5-10 the objective function of the third model is $69034.To evaluate both the p-robust and robust models two performance measures are used: the mean and the standard deviation of the objective function under random realizations.Additionally, we vary the probust parameter between [0 1] and calculate mean and standard deviation for p-robust and robust models.The results show the p-robust model gained the solutions with both higher quality and lower standard deviation than robust model for fixed, moving, operational, transportation and inventory costs.
In most problems, the p-robust approach dominates the robust model with respect to the mean of the cost objective function value and its standard deviation.These results are seen in Table 13.However, because of simulation in two cases the mean of the robust model is better than the p-robust model which are shown with a different color.The results imply that the p-robust strategy has a better performance in low values for p-robust parameters.As seen in Fig. 6, when p-robust parameter increases the mean of the objective function of p-robust model is closer to this objective in robust model.To determine the sensitivity of the objective functions' value to variations in robust parameter, sensitivity analysis experiment is performed.Fig. 8 shows the sensitivity of the proposed model's objective functions to Total cost -$ variations in robust parameter.Based on the proposed model with increasing robust parameter, feasible region increases.Therefore, we expect that the increasing of the mentioned parameter improves both objective functions.

Conclusion and Future Research
In this paper a robust model for blood supply chain was presented in emergency situations to minimize total cost.This model determined location and distribution decisions for an uncertain environment and a multi-period network.The location decisions consist of the number and location of temporary and permanent blood facilities, and the number and location of laboratories.Distribution decisions involve the quantity of transported blood between the components.In order to improve the application of the model against unforeseen events and possible disruption among routes, a p-robust approach was used.To evaluate the application of the robust model, real data was applied and location-allocation decisions were determined.We presented different sensitivity analysis experiments from which important implications were drawn.For example, we demonstrated how the total cost of the supply chain can be balanced against unsatisfied demands.In addition, we showed how donation capacity and demand rate effect the objective function.In the last part of our numerical example, we compared the "robust" and "p-robust" models' performance by their objective functions' mean and standard deviation.The results explained that "p-robust" model dominated the "robust" model.This comparison also showed "p-robust" model performance is far better in lower levels of the p-robust parameter.
In brief, our contributions can be summarized as follows: 1. We developed a multi-period robust model for blood supply chain which captured uncertainty in the value of some input data.2. The proposed model consists of all components in a given blood supply chain which are donors, blood facilities, laboratories, and hospitals.3. We developed a p-robust model to consider possible damages among routes after earthquake occurrences.
Like other studies our paper is not without any defection.In this paper we just considered one decision maker which controlled the whole supply chain, in the real world, however, different Decision Makers paly roles in such a supply chain which has different goals.Future research can study this situation by using multi-level programming.Also, presenting a new solution technique which can solve the model within a reasonable length of time, can be a logical set point for future researches.Finally, considering more objective functions will be advantageous for this blood supply chain.Because of the emergency situation, considered in this paper, a reliable network is important.Developing a model which increases the reliability of the model helps decision makers more.

Fig. 1 .
Fig. 1.Schematic form of blood supply chain network

Fig. 3 .
Fig. 3. Districts of Tehran and potential sites of processing labs

Fig. 7 .
Fig. 7. Sensitivity of total costs by variation the donation capacity

Fig. 8 .
Fig. 8. Sensitivity of the proposed model to variations in robust parameter Unit of operational costs of gathering blood at hospital k from lab q in period t under scenario s ts jl v Cost of moving mobile blood facility from point l to point j in period t under scenario s ts ij O Unit of operational costs of gathering blood at point j from donor i in period t under scenario s ts jq O Unit of operational costs of gathering blood at lab q from point j in period t under scenario s ij d Distance between point j and donor i qk d  Distance between hospital k and lab q jq d Distance between lab q and point j k h Unit of inventory cost at hospital k ts i m Maximum donation capacity of each donor i in period t under scenario s kp u Total capacity of hospital k to hold blood product p j T Duration which bloods remain in point j q T  Duration which blood products remain in lab q ts j C Capacity of a permanent blood facility at point j in period t under scenario s ts j b Capacity of a mobile blood facility at point j in period t under scenario s ts q Cbb Capacity of lab q in period t under scenario s s p Possibility of scenario s occurrence TT Maximum time that blood products should be arrived in hospitals V Average velocity of transportation vehicles M A very large number ts kp D Demand of blood product p at hospital k in period t under scenario s Decision variables j X If a permanent facility is located in point j equal to 1, otherwise 0 q Y If a lab is located in point q equal to equal to 1, otherwise 0 ts ij y If point j is assigned to donor i in period t under scenario s equal to 1, otherwise 0 ts jq y If lab q is assigned to point j in period t under scenario s equal to 1, otherwise 0 ts qk y If hospital k is assigned to lab q in period t under scenario s equal to 1, otherwise 0 ts jl Z If a mobile blood facility is located at point l in period t-1 and moves to point j in period t equal to 1, otherwise 0 ts ijq Q Quantity of gathered blood at point j from donor i and transported to lab q in period t under scenario s ts qkp Q Quantity of transported blood product p in lab q to hospital k in period t under scenario s ts kp I Quantity of blood product p in hospital k in period t under scenario s ts kp  Unsatisfied demand of blood product p in hospital k in period t under scenario s

Table 5
Allocated donors to mobile and permanent facilities

Table 6
Quantity of gathered bloods in each facility at first period under each scenario

Table 7
Quantity of gathered bloods in each processing lab at each period under each scenario

Table 10
Uniform distribution used in sensitivity analysis of demand for each product

Table 11
Uniform distribution used in sensitivity analysis of donation capacity

Table 12
Values of the objective functions for four scenarios

Table 13
Summary of test results of the second objective function value and the standard deviation of both models