Robust stochastic multi-choice goal programming for blood collection and distribution problem with real application

Abstract

Blood transfusion services are a vital section component of the healthcare system all over the world. Literature on studying and modeling of these systems is surprisingly sparse. In this paper, we expand a generalized network optimization model for the complex supply chain of blood, which is a regionalized blood bank system. In this paper; the purpose of blood is red blood cells (RBC). This system consists of collection sites, testing and processing facilities, storage facilities, distribution centers, as well as points of demand, which are classically, include hospitals. Our major contribution is to develop a novel Hybrid stochastic programming, multi-choice goal programming and robust optimization (SMCGR) approaches to simultaneously model two different types of uncertainties by including stochastic scenarios for total blood donations and polyhedral uncertainty sets for demands. Real numerical studies are implemented to verify our mathematical formulation and also show the benefits of the SMCGR approach. The performance improvements achieved by the valid inequalities and Pareto-optimal cuts are demonstrated in real world application.

Keywords

Blood supply chain multi-choice goal programming uncertainty condition robust optimization

1 Introduction

Blood is critical for survival of the human beings while carries vital substances into cells within the body. It consists of two major components: a) plasma, which is approximately 55% of the total amount of blood, and b) cellular elements. No chemical composite can be replaced with blood. Thus, blood transfusion is required from one person to another in several life-threatening circumstances, such as organ transplantation, childbirth, treatment forcancer, leukemia, and anemia [1]. The aim of the blood supply chain is supplying acceptable safe blood to hospitals. It is paramount that blood is accessible at hospitals for transfusion purposes since a shortage may endanger the life of patients [2].

An optimal blood collection and distribution network could be organized by using mathematical formulation in order to optimize the blood supply chain network design (BSCND) problem [3].

In Iran, blood can be collected by mobile teams (MTs), blood collection centers (BCCs), blood collection and processing centers (BCPCs) and blood transfusion center (BTC).MTs and BCCs only collect the donated blood. BCPCs in addition to blood collection, also attempted to prepare blood products. But they don’t blood tests. BTCs located in center of every province.

Details of the inbound process of blood donation and other steps of collection are shown in Fig. 1. Firstly, donors refer to MTs, BCCs, BCPCs and BTC for donation. This step is accompanied by filling registration form; undergoing a short examination and donation. Secondly, the collected bloods in MTs and BCCs are transferred to the BCPCs or BTC at the end of defined period. BCCs and MTs reduce distance with donors, but they have fewer capacities. Thirdly, one blood tube sample is sent to BTC for several tests such as HIV and Hepatitis. In Iranian blood transfusion organization (IBTO) BCPCs and BTC are responsible for blooddistribution.

Fig.1

Iran’s blood supply chain network.

BCPCs should be send tube sample of collected blood for BTC and receive online results. It should be noted that blood bags and tube samples from BCCs are sent by an equipped vehicle to the BCPCs to prevent corruption of collectedblood.

Zahiri et al. developed blood collection management, but in their article BTC, BCPCs are not considered. In this paper allocation of MTs, BCCs to BCPCs and BCPCs to BTC is determined[4].

This paper contributes to the BSCND design literature by developing a novel Hybrid stochastic programming, multi-choice goal programming and robust optimization (SMCGR) approaches and also devising an efficient solution procedure. Specifically, a mathematical model is developed for amulti-period, single-product and capacitated BSCND. The strategic decisions including locations of facilities as well as the tactical decisions including blood collection centers and coverage these centers by BTC are determined to minimize the expected costs. The major contributions can be summarized asfollows:

A novel BSCND design model as a mixed-integer linear program (MILP) to integrate both strategic and tactical decisions with flexibility to cover varying proportions of demands and blood donors based IBTO structure.

SMCGR approach to simultaneously model two different types of uncertainties including stochastic scenarios for total blood donations and polyhedral uncertainty sets for demands.

The reminder of the paper is organized as follows. Section 2 reviews the relevant literature, briefly. The proposed model and its mathematical formulation are developed in Section 3. The case study is presented in Section 4. Computational results are presented in Section 5. Section 6 concludes the paper and proposes the future research.

2 Literature review

In this section the Literature review of Goal programming (GP) approach and BSCND is discussed.

2.1 Goal programming approach

GP has been used widely in literature for solving different multi-objective optimization problems. This method was firstly introduced by Charnes and Cooper for solving various types of multi-objectiveproblems [5]. Most of decision makers are interested in selecting lower values for aspiration level. Therefore, multi-choice version of GP technique (MCGP) was firstly introduced by Chang was firstly developed to tackle with this problem [6]. The main reasons behind using this technique is to consider different aspiration levels for the goals and avoid assigning lower values to each goal’s aspiration level. This technique is chiefly designed to consider single and multiple aspiration levels for local and global areas in order to find global optimal solution. Overall structure of this technique is presented as follow: $\begin{matrix} Min \sum_{i = 1}^{n} w_{i} (d_{i}^{+} + d_{i}^{-}) \\ S . T; \\ h_{k} (x) = (⩽ or ⩾) 0 \forall k = 1, 2, \dots, q \\ f_{i} (x) + d_{i}^{+} - d_{i}^{-} = \sum_{j = 1}^{m} b_{ij} s_{ij} (B) \forall i = 1, 2, \dots, n \\ x \in X \\ d_{i}^{+}, d_{i}^{-} ⩾ 0 \forall i = 1, 2, \dots, n \\ s_{ij} (B) \in R_{i} (x) \forall i = 1, 2, \dots, n \end{matrix}$ (1)

Where, ith goal’s jth aspiration level, function of binary serial number and function of resource boundaries are respectively shown by b _ij, s _ij (B) and R _i (x). Moreover, overall structure of two other MCGP based mathematical models proposed by Chang is[6].

The less is better $\begin{matrix} Min \sum_{i = 1}^{n} w_{i} (d_{i}^{+} + d_{i}^{-}) + α_{i} (e_{i}^{+} + e_{i}^{-}) \\ S . T; \\ h_{k} (x) = (⩽ or ⩾) 0 \forall k = 1, 2, \dots, q \\ f_{i} (x) + d_{i}^{+} - d_{i}^{-} = y_{i} \forall i = 1, 2, \dots, n \\ y_{i} + e_{i}^{+} - e_{i}^{-} = b_{i min} \forall i = 1, 2, \dots, n \\ b_{i min} ⩽ y_{i} ⩽ b_{i max} \\ x \in X \\ d_{i}^{+}, d_{i}^{-}, e_{i}^{+}, e_{i}^{-} ⩾ 0 \forall i = 1, 2, \dots, n \end{matrix}$ (2)

The more is better $\begin{matrix} Min \sum_{i = 1}^{n} w_{i} (d_{i}^{+} + d_{i}^{-}) + α_{i} (e_{i}^{+} + e_{i}^{-}) \\ S . T; \\ h_{k} (x) = (⩽ or ⩾) 0 \forall k = 1, 2, \dots, q \\ f_{i} (x) + d_{i}^{+} - d_{i}^{-} = y_{i} \forall i = 1, 2, \dots, n \\ y_{i} + e_{i}^{+} - e_{i}^{-} = b_{i max} \forall i = 1, 2, \dots, n \\ b_{i min} ⩽ y_{i} ⩽ b_{i max} \\ x \in X \\ d_{i}^{+}, d_{i}^{-}, e_{i}^{+}, e_{i}^{-} ⩾ 0 \forall i = 1, 2, \dots, n \end{matrix}$ (3)

Where, continuous variable, lower and upper bound of ith aspiration level, positive and negative deviations defined as |y_i - b_imax| in less is better model and |y_i - b_imax| in more is better formulations and weight of sum of positive and negative deviations are respectively presented by y_i, b_imin, b_imax, $e_{i}^{+}$ , $e_{i}^{+}$ and α_i.

MCGP technique has been used in literature for modeling different real world multi-objective problems. Ustun developed a conic scalarizing based MCGP model [7]. The model had two main contributions including reducing auxiliary constraints and additional variables and ensuring obtaining global optimal solution that can guaranty feasibility of obtained solutions. They presented some examples and test problems to show usefulness of the proposed model. Paksoy and Chang developed a MCGP model for multi-period multi- stage supply chain problem [8]. The proposed mixed integer programming formulation aimed to minimize three different goals including transportation costs, setup costs and inventory costs. They used LINDO software to solve proposed model. Bankian-Tabrizi, Shahanaghi, and Saeed Jabalameli developed a new formulation for fuzzy MCGP technique [9]. The proposed formulation can be used both for modeling real world problems and giving understanding about the solutions of the recently developed fuzzy MCGP problems. Pal and Kumar developed a revised MCGP technique for modeling and solving economic environmental power generation and dispatch problems [10]. They converted proposed nonlinear model into a linear model and used GP technique for solving the developed linear model. Patro, Acharya, Biswal, and Acharya used two methods of Vandermonde’s interpolating polynomial binary variables and the least square approximation method to develop a novel mathematical formulation of MCGP problem. They used LINGO 11 software to solve proposed model and find the model’s global optimalsolutions [11].

2.2 BSCND

Blood product management is one of the very important and challenging healthcare problems. Belien and Force categorized blood supply chain management studies. Their review article indicates the increasing status of this subject [12]. Pieskalla’s article is thorough analysis on the supply chain operations of blood banking [13].

Several authors have used integer optimization models such as facility location, allocation, set covering, and routing to deal with the optimization/design of supply chains of blood or other perishable critical products (see [14 –17]). In addition, inventory management techniques (see [18]), as well as simulation methods (see [19, 20]) have been used to handle blood banking systems. Haijema et al. used a Markov dynamic programming and simulation approach with data from a Dutch blood bank [21]. The nature of the blood supply chains established across the world is not consistent. Quaranta et al differ in the structure of hospitals, the type of supply, pricing for blood, the distribution of blood, and the handling of shortages In other hand, efficient management of blood supply chain and logistics can reduce costs [22]. In addition to reducing costs, blood tests to prevent transmission of disease are very important. Transfusion-transmitted infections have been a worry, especially in the 1980 s with the recognition of transfusion-associated HIV and the risk of transmission of hepatitis C virus [23].

For easily by Table 1 provide the related research papers in the literature based on the characteristics of supply chain design network (SCDN) and BSCDN problems. We explain the columns of Table 1 in line with the contribution of this research as follows:

Table 1
A classification of BSCND problems

References Modeling approach Time-period Solution technique Uncertainty Capacitated facilities Collection centers Testing centers Single Multi Exact simulation heuristic Meta-heuristic D FP SP RP

[34] – – – – √ – – – – – – – √ –

[16] BNLGP √ – √ – – – √ – – – – √ –

[35] MINLP – √ – – √ – √ – – – – √ –

[36] – – – – √ – – – – – – – √ –

[37] MILP – √ √ – – – – – √ – √ √ –

[30] MILP – √ √ – – – – – – √ √ √ –

[38] MILP – √ – √ – – – – – – – √ –

[4] MILP – √ √ – – – – √ – √ √ √ –

[39] MILP – √ – √ √ – – – – – √ – √

[40] MILP – √ √ – – √ – – √ – – – –

Our work MILP – √ √ – – – – – √ – √ √ √

References	Modeling approach	Time-period	Solution technique	Uncertainty	Capacitated facilities	Collection centers	Testing centers	Single	Multi	Exact	simulation	heuristic	Meta-heuristic	D	FP	SP	RP
[34]	–	–	–	–	√	–	–	–	–	–	–	–	√	–
[16]	BNLGP	√	–	√	–	–	–	√	–	–	–	–	√	–
[35]	MINLP	–	√	–	–	√	–	√	–	–	–	–	√	–
[36]	–	–	–	–	√	–	–	–	–	–	–	–	√	–
[37]	MILP	–	√	√	–	–	–	–	–	√	–	√	√	–
[30]	MILP	–	√	√	–	–	–	–	–	–	√	√	√	–
[38]	MILP	–	√	–	√	–	–	–	–	–	–	–	√	–
[4]	MILP	–	√	√	–	–	–	–	√	–	√	√	√	–
[39]	MILP	–	√	–	√	√	–	–	–	–	–	√	–	√
[40]	MILP	–	√	√	–	–	√	–	–	√	–	–	–	–
Our work	MILP	–	√	√	–	–	–	–	–	√	–	√	√	√

Uncertainty: Some research papers such as Uster et al and Ozceylan and Paksoy [24] suppose that all the parameters in SCND are known with certainty. Uster et al expand a model for deterministic multi-product CLSCND problem [25]. Some researchers like Listes embed uncertainty into the optimization model while others consider fuzzy parameters [26]. Bidhandi and Yusuff present a two-stage linear programming model for multi-product and single-period forward SCND problem; they suppose customer demand, operational cost, and capacity of the facilities as stochastic parameters [27]. Roghanian and Kamandanipour expand a fuzzy-random programming model for forward/reverse SCND problem; the authors consider the demand and rate of return as fuzzy random variables [28]. In BSCDN problems zahiri develop robust possibilistic programming for Blood collection management [4]. Based on the type of uncertainty programming approach used to deal whit uncertain data, this column including deterministic (D), fuzzy programming (FP), stochastic programming (SP) and robust programming (RP).

Capacitated facilities: Considering uncapacitated facilities is a simplifying assumption of conventional research papers. In order to make this assumption many researchers suppose capacitated facilities and assign a maximum capacity level to them [29]. But in BSCND Using vehicles to transport blood bags from BCCs to BCPCs and transport blood samples from BCPCs to BTCs are essential.

Time-period: This paper considers flow between facilities in multi-period context, which is more realistic than single period statement because the decisions made in each period change the subsequent decisions in the next periods; this assumption is seldom considered in the literature. In BSCDN area, some authors like Jabbarzadeh et al. [30], Zahiri et al.[4] consider the problem in multi-period context.

Solution technique: High implementation cost of SCND and BSCND decisions emphasizes the importance of accurate decision making. Therefore, using exact solution methods to solve SCND and BSCND problems is more efficient than solutions with even minor errors. Some authors such as Listes [26], Hasani et al. [31] propose exact methods like branch-and-cut and branch-and-bound to solve the problem. In other hand Jyrki et al. [32] and Blake and Hardy [33] apply simulation to solve the BSCND problem. Some researchers apply meta-heuristic methods such as Genetic Algorithm (GA), Tabu Search (TS), and Memetic Algorithm (MA) to solve the problem. Table 1 categorizes the related research papers in the literature based on the characteristics of the BSCND problems.

3 Model formulation

Most BSCND models in the literature are trying to minimize the total costs. However, one of our concerns in this research is to minimize time blood transfusion.

The main concern of this paper is to design the BSCND in the presence of uncertainty. Two unlike types of uncertainty are present; one for total blood donation and the other for totaldemands.

Figure 2 shows that the first, stochastic model of collecting and testing of blood provided. In step 2,with multi-choice goal programming, bi-objective model is conserved to single objective. Then in step 3 robustness model is performed.

Fig.2

Geographical of EAPI.

Figure 2: modeling Process

Step 1. Present two-stage stochastic bi-objective model

The supply chain under study comprises blood donors, MTs, BCCs, BCPCs and BTC. The location of MTs may vary from one period to another. But BCCs, BCPCs and BTCs have fixed location. Blood can be donated at every four centers within a certain geographical distance. The blood collected in MTs and BCCs is transferred to BCPCs and or BTC where the blood transfusion process is completed. These centers will then fulfill the blood demand of hospitals and medical centers. BCPCs are capable of providing all transfusion processes and services except QA (Quality Assurance) and QC (Quality Control) departments’ and viral testing is performed in BTC. MTs, BCCs may not present a full range of services and only blood collection could be performed in BCCs.

The problem is formulated as a bi-objective stochastic model to design a blood supply chain resilient to different demand scenarios. The first objective minimizes the total supply chain costs including the cost of establishing BCCs and BCPCs cost, relocating MTs cost, operational cost and delivering blood cost. The second objective minimizes the average delivery time from MTs and BCCs to BCPCs and BTCs. The model tries to decide the following decisions at each period of the planning horizon:

The location of MTs, BCCs, BCPCs under each scenario.

The quantity of donated blood to be collected at each facility under each scenario.

The quantity of donated blood to be transported from MTs, BCCs to BCPCs or BTC under each scenario.

The quantity of blood tube samples to be trans-ported from BCPCs to BTC under each scenario.

How to cover of MTs and BCCs by BCPCs and BTC.

The problem is formulated as a two-stage stochastic MILP model [41]. In the two-stage model, the decision variables are classified in two groups: the first and second stages, as follows.

First stage: certainty in strategic decisions

First stage decision variables are independent from scenarios. It means that such decisions are made before realization of uncertain parameters. These variables are related to the decisions regarding location and allocation of BCPCs toBTCs.

Second stage: uncertainty in tactical/operational decisions. Second stage decision variables are based on scenarios. In other words, after deciding about the first stage decision variables, a random event occurs which affects the second-stage decisions. First, the random parameters including demand and unit operational cost of blood collection by BCPCs and BCCs are realized. Then, based on the decisions made at the first stage, we decide about the amount of shipments between MTs and BCCs, BCPCs and BCCs in the second stage.

Indices

I Index of categories of donors (i = 1, 2 …, I)

J Index of candidate locations for MTs (j = 1, 2 …, J and J1, J2 ∈ J)

K Index of candidate locations for BCCs (k = 1, 2 …, K)

L Index of candidate locations for BCPCs (l = 1, 2 …, L)

M Index of available locations for BTCs (m = 1, 2 …, M)

T Index of time periods (t = 1, 2 …, T)

S Index of scenario (s = 1, 2 …, S)

Parameters

cm_j₁,j₂ Cost of moving a MT from location j₁ to j₂ in the two next periods

cl1_k Cost of locating a BCCs in candidate location k

cl2_l Cost of locating a BCPCs in candidate location l

cd1_jlt Unit cost of delivering blood pack from a MT located in j to BCPC located in l at period t

cd2_klt Unit cost of delivering blood pack from a BCCs located in k to BCPC located in l at period t

cd3_jmt Unit cost of delivering blood pack from a MT located in j to BTC located in m at period t

cd4_kmt Unit cost of delivering blood pack from a BCCs located in k to BTC located in m at period t

cd5_lmt Unit cost of delivering blood sample from a BCPC located in l to BTC located in m at period t

oc1_jt Unit operational cost of collecting blood by MT in location j from donors at period t

oc2_kt Unit operational cost of collecting blood by BCCs in location k from donors at period t

oc3_lt Unit operational cost of testing blood by BCPCs in location l at period t

oc4_mt Unit operational cost of testing blood by BTC in location m at period t

d1_ij Distance between the center of category i of donors and candidate MT j

d2_ik Distance between the center of category i of donors and candidate BCC k

d3_il Distance between the center of category i of donors and candidate BCPC l

d4_im Distance between the center of category i of donors and candidate BTC m

d5_jl Distance between the of candidate MT j and candidate BCPC l

d6_jm Distance between the of candidate MT j and candidate BTC m

d7_kl Distance between the of candidate BCC k and candidate BCPCs l

d8_km Distance between the of candidate BCC k and candidate BTC m

d9_lm Distance between the of candidate BCPC l and candidate BTC m

cr1₀ Coverage radius of MT (if d_ij < cr1₀, i covered by j)

cr2₀ Coverage radius of BCCs (if d_ik < cr2₀, i covered by k)

cr3₀ Coverage radius of BCPCs (if d_il < cr3₀, i covered by l)

cr4₀ Coverage radius of BTC (if d_im < cr4₀, i covered by m)

cr5₀ Coverage radius of BCPC (if d_jl < cr5₀ or d_kl < cr5₀, j or k covered by m)

cr6₀ Coverage radius of BTC (if d_jm < cr6₀ or d_km < cr6₀ or d_lm < cr6₀, j or k or l covered by m)

t1_ijts Time between the center of category i of donors and candidate MT j at period t, under scenario s

t2_ikts Time between the center of category i of donors and candidate BCC kat period t, under scenario s

t3_ilts Time between the center of category i of donors and candidate BCPC lat period t, under scenario s

t4_imts Time between the center of category i of donors and candidate BTC mat period t, under scenario s

t5_jlts Time between the of candidate MT j and candidate BCPC lat period t, underscenario s

t6_jmts Time between the of candidate MT j and candidate BTC m at period t, under scenario s

t7_klts Time between the of candidate BCC k and candidate BCPCs l at period t, under scenario s

t8_kmts Time between the of candidate BCC k and candidate BTC m at period t, under scenario s

t9_lmts Time between the of candidate BCPC l and candidate BTC mat period t, under scenario s

ca1_j Capacity of MT j in period t under scenario s

ca2_k Capacity of BCC k in period t under scenario s

ca3_l Capacity of BCPC l in period t under scenario s

ca4_m Capacity of BTC m in period t under scenario s

$B {\hat{d}}_{its}$ Nominal volume of Blood donation for category i of donors in period t and under scenario s

$T {\hat{d}}_{ts}$ Nominal total demand of blood in period t and scenario s

pc^Bd Penalty cost per unit of uncollected blood of donors

pc^Td Penalty cost per unit of non-satisfied demands

sc^Bd Waste cost per unit of excess amounts of flow over blood collected from donors

sc^Td Surplus cost per unit of excess amounts of flow over demands

PT_ts Maximum time that blood can be used before perishing in period t under scenario s

π_s probability of scenario s occurrence

Mc total acceptance of blood donated by donors

M′ A large number

Variables

x1_ijts 1; if category i of donors is assigned a MT j in period t and under scenario s, and 0 otherwise

x2_ikts 1; if category i of donors is assigned a BCC k in period t and under scenario s, and 0 otherwise

x3_ilts 1; if category i of donors is assigned a BCPC l in period t and under scenario s, and 0 otherwise

x4_imts 1; if category i of donors is assigned a BTC m in period t and under scenario s, and 0 otherwise

x5_jlts 1; if MT j is referred to BCPC l in period t and under scenario s, and 0 otherwise

x6_klts 1; if BCC k is referred to BCPC l in period t and under scenario s, and 0 otherwise

x7_jmts 1; if MT j is referred to BTC m in period t and under scenario s, and 0 otherwise

x8_kmts 1; if BCC k is referred to BTC m in period t and under scenario s, and 0 otherwise

x9_lmts 1; if BCPC l is referred to BTC m in period t and under scenario s, and 0 otherwise

w_{j
₁
j
₂
ts} 1; if a MT is assigned to location j_1 in period t-1 and moves to location j_2 in period t under scenario s, and 0 otherwise

z1_kts 1; if a BCC is located in location k at period t under scenario s, and 0 otherwise

z2_lts 1; if a BCPC is located in location l at period t under scenario s and 0 otherwise

BV1_ijts The blood volume that category i of donors donate to a MT in location j in period t under scenario s

BV2_ikts The blood volume that category i of donors donate to a BCC in location k in period t under scenario s

BV3_ilts The blood volume that category i of donors donate to a BCPC in location l in period t under scenario s

BV4_imts The blood volume that category i of donors donate to a BTC in location m in period t under scenario s

BV5_jlts The blood volume delivered from a MT located in location j to a BCPC located in location l in period t under scenario s

BV6_jmts The blood volume delivered from a MT located in location j to a BTC located in location m in period t under scenario s

BV7_klts The blood volume delivered from a BCC located in location k to a BCPC located in location l in period t under scenario s

BV8_kmts The blood volume delivered from a BCC located in location k to a BTC located in location m in period t under scenario s

BV9_lmts blood samples sent from a BCPC located in location l to a BTC located in location m in period t under scenario s

The first objective function is calculated by Equation (1). The detail of the objective function is presented in Equations (2 to 5), which minimizes the total cost of establishing BCCs, BCPCs, relocation of MTs in each period and cost of delivering blood packs from MTs, BCCs to BCPCs and BTCs throughout the planning horizon. $Min Z_{1} (Total cost) = \sum_{s} π_{s} {(Z}_{11} {+ Z}_{12} {+ Z}_{13} {+ Z}_{14})$ (4) $\begin{matrix} Z_{11} (establishing BCCs and BCPCs cost) = \\ \sum_{k} \sum_{t} \sum_{s} {cl 1}_{k} {Z 1}_{kts} + \sum_{l} \sum_{t} \sum_{s} {cl 2}_{l} {Z 2}_{lts} \end{matrix}$ (5) $\begin{matrix} Z_{12} (Relocating MTs cost) = \\ \sum_{j_{1}} \sum_{j_{2}} \sum_{t} \sum_{s} {cm}_{j_{1} {, j}_{2}} W_{j_{1} j_{2} ts} \end{matrix}$ (6) $\begin{matrix} Z_{13} (operational cost) = \sum_{i} \sum_{j} \sum_{t} \sum_{s} {oc 1}_{jt} {BV 1}_{ijts} \\ + \sum_{i} \sum_{k} \sum_{t} \sum_{s} {oc 2}_{kt} {BV 2}_{ikts} \\ + \sum_{i} \sum_{l} \sum_{t} \sum_{s} {oc 3}_{lt} {BV 3}_{ilts} \\ + \sum_{i} \sum_{m} \sum_{t} \sum_{s} {oc 4}_{mt} {BV 4}_{imts} \end{matrix}$ (7) $\begin{matrix} Z_{14} (delivering blood cost) = \\ \sum_{j} \sum_{l} \sum_{t} \sum_{s} {cd 1}_{jlt} {BV 5}_{jlts} + \sum_{k} \sum_{l} \sum_{t} \sum_{s} {cd 2}_{klt} {BV 7}_{klts} \\ + \sum_{j} \sum_{m} \sum_{t} \sum_{s} {cd 3}_{jmt} {BV 6}_{jmts} + \\ \sum_{k} \sum_{m} \sum_{t} \sum_{s} {cd 4}_{kmt} {BV 8}_{kmts} + \\ \sum_{l} \sum_{m} \sum_{t} \sum_{s} {cd 5}_{lmt} {BV 9}_{lmts} \end{matrix}$ (8)

The second Objective function minimizes the sum of times that donated blood remains in the network. We now formulate the second objective function in Equation (6). $Min Z_{2} (perishing time) = \sum_{s} \sum_{t} π_{s} {PT}_{ts}$ (9)

Constraint set (7) enforces that only an existing MT can be moved to another location in the successive period. $\sum_{j_{2}} W_{j_{1} j_{2} ts} ⩽ \sum_{j} W_{j_{1} j_{2 (} t - 1) s} \forall t ⩾ 2, j, s$ (10)

Constraint (8) ensures that in each period, at most one MT can be transported to other candidate temporary locations to the candidate location j₂. $\sum_{j_{1}} W_{j_{1} j_{2} ts} ⩽ 1, \forall j_{2}, j, s$ (11)

Constraint (9) enforces that every category of donors, can donate only one of the MT, BCC, BCPC or BTC: $\begin{matrix} \sum_{j} x 1_{ijts} + \sum_{k} x 2_{ikts} + \sum_{l} x 3_{ilts} \\ + \sum_{m} x 4_{imts} ⩽ 1, \forall t, i, s \end{matrix}$ (12)

Constraint (10) guarantees that each category i of donors can be assigned to a MT located in candidate location j if it is in coverage radius and there is already a MT in that location. Constraint (11) enforces that each category i of donors can be assigned to a BCC located in candidate location k if it is in coverage radius and there is already a BCC in that location. Constraint (12) enforces that each category i of donors can be assigned to a BCPC located in candidate location l if it is in coverage radius and there is already a BCPC in that location. Constraint (13) enforces that each category i of donors can be assigned to a BTC located in candidate location m if it is in coverage radius and there is already a BTC in that location. $d 1_{ij} ⩽ cr 1_{0} \sum_{j_{1}} W_{j_{1} j_{2} ts}, \forall i, j, t, s$ (13) $x 2_{ikts} d 2_{ik} ⩽ cr 2_{0} Z 1_{kts}, \forall i, k, t, s$ (14) $x 3_{ilts} d 3_{il} ⩽ cr 3_{0} Z 2_{lts}, \forall i, l, t, s$ (15) $x 4_{imts} d 4_{im} ⩽ cr 4_{0}, \forall i, m, t, s$ (16)

Constraints (14, 15) determine that each MT, BCC can be assigned to a BCPC located in candidate location k if it is in coverage radius and there is already a BCPC in that location. Constraint (14) ensures that each BCPC should be assigned to a BTC located in location l if it is in coverage radius and there is already a BTC in that location. $x 5_{jlts} d 5_{jl} ⩽ cr 5_{0} Z 2_{lts}, \forall j, l, t, s$ (17) $x 6_{klts} d 7_{kl} ⩽ cr 5_{0} Z 2_{lts}, \forall k, l, t, s$ (18)

Constraints (16–18) determine that each MT, BCC, BCPC can be assigned to a BTC located in candidate location m if it is in coverage radius and there is already a BTC in that location. $x 7_{jmts} d 6_{jm} ⩽ cr 6_{0}, \forall j, m, t, s$ (19) $x 8_{kmts} d 8_{km} ⩽ cr 6_{0}, \forall k, m, t, s$ (20) $x 9_{lmts} d 9_{lm} ⩽ cr 6_{0}, \forall l, m, t, s$ (21)

Constraint (19) ensures that each MT should be assigned to one BCPC or one BTC. $\sum_{m} x 7_{jmts} + \sum_{l} x 5_{jlts} = 1, \forall j, t, s$ (22)

Constraint (20) guarantees that each BCC should be assigned to one BCPC or one BTC. $\sum_{m} x 8_{kmts} + \sum_{l} x 6_{klts} = 1, \forall k, t, s$ (23)

Constraint (21) ensures that each BCPC should be assigned to one BTC. $\sum_{m} x 9_{lmts} = 1, \forall l, t, s$ (24)

Constraints (22–30) cover that blood flow from MT, BCCs to BCPCs and BTC, from donor categories to MT, BCCs, BCPCs, BTC and also from BCPCs to BTCs if these facilities have been established in relevant candidate locations. $BV 1_{ijts} ⩽ M^{'} x 1_{ijts}, \forall i, j, t, s$ (25) $BV 2_{ikts} ⩽ M^{'} x 2_{ikts}, \forall i, k, t, s$ (26) $BV 3_{ilts} ⩽ M^{'} x 3_{ilts}, \forall i, l, t, s$ (27) $BV 4_{imts} ⩽ M^{'} x 4_{imts}, \forall i, m, t, s$ (28) $BV 5_{jlts} ⩽ M^{'} x 5_{jlts}, \forall j, l, t, s$ (29) $BV 6_{jmts} ⩽ M^{'} x 7_{jmts}, \forall j, m, t, s$ (30) $BV 7_{klts} ⩽ M^{'} x 6_{klts}, \forall k, l, t, s$ (31) $BV 8_{kmts} ⩽ M^{'} x 8_{kmts}, \forall k, m, t, s$ (32) $BV 9_{lmts} ⩽ M^{'} x 9_{lmts}, \forall l, m, t, s$ (33)

Constraint (31–34) restricts the maximum capacity of MT, BCC, BCPC and BTC: $\sum_{i} BV 1_{ijts} ⩽ ca 1_{j}, \forall j, t, s$ (34) $\sum_{i} BV 2_{ikts} ⩽ ca 2_{k}, \forall k, t, s$ (35) $\begin{matrix} \sum_{i} BV 3_{ilts} + \sum_{j} BV 5_{jlts} \\ + \sum_{k} BV 7_{klts} ⩽ ca 3_{l}, l, t, s \end{matrix}$ (36) $\begin{matrix} \sum_{i} BV 4_{imts} + \sum_{j} BV 6_{jmts} \\ + \sum_{k} BV 8_{kmts} ⩽ ca 3_{l}, \forall m, t, s \end{matrix}$ (37)

Constraint (35) shows the nominal volume of blood donated by each category of donors in each period. $\begin{matrix} \sum_{j} BV 1_{ijts} + \sum_{k} BV 2_{ikts} + \sum_{l} BV 3_{ilts} \\ + \sum_{m} BV 4_{imts} = {\hat{Bd}}_{its} \forall i, t, s \end{matrix}$ (38)

Constraint (36) guarantees the blood packs can be delivered only from a MT if it has already been opened. $x 5_{jlts} + x 7_{jmts} ⩽ \sum_{j_{1}} W_{j_{1} j_{2} ts}, \forall j, l, m, t, s$ (39)

Constraint (37) indicates that Mc percent of accepted blood samples by BTCs should satisfy total demand. $Mc \sum_{l, m} BV 9_{lmts} = {\hat{Td}}_{ts}, \forall t, s$ (40)

Constraint (38) assures that total volume of collected blood by MTs at each period should be delivered to BCPCs and BTCs. $\begin{matrix} \sum_{i} \sum_{j} BV 1_{ijts} = \sum_{j} \sum_{l} BV 5_{jlts} \\ + \sum_{j} \sum_{m} BV 6_{jmts}, \forall t, s \end{matrix}$ (41)

Constraint (39) assures that total volume of collected blood by BCCs at each period should be delivered to BCPCs and BTCs. $\begin{matrix} \sum_{i} \sum_{k} BV 2_{ikts} = \sum_{k} \sum_{l} BV 7_{klts} \\ + \sum_{k} \sum_{m} BV 8_{kmts}, \forall t, s \end{matrix}$ (42)

Constraint (40) assures that total volume of collected blood by BCPCs at each period for testing should be blood samples delivered to BTCs. $\begin{matrix} \sum_{i} \sum_{l} BV 3_{ilts} + \sum_{j} \sum_{l} BV 5_{jlts} \\ + \sum_{k} \sum_{l} BV 7_{klts} = \sum_{l} \sum_{m} BV 9_{lmts}, \forall t, s \end{matrix}$ (43)

Constraints (41–43) ensures the total time that each donated blood is in the system doesn’t exceed its related expiration time. $\begin{matrix} x 1_{ijts} t 1_{ijts} + x 2_{ikts} t 2_{ikts} + x 3_{ilts} t 3_{ilts} \\ + x 4_{imts} t 4_{imts} + x 5_{jlts} t 5_{jlts} + x 6_{klts} t 7_{klts} \\ + x 7_{jmts} t 6_{jmts} + x 8_{kmts} t 8_{kmts} \\ + x 9_{lmts} t 9_{lmts} ⩽ {PT}_{ts}, \forall i, j, k, l, m, t, s \end{matrix}$ (44) $\begin{matrix} x 1_{ijts}, x 2_{ikts}, x 3_{ilts}, x 4_{imts}, x 5_{jlts}, x 6_{klts}, \\ x 7_{jmts}, x 8_{kmts}, x 9_{lmts}, W_{j_{1} j_{2} ts}, Z 1_{kts}, Z 2_{lts} = 0 or 1 \end{matrix}$ (45) $\begin{matrix} BV 1_{ijts}, BV 2_{ikts}, BV 3_{ilts}, BV 4_{imts}, BV 5_{jlts}, \\ BV 6_{jmts}, BV 7_{klts}, BV 8_{kmts}, BV 9_{lmts} ⩾ 0 \end{matrix}$ (46)

Step 2. Conversion to a single-objective model

We apply MCGP to change the bi-objective model presented in step 1 to a single – objective optimization model. $\begin{matrix} Min [W_{1} (d_{1}^{+} + d_{1}^{-}) + α_{1} (e_{1}^{+} + e_{1}^{-})] \\ + [W_{2} (d_{2}^{+} + d_{2}^{-}) + α_{2} (e_{2}^{+} + e_{2}^{-})] \end{matrix}$ (47)

Subject to:

Constraints (7–43) $Z_{1} + d_{1}^{+} - d_{1}^{-} = y_{1}$ (48) $Z_{2} + d_{2}^{+} - d_{2}^{-} = y_{2}$ (49) $y_{1} + e_{1}^{-} - e_{1}^{+} = b_{1 . min}$ (50) $(y_{2} + e_{2}^{+} = b_{2. m i n})$ (51) $b_{1 . min} ⩽ y_{1}$ (52) $b_{2 . min} ⩽ y_{2}$ (53) $y_{1} ⩽ b_{1 . max}$ (54) $y_{2} ⩽ b_{2 . max}$ (55) $d_{1}^{+}, d_{1}^{-}, e_{1}^{+}, e_{1}^{-}, d_{2}^{+}, d_{2}^{-}, e_{2}^{+}, e_{2}^{-} ⩾ 0$ (56)

Step 3. robustness model

Robust optimization (RO) approach offered by Bertsimas and Simas a introduction to recitation our SMCGR formulation [42]. Consider the linear program (LP) where C is an n-vector, A is a m×n matrix, and b is an m-vector: $MinCx, s . t; Ax ⩽ b, x ⩾ 0$ (57)

Assume uncertainty only affects the elements of matrix A. That is, consider a particular row I of A and let ji symbolize the set of coefficients in row I of A subject to uncertainty. Each data element ${\tilde{a}}_{ij}$ , j ∈ j_i is modeled as a bounded and independent random variable taking value in an interval $[\hat{a} ij - a_{ij}, \hat{a} ij + a_{ij}]$ where ${\hat{a}}_{ij}$ is the nominal value and a_ij is the maximum deviation from this nominal value. With this assumption, LP (54) is reformulated as: $\min C x, s . t \max_{\forall {\tilde{a}}_{ij} \in j i} (\underset{j}{\sum {\tilde{a}}_{ij} x}) \leq b_{i} \forall i, x \geq 0$ (58)

Next, we define a scaled deviation $z_{ij} = ({\tilde{a}}_{ij} - {\hat{a}}_{ij}) / a_{ij}$ note that ${\tilde{a}}_{ij}, {\hat{a}}_{ij}$ and a_ij signify the uncertain value, its nominal value and its maximum deviation from the nominal value, respectively. It is un-likely that all of the uncertain input ${\tilde{a}}_{ij}$ , j_i will understand their worst-case values simultaneously. Thus, a maximum number of parameters that can deviate from their nominal values for each constraint i is considered as γ_i, called the budget of uncertainty, where γ_i ∈ [0, |J_i|]. The aggregated scaled deviation of uncertain parameters for constraint I is bounded as ∑_j∈ji|z_ij| ⩽ γ_i, ∀ i.

The budget of uncertainty plays a critical role in adjusting the solution’s level of conservatism against the robustness. If γ_i = 0, it decreases to the nominal formulation where there is no protection against uncertainty. If γ_i = |J_i|, the ith constraint is completely kept against the worst-case realization of uncertain parameters. Finally, if γ_i ∈ (0, |J_i|) the decision maker considers a tradeoff between conservatism and cost of the solution against the level of protection against constraint violation. Based on this definition, the set J_i is defined as $J_{i} = {{\tilde{a}}_{ij} | {\tilde{a}}_{ij} = {\hat{a}}_{ij} + z_{ij} a_{ij} \forall i, j, z \in Ω}$ where $Ω = {z | \sum_{j = 1}^{n} z_{ij} | ⩽ γ_{i} | z_{ij} | ⩽ 1 \forall i}$ . Therefore; $\begin{matrix} \sum_{j = 1} {\tilde{a}}_{ij} x_{j} = \sum_{j = 1} ({\hat{a}}_{ij} + z_{ij} a_{ij}) x_{j} \\ = \sum_{j = 1} {\hat{a}}_{ij} x_{j} + \sum_{j = 1} z_{ij} a_{ij} x_{j} \end{matrix}$ (59)

LP (55) can be reformulated; $\begin{array}{l} Min Cx, s . t; \sum_{j} {\hat{a}}_{ij} x_{j} + \\ _{z_{i} \in Ω_{i}}^{\max} (\sum_{j} z_{ij} a_{ij} x_{j}) ⩽ b_{i} \forall i, x ⩾ 0 \end{array}$ (60)

The lower level problem $max_{z_{1} \in Ω_{1}} (\sum_{j} z_{ij} a_{ij} x_{j})$ for a given vector x* is equivalent to LP $max \sum_{j} z_{ij} a_{ij} x_{j}^{*} s . t . \sum_{j = 1}^{n} z_{ij} ⩽ γ_{i} \forall i . 0 ⩽ z_{ij} ⩽ 1 \forall J \in J_{i}$ (61)

Then by introducing the dual variables α_i and β_ij, the dual of LP (58) is: $\begin{matrix} Min γ_{i} α_{i} + \sum_{J \in J_{i}} β_{ij}, s . t α_{i} + β_{ij} ⩾ a_{ij} x_{j}^{*}, \\ \forall i, J_{i}, β_{ij} ⩾ 0, \forall i α_{i} ⩾ 0 \end{matrix}$ (62)

The dual (59) is applied to LP (57) to obtain the robust counterpart of LP (44): $\begin{matrix} Min Cx, s . t; {\hat{a}}_{i} x - γ_{i} α_{i} - \\ \sum_{J \in J_{i}} β_{ij} ⩽ b_{i} \forall i, α_{i} + β_{ij} ⩾ a_{ij} x_{j}, \\ \forall i, J \in J_{i}, β_{ij}, α_{i} ⩾ 0, \forall i, j \in j_{i} \end{matrix}$ (63)

This RO approach provides an efficient way to decide bounds on the probability of violation of each constraint. Let $x_{j}^{*}$ be the robust solution, and then the violation probability of the ith constraint is calculated by: $Pr (\sum_{j} {\bar{a}}_{ij} x_{j}^{*} < b_{i}) ⩽ 1 - \emptyset (\frac{(γ_{i} - 1)}{\sqrt{| J_{i} |}}$ (64)

Where φ is the standard normal cumulative distribution function. This upper bound presents a way of assigning a proper budget of un-certainty to each constraint when our uncertain parameters are in-dependent and symmetrically distributed random variables in their associated uncertaintyset.

In our SMCGR approach for BCSND, we define polyhedral uncertainty sets for blood donation volume of each category of donors and Total demand of blood in each period. To develop the uncertainty sets, first we define the positive and negative deviation percentages from the nominal scenario for blood donation volume and total demand, respectively, as (62, 63): $\begin{matrix} τ {Bd}_{i}^{ts +} = \frac{{Bd}_{i}^{ts} - {B \hat{d}}_{i}^{ts}}{Δ {Bd}_{i}^{ts +}}, if {Bd}_{i}^{ts} > {B \hat{d}}_{i}^{ts} \\ τ {Bd}_{i}^{ts -} = \frac{{Bd}_{i}^{ts} - {B \hat{d}}_{i}^{ts}}{Δ {Bd}_{i}^{ts -}}, if {Bd}_{i}^{ts} < {B \hat{d}}_{i}^{ts} \end{matrix}$ (65)

$\begin{matrix} τ {Td}_{t}^{s +} = \frac{{Td}_{t}^{s} - T {\hat{d}}_{t}^{s}}{Δ {Td}_{t}^{s +}}, if {Td}_{t}^{s} > T {\hat{d}}_{t}^{s} \\ τ {Td}_{t}^{s -} = \frac{{Td}_{t}^{s} - T {\hat{d}}_{t}^{s}}{Δ {Td}_{t}^{s -}}, if {Td}_{t}^{s} < T {\hat{d}}_{t}^{s} \end{matrix}$ (66)

Then, the uncertainty sets of Blood donation volume of each category of donors and Total demand of blood in each period are: $\begin{matrix} J_{s}^{Bd} = \\ {{Bd}_{i}^{ts} | \begin{matrix} {Bd}_{i}^{ts} = {B \hat{d}}_{i}^{ts} + Δ {Bd}_{i}^{ts +} \times τ {Bd}_{i}^{ts +} - Δ {Bd}_{i}^{ts -} \times τ {Bd}_{i}^{ts -} \\ \forall k, t, \forall τ {Bd}_{i}^{ts +}, τ {Bd}_{i}^{ts -} \in k^{Bd} \end{matrix}} \end{matrix}$ (67)

Where; $\begin{matrix} k^{Bd} = \\ {τ {Bd}_{i}^{ts +}, τ {Bd}_{i}^{ts -} | \begin{matrix} 0 ⩽ τ {Bd}_{i}^{ts +} ⩽ 1, 0 ⩽ τ {Bd}_{i}^{ts -} ⩽ 1, \\ \sum_{k} \sum_{t} ({Bd}_{i}^{ts +} + τ {Bd}_{i}^{ts -}) ⩽ γ_{s}^{Bd} \end{matrix}} \end{matrix}$ (68)

Similarly; $\begin{matrix} J_{s}^{Td} = \\ {{Td}_{t}^{s} | \begin{matrix} {Td}_{t}^{s} = T {\hat{d}}_{t}^{s} + Δ {Td}_{t}^{s +} \times τ {Td}_{t}^{s +} - Δ {Bd}_{i}^{ts -} \times τ {Td}_{t}^{s -} \\ \forall k, t, \forall τ {Td}_{t}^{s +}, τ {Td}_{t}^{s -} \in k^{Td} \end{matrix}} \end{matrix}$ (69) $\begin{matrix} k^{Td} = \\ {τ {Td}_{t}^{s +}, τ {Td}_{t}^{s -} | \begin{matrix} 0 ⩽ τ {Td}_{t}^{s +} ⩽ 1, 0 ⩽ τ {Td}_{t}^{s -} ⩽ 1, \\ \sum_{k} \sum_{t} (τ {Td}_{t}^{s +} + τ {Td}_{t}^{s -}) ⩽ γ_{s}^{Td} \end{matrix}} \end{matrix}$ (70)

The parameter $γ_{s}^{Bd}$ in (66) is the budget of uncertainty for blood donation volume in scenario s via which we can constrain the number of periods in which the blood donation volume may deviate from its nominal value. Similar definitions apply to the polyhedral uncertainty sets for total demand in(68).

Allowing for this uncertainty implies that constraints (35) and (37) may not be satisfied. In the SMCGR, we relax these constraints and penalize their violation in the objectivefunction.

Our purpose is to minimize the worst-case costs associated with violations of (35) and (37). To incorporate the uncertainty sets (63) and (65) in the stochastic formulation (7–34), (36), (38–43), (46–53) we isolate the objective function terms containing random Blood donation volume and Total demand parameters for scenario s as the following nonlinear expression: $\begin{matrix} {PC}_{s} (BV) = \max_{{Bd}_{i}^{ts} ɛ J_{s}^{Bd}} [\sum_{t} \sum_{i} ({Bd}_{i}^{ts} - \sum_{j} {BV 1}_{ijts} - \\ \sum_{k} {BV 2}_{ikts} - \sum_{l} {BV 3}_{ilts} - \sum_{m} {BV 4}_{imts}) {\times SC}^{Bd}, \\ \sum_{i} \sum_{t} (\sum_{j} {BV 1}_{ijts} + \sum_{k} {BV 2}_{ikts} + \sum_{l} {BV 3}_{ilts} \\ + \sum_{m} {BV 4}_{imts} {- Bd}_{i}^{ts}) {\times PC}^{Bd}] + \end{matrix}$ (71) $\begin{matrix} \max_{{Td}_{t}^{s} ɛ J_{s}^{Td}} [\sum_{t} ({Td}_{t}^{s} - Mc \sum_{l} \sum_{m} {BV 9}_{lmts}) {\times SC}^{Td}, \\ \sum_{t} (Mc \sum_{l} \sum_{m} {BV 9}_{lmts} {- Td}_{t}^{s}) {\times PC}^{Bd}] \end{matrix}$ (72)

This term represents the worst-case value for penalty, waste and surplus costs. We reformulate this nonlinear optimization problem (69) as the following LP for each scenario s by defining auxiliary variables ZZ1 s and ZZ2s: $min_{z 1_{s}, z 2_{s}} {PC}_{s} (BV) = zz 1_{s} + zz 2_{s}$ (73) $\begin{matrix} \sum_{i} \sum_{t} ({Bd}_{i}^{ts} - \sum_{j} BV 1_{ijts} - \sum_{k} BV 2_{ikts} \\ - \sum_{l} BV 3_{ilts} - \sum_{m} BV 4_{imts}) \times {SC}^{Bd} ⩽ zz 1_{s}, \\ \forall {Bd}_{i}^{ts} ɛ J_{s}^{Bd} \end{matrix}$ (74) $\begin{matrix} \sum_{i} \sum_{t} (\sum_{j} BV 1_{ijts} + \sum_{k} BV 2_{ikts} + \sum_{l} BV 3_{ilts} \\ + \sum_{m} BV 4_{imts} - {Bd}_{i}^{ts}) \times {PC}^{Bd} ⩽ zz 1_{s}, \forall {Bd}_{i}^{ts} ɛ J_{s}^{Bd} \end{matrix}$ (75) $\begin{matrix} \sum_{t} ({Td}_{t}^{s} - Mc \sum_{l} \sum_{m} BV 9_{lmts}) \times {SC}^{Td} \\ ⩽ zz 2_{s}, \forall {Td}_{t}^{s} ɛ J_{s}^{Td} \end{matrix}$ (76) $\begin{matrix} \sum_{t} (Mc \sum_{l} \sum_{m} BV 9_{lmts} - {Td}_{t}^{s}) \times {PC}^{Bd} \\ ⩽ zz 2_{s}, \forall {Td}_{t}^{s} ɛ J_{s}^{Td} \end{matrix}$ (77) $zz 1_{s}, zz 2_{s} ⩾ 0$ (78)

The constraints (70–74) should be satisfied for all realizations of the uncertain Blood donation volume and total demand in their polyhedral uncertainty sets. We find their robust counterparts, explained in detail for constraint (70). From the set definition (64), we can rewrite the constraints (60) as: $\begin{matrix} max_{{Bd}_{i}^{ts} ɛ J_{s}^{Bd}} [\sum_{i} \sum_{t} ({Bd}_{i}^{ts} - \sum_{j} BV 1_{ijts} - \sum_{k} BV 2_{ikts} \\ - \sum_{l} BV 3_{ilts} - \sum_{m} BV 4_{imts})] \times {SC}^{Bd} ⩽ ZZ 1_{s} \end{matrix}$ (79) $\begin{matrix} \sum_{i} \sum_{t} ({B \hat{d}}_{i}^{ts} - \sum_{j} BV 1_{ijts} - \sum_{k} BV 2_{ikts} \\ - \sum_{l} BV 3_{ilts} - \sum_{m} BV 4_{imts}) \times {SC}^{Bd} \\ + max_{\forall τ {Bd}_{i}^{ts +}, τ {Bd}_{i}^{ts -} \in k^{Bd}} [\sum_{i} \sum_{t} (Δ {Bd}_{i}^{ts +} \times τ {Bd}_{i}^{ts +} \\ - Δ {Bd}_{i}^{ts -} \times τ {Bd}_{i}^{ts -})] \times {SC}^{Bd} ⩽ ZZ 1_{s} \end{matrix}$ (80)

In this constraint, we optimize over the positive and negative deviation percentages from nominal scenario for uncertain blood donation volume. We expand the maximization problem in (78) considering constraints from polyhedral uncertainty sets as follows: $\begin{matrix} \underset{\forall τ {Bd}_{i}^{ts +}, τ {Bd}_{i}^{ts -} \in k^{Bd}}{- min} [\sum_{i} \sum_{t} (- Δ {Bd}_{i}^{ts +} \times τ {Bd}_{i}^{ts +} \\ + Δ {Bd}_{i}^{ts -} \times τ {Bd}_{i}^{ts -}) \times {SC}^{Bd}] \end{matrix}$ (81)

$s . t : τ {Bd}_{i}^{ts +} ⩾ - 1, \forall t, i : δ 1_{i}^{ts}$ (82) $τ {Bd}_{i}^{ts -} ⩾ - 1, \forall t, i : δ 2_{i}^{ts}$ (83) $\sum_{i} \sum_{t} (- τ {Bd}_{i}^{ts +} - τ {Bd}_{i}^{ts -}) ⩾ - γ_{s}^{Bd}, : δ 3^{Bd}$ (84) $τ {Bd}_{i}^{ts +}, τ {Bd}_{i}^{ts -} ⩾ 0$ (85)

Then we take the dual as:

$max_{δ 1_{i}^{ts}, δ 2_{i}^{ts}, δ 3^{Bd}} [- γ_{s}^{Bd} \times δ 3^{Bd} - \sum_{i} \sum_{t} (δ 1_{i}^{ts} + δ 2_{i}^{ts})]$ (86) $s . t : - δ 1_{i}^{ts} - δ 3^{Bd} ⩽ - Δ {Bd}_{i}^{ts +}, \forall i, t$ (87) $- δ 2_{i}^{ts} - δ 3^{Bd} ⩽ - Δ {Bd}_{i}^{ts -}, \forall i, t$ (88) $δ 2_{i}^{ts}, δ 2_{i}^{ts}, δ 3^{Bd} ⩾ 0, \forall i, t$ (89)

According to strong duality theory, Constraint (86) is actually redundant, we can remove $δ 2_{i}^{ts}$ and replace the (83) without $δ 2_{j}^{ts}$ in the constraint (77) and, hence, the robust counterpart of constraint (71) is equivalent to the system of inequalities: $\begin{matrix} (\sum_{i} \sum_{t} ({B \hat{d}}_{i}^{ts} + δ 1_{i}^{ts}) + δ 3^{Bd} \times γ_{s}^{Bd} - \sum_{i, j, t} BV 1_{ijts} - \\ \sum_{i, k, t} BV 2_{ikts} - \sum_{i, l, t} BV 3_{ilts} - \sum_{i, m, t} BV 4_{imts}) \times {SC}^{Bd} \\ ⩽ ZZ 1_{s} \end{matrix}$ (90) $s . t : δ 1_{i}^{ts} + δ 3^{Bd} ⩾ Δ {Bd}_{i}^{ts +}, \forall i, t$ (91) $δ 2_{i}^{ts}, δ 3^{Bd} ⩾ 0, \forall i, t$ (92)

The robust counterpart of the other constraints is found by the same procedure. Finally, our hybrid stochastic, MCGP and robust formulation of this BSCND problem is:

$\begin{matrix} Min Z 3 = [W_{1} (d_{1}^{+} + d_{1}^{-}) + α_{1} (e_{1}^{+} + e_{1}^{-})] + \\ [W_{2} (d_{2}^{+} + d_{2}^{-}) + α_{2} (e_{2}^{+} + e_{2}^{-})] - ZZ 1_{s} - - - ZZ 2_{s}) \end{matrix}$ (93)

s.t: Constrains (7–34), (36), (38–44), (45–53)

$\begin{matrix} (\sum_{i} \sum_{t} ({B \hat{d}}_{i}^{ts} {+ δ 1}_{i}^{ts}) {+ δ 3}^{Bd} {\times γ}_{s}^{Bd} - \sum_{i, j, t} {BV 1}_{ijts} - \\ \sum_{i, k, t} {BV 2}_{ikts} - \sum_{i, l, t} {BV 3}_{ilts} - \sum_{i, m, t} {BV 4}_{imts}) {SC}^{Bd} \\ ⩽ {ZZ 1}_{s}, \forall s \end{matrix}$ (94) $\begin{matrix} (\sum_{i} \sum_{t} ({B \hat{d}}_{i}^{ts} {- δ 2}_{i}^{ts}) {- δ 4}^{Bd} {\times γ}_{s}^{Bd} - \sum_{i, j, t} {BV 1}_{ijts} - \\ \sum_{i, k, t} {BV 2}_{ikts} - \sum_{i, l, t} {BV 3}_{ilts} - \sum_{i, m, t} {BV 4}_{imts} {PC}^{Bd}) \\ ⩾ {- ZZ 1}_{s}, \forall s \end{matrix}$ (95) $\begin{matrix} (\sum_{t} (T {\hat{d}}_{t}^{s} + δ 5_{s}^{t}) + δ 7^{Td} \times γ_{s}^{Td} \\ - Mc \sum_{l, m, t} BV 9_{lmts}) S^{Td} ⩽ ZZ 2_{s}, \forall s \end{matrix}$ (96) $\begin{matrix} (\sum_{t} (T {\hat{d}}_{t}^{s} - δ 6_{t}^{s}) - \\ δ 8^{Td} \times γ_{s}^{Td} Mc \sum_{l, m, t} BV 9_{lmts}) {PC}^{Td} ⩾ - ZZ 2_{s}, \forall s \end{matrix}$ (97) $δ 1_{i}^{ts} + δ 3^{Bd} ⩾ Δ {Bd}_{i}^{ts +}, \forall i, t, s$ (98) $δ 2_{i}^{ts} + δ 4^{Bd} ⩾ Δ {Bd}_{i}^{ts -}, \forall i, t, s$ (99) $δ 5_{s}^{t} + δ 7^{Bd} ⩾ Δ {Td}_{t}^{s +}, \forall i, t, s$ (100) $δ 6_{t}^{s} + δ 8^{Bd} ⩾ Δ {Td}_{t}^{s -}, \forall i, t, s$ (101) $\begin{matrix} ZZ 1_{s}, ZZ 2_{s}, δ 1_{i}^{ts}, δ 2_{i}^{ts}, δ 3^{Bd}, δ 4^{Bd}, \\ δ 5_{s}^{t}, δ 6_{t}^{s}, δ 7^{Bd}, δ 8^{Bd} ⩾ 0 \forall i, t, s \end{matrix}$ (102)

In this formulation, the parameters $γ_{s}^{Bd}$ and $γ_{s}^{Td}$ control the tradeoff between the robustness and the level of conservatism of the obtained solution at each scenario by restricting the Blood donation volume and total demand deviate from the nominal scenario in their associated uncertainty sets. As a result, higher values for the parameters $γ_{s}^{Bd}$ and $γ_{s}^{Td}$ increase the level of robustness at the expense of a lower expected profit.

3 Case study

In this section, we apply a case study to implement the applicability of the proposed model. For this reason, we collected the essential data onto East Azerbaijan Province of Iran (EAPI). EAPI is located in north west of Iran. Figure 2 demonstrates cities location of EAPI (adapted from Wikipedia). Now in this province two BCPCs located that in Maragheh (C16) and Miyanehis (C20). Furthermore some MTs and BCCs exist in some cities. Location of BTC in the center of EAPI is determined. In blood transfusion network some MTs and BCCs in cities, for delivering blood packs can refer to other center of province. This feasiblity for keeping blood from pershiblity. According to expert’s of blood tranfusion center and various population of cities 23 candidate locations for MTs, 16 candidate locations for BCCs,8 candidate locations for BCPCs and 60 groups of donors is identified. Table 2 (in appendix) shows the travel distance between each cities of center of province and other neighboard centers of province. As well as this table demonstrates distance between each cities of center of EAPI and other neighbored centers of province. Also Table 3 (in appendix) to Tables 4–6 shows blood delivering costs.

Table 2
Travel distance between each cities of center of province and other neighbored centers of province (km)

Cities City codes Population C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 Tabriz Ardabil Oromieh Zanjan

Marand C1 240000 0 73 171 286 257 257 190 184 74 66 114 129 176 201 227 221 180 112 214 252 233 73 311 208 369

Jolfa C2 55000 73 0 126 353 235 249 182 256 146 138 186 201 249 273 299 293 279 211 286 324 299 138 376 217 434

Varzaqan C3 49000 171 126 0 153 124 107 56 116 110 165 150 152 200 224 250 244 190 122 197 235 216 91 209 237 354

Khodaafrin C4 34000 286 353 153 0 46 124 153 215 219 332 251 276 358 381 411 398 280 252 285 313 346 219 268 404 498

kaleybar C5 92000 257 235 124 46 0 88 67 125 183 255 223 238 286 310 263 330 272 208 202 321 301 173 223 317 410

Horand C6 4000 257 249 107 124 88 0 67 127 183 255 223 238 296 310 336 330 276 208 204 361 297 169 219 315 407

Ahar C7 155000 190 182 56 153 67 67 0 60 116 188 156 166 219 243 229 263 205 141 137 254 238 112 163 258 350

Heris C8 71000 184 256 116 215 125 127 60 0 110 182 150 165 213 237 263 257 263 76 77 189 183 91 169 237 322

Tabriz C9 1652000 74 146 110 219 183 183 116 110 0 72 40 55 103 127 153 147 133 65 140 178 159 0 228 150 305

Shabestar C10 127000 66 138 165 332 255 255 188 182 72 0 122 127 175 199 225 219 205 137 214 250 233 72 290 180 368

Osku C11 77000 114 186 150 251 223 223 156 150 40 122 0 195 243 102 128 122 173 105 180 218 191 36 248 127 326

Azarshahr C12 104000 129 201 152 276 238 238 166 165 55 127 195 0 48 72 98 92 188 120 195 233 187 58 290 124 348

Ajabshir C13 86000 176 249 200 358 286 296 219 213 103 175 243 48 0 34 50 44 144 168 243 219 147 98 310 163 300

Bonab C14 132000 201 273 224 381 310 310 243 237 127 199 102 72 34 0 26 20 120 192 267 195 125 121 353 187 277

Malekan C15 105000 227 299 250 411 263 336 229 263 153 225 128 98 50 26 0 47 286 218 293 331 120 145 333 165 279

Margheh C16 245000 221 293 244 398 330 330 263 257 147 219 122 92 44 20 47 0 100 212 287 175 108 139 323 204 261

Hashtrud C17 68000 180 279 190 280 272 276 205 263 133 205 173 188 144 120 286 100 0 68 143 75 61 119 225 264 197

Bostanabad C18 101000 112 211 122 252 208 208 141 76 65 137 105 120 168 192 218 212 68 0 75 113 114 58 261 204 250

Sarab C19 138000 214 286 197 285 202 204 137 77 140 214 180 195 243 267 293 287 143 75 0 188 183 128 87 273 319

Miyaneh C20 200000 252 324 235 313 321 361 254 189 178 250 218 233 219 195 331 175 75 113 188 0 86 165 153 310 133

charoymaq C21 36000 233 299 216 346 301 297 238 183 159 233 191 187 147 125 120 108 61 114 183 86 0 193 234 300 177

Table 3

Costs of moving a MT between locations (million Rials)

	j1	j2	j3	j4	j5	j6	j7	j8	j9	j10	j11	j12	j13	j14	j15	j16	j17	j18	j19	j20	j21	j22	j23
j1	0	7.3	17.1	28.6	25.7	19	18.4	7.4	7.7	6.6	11.4	12.9	17.6	20.1	22.7	22.1	23	18.1	11.2	21.4	25.2	26	23.5
j2	7.3	0	12.6	35.3	23.5	18.2	25.6	14.6	15.2	13.8	18.6	20.1	24.9	27.3	29.9	29.3	30.1	27.9	21.1	28.6	32.5	33	30.2
j3	17.1	12.6	0	15.3	12.4	5.9	12	11.2	11.5	16.5	15	15.2	20.1	22.4	25	24.7	25	19	12.2	19.7	23.5	24	22
j4	28.6	35.3	15.3	0	4.6	15.4	21.4	21.6	22	33.4	25.5	27.7	35.6	38.4	41.5	39.8	40.1	28.1	25.5	28.7	31.5	31.9	34.9
j5	25.7	23.5	12.4	4.6	0	6.9	12.9	18.5	19	33.6	25.2	27.9	36.1	38.2	41.5	39.9	40.2	20.7	20.2	20.5	32.2	32.5	30.4
j6	19	18.2	5.9	15.4	6.9	0	12.7	18.4	19	25.8	22.7	24	29.9	31.2	33.8	34	33.5	27	20	19.9	36	35.9	29.5
j7	18.4	25.6	12	21.4	12.9	12.7	0	11	11.2	18.3	15.2	16.7	21.5	23.8	25.9	25.8	26	26.3	7.5	7.9	19	19.2	18.5
j8	7.4	14.6	11.2	21.6	18.5	18.4	11	0	0.5	7.2	5.5	10.3	12.8	15.8	14.9	15.2	16.2	13	6.5	14	17.8	18	16.1
j9	7.7	15.2	11.5	22	19	19	11.2	0.5	0	7.5	5.8	10.6	13.1	16.1	15.2	15.5	16.5	13.3	6.8	14.3	18.1	18.3	16.4
j10	6.6	13.8	16.5	33.4	33.6	25.8	18.3	7.2	7.5	0	4	5.5	10.3	12.7	15.3	14.7	15	13.6	7	22	25	25.5	24
j11	11.4	18.6	15	25.5	25.2	22.7	15.2	5.5	5.8	4	0	19.5	24.4	10.3	12.9	12.7	12.5	14	10.5	18	21.8	22.3	19.5
j12	12.9	20.1	15.2	27.7	27.9	24	16.7	10.3	10.6	5.5	19.5	0	4.8	7.5	10.1	9.5	9.6	19	12.5	19.9	24	24.5	18.9
j13	17.6	24.9	20.1	35.6	36.1	29.9	21.5	12.8	13.1	10.3	24.4	4.8	0	3.4	5.1	4.4	4.9	16.9	24.5	22	22.5	24.5	18.7
j14	20.1	27.3	22.4	38.4	38.2	31.2	23.8	15.8	16.1	12.7	10.3	7.5	3.4	0	2.6	2	2.5	12.2	19.5	26.7	19.7	20.1	12.8
j15	22.7	29.9	25	41.5	41.5	33.8	25.9	14.9	15.2	15.3	12.9	10.1	5.1	2.6	0	4.7	5.1	28.6	21.8	29.5	33.3	33.8	12.2
j16	22.1	29.3	24.7	39.8	39.9	34	25.8	15.2	15.5	14.7	12.7	9.5	4.4	2	4.7	0	0.5	29.1	22.3	30	33.8	34.3	12.7
j17	23	30.1	25	40.1	40.2	33.5	26	16.2	16.5	15	12.5	9.6	4.9	2.5	5.1	0.5	0	10	21.2	28.9	17.7	18.1	11
j18	18.1	27.9	19	28.1	20.7	27	26.3	13	13.3	13.6	14	19	16.9	12.2	28.6	29.1	10	0	6.9	14.5	7.7	8.1	6.2
j19	11.2	21.1	12.2	25.5	20.2	20	7.5	6.5	6.8	7	10.5	12.5	24.5	19.5	21.8	22.3	21.2	6.9	0	7.5	11.5	12	11.8
j20	21.4	28.6	19.7	28.7	20.5	19.9	7.9	14	14.3	22	18	19.9	22	26.7	29.5	30	28.9	14.5	7.5	0	17.9	18.2	19
j21	25.2	32.5	23.5	31.5	32.2	36	19	17.8	18.1	25	21.8	24	22.5	19.7	33.3	33.8	17.7	7.7	11.5	17.9	0	0.5	9
j22	26	33	24	31.9	32.5	35.9	19.2	18	18.3	25.5	22.3	24.5	24.5	20.1	33.8	34.3	18.1	8.1	12	18.2	0.5	0	8.6
j23	23.5	30.2	22	34.9	30.4	29.5	18.5	16.1	16.4	24	19.5	18.9	18.7	12.8	12.2	12.7	11	6.2	11.8	19	9	8.6	0

Table 4

Costs of delivering blood packs from MT to BCPC (million Rials)

l1	l2	l3	l4	l5	l6	l7	l8
j1	0.5	28.7	23.9	9.8	11.8	28.6	20.5	29.8
j2	12	25.4	25.2	25.5	15.8	16.2	24.6	12.7
j3	7.4	20.9	7.3	22.4	23.1	25.9	28.9	10.3
j4	12	11.2	23.4	12.9	26.7	26.6	21.7	20.7
j5	19.2	23.9	12.3	24.5	9.2	22.3	15.6	17.5
j6	24.3	0.4	16.1	13.4	9.8	29.3	14.1	21.5
j7	21.8	21.7	28	12.1	11.4	12	20.2	19.5
j8	9.2	21.4	0.9	0.7	0.5	7.3	7.7	21.7
j9	16.9	29.8	1.1	1	0.9	21.7	12	27.4
j10	25.8	18	22.8	14.1	23.9	0.7	22.3	11.6
j11	21.8	10.7	10.1	29.1	15.6	20.3	18.6	29.3
j12	25.7	16.3	19.1	22.5	21.9	19.1	0.5	24.5
j13	25.2	29.7	23.9	27.3	17.1	25.2	19.1	22.4
j14	16.9	8	28.7	10.4	22.3	12.6	22	0.6
j15	20.1	29.3	14.9	8.4	29.4	7.6	8.6	9
j16	15.9	21.1	10.6	19.2	20.6	21.4	14.3	11.4
j17	13.8	13.4	23.9	24.1	29.9	12.3	22.7	26
j18	9.8	15.3	26.8	13.6	21.6	13.5	24.8	17.7
j19	25.8	8	17.8	9.9	22.1	14.6	21.8	13.3
j20	15.1	24.1	25	16.8	29.9	14.3	16.6	13.2
j21	22	10.9	29.3	18.8	29.5	17.5	13.6	15.9
j22	13.4	21.4	24.8	27.9	22.3	17.1	11.9	27.7
j23	20.3	10.5	14.1	10.6	10.2	22.5	24.1	15.8

Table 5

Costs of delivering blood packs from BCCs to BCPCs (million Rials)

l1	l2	l3	l4	L5	L6	L7	L8k1	17	12	22	11	8.3	20.6	24.8	28.1
k2	34	24	30	26	12.2	16.3	23.5	17.8
k3	0.9	18	17	8.2	27.9	12.5	9.2	15.6
k4	0.5	27	11	18	20.4	11.1	8.7	29.1
k5	0.7	21	30	28	30	23.8	20.9	13.1
k6	15	16	33	31	24.4	21.9	29.5	26.8
k7	33	35	32	14	10.6	20.8	25	17.5
k8	13	20	29	6.9	27.7	15.4	19.2	20.9
k9	14	22	33	33	19.2	29.7	13.8	12
k10	8.2	31	26	33	25.2	10.5	8	12
k11	13	29	19	28	17.6	25.4	22.9	24.1
k12	22	18	17	10	12.4	24.2	29.2	22.7
k13	31	17	28	16	11.1	22.5	16.3	18.7
k14	8.2	12	17	17	23.7	12.3	16.4	17.2
k15	19	27	28	26	22.4	19.3	19.8	9.4
k16	15	7.7	12	27	8.3	20.6	24.8	28.1

Table 6

Costs of delivering blood packs from BCCs to BTC (million Rials)

m1	m2	m3	m4
(Tabriz)	(Ardabil)	(Oromieh)	(zanjan)k1	17	12	22	11
k2	34	24	30	26
k3	0.9	18	17	8.2
k4	0.5	27	11	18
k5	0.7	21	30	28
k6	15	16	33	31
k7	33	35	32	14
k8	13	20	29	6.9
k9	14	22	33	33
k10	8.2	31	26	33
k11	13	29	19	28
k12	22	18	17	10
k13	31	17	28	16
k14	8.2	12	17	17
k15	19	27	28	26
k16	15	7.7	12	27

Three scenarios are also defined in this study. The first scenario takes into account low-donation conditions, such as the start of the New Year. Where the donation rate is reduced by 20%. The second scenario covers the normal conditions. Finally, the third scenario is defined for high-donation conditions. Where the donation rate is increased by 20%.

4 Computational results

The present case study was conducted using data from the 2014/01/04 to 2016/30/03. The aim was to analyze whether the proposed model could improve BSCND in EAPI.

To solve the proposed model of SMCGR we used, an Intel (Core i7) 1.252 GHz processor, with 8 GB RAM and max turbo incidence and operationalsystem from Microsoft 64 bits. The model was solved by GAMS 23.6.2 software with the optimization solver CPLEX 12.2.1. The total time consumed for optimization of SMCGR model was 1.2 minutes for each scenario. For this reason, dimensions of the proposed mathematical model are addressed in Table 7.

Table 7
Dimensions of the proposed mathematical model

|I| × |J| × |K| × |L| × |M| × |T| × |S| Binary variables Continuous variables60 × 23 × 16 × 8 ×4 × 4 ×3 11792 93323

Table 8

Optimal movement of MTs

priods

T = 1

T = 2

T = 3

T = 4

(2014/01/04 to 2014/30/09)

(2014/01/10 to 2015/30/03)

(2015/01/04 to 2015/30/09)

(2015/01/10 to 2016/30/03)

scenarios

S = 1

S = 2

S = 3

S = 1

S = 2

S = 3

S = 1

S = 2

S = 3

S = 1

S = 2

S = 3

J1 → j2(J* = 2,4,8,12,20,22)

2→4

•

2→12

•

2→18

•

2→20

•

2→22

•

4→2

•

4→12

•

4→18

•

4→20

•

4→22

•

12→2

•

12→4

•

12→18

•

12→20

•

12→22

•

18→2

•

18→4

•

18→12

•

18→20

•

18→22

•

20→2

•

20→4

•

20→12

•

20→18

•

20→20

•

22→2

•

22→4

•

22→12

•

22→18

•

22→20

•

Tables 8, 9 shows optimal movement of MTs and optimal location of BCCs and BCPCs.

Table 9

Optimal location of MTs, BCCs and BCPCs

Periods		T = 1			T = 2			T = 3			T = 4
scenarios	S = 1	S = 2	S = 3	S = 1	S = 2	S = 3	S = 1	S = 2	S = 3	S = 1	S = 2	S = 3
Z1_kts	3,5	3,5	3,5	3,5	3,5	3,5	3,5	3,5	3,5	3,5,6	3,5,6	3,5,6
Z2_kts	6,8	6,8	6,8	6,8	6,8	6,8	6,8	6,8	6,8	6,8	6,8	6,8

Table 10 shows reference location (RL) by donors groups (DG) that can be including a MT, BCC, BCPC or BTC. Finally Tables 11, 12 shows coverge of MTs & BCCs by BCPCs& BTCs for sending blood packs and coverage of BCPCs by BTCs for sending blood samples.

Table 10

Reference location by donors

DG	RL	DG	RL	DG	RL
i1	J2	i21	J12	i41	L6
i2	K3	i22	J12	i42	L6
i3	K3	i23	L6	i43	L6
i4	K3	i24	L6	i44	L6
i5	J4	i25	L6	i45	J20
i6	J4	i26	L6	i46	J20
i7	K3	i27	L6	i47	J20
i8	J4	i28	L6	i48	J20
i9	M1	i29	L6	i49	J20
i10	M1	i30	L6	i50	K6
i11	M1	i31	J12	i51	K6
i12	J8	i32	J12	i52	L8
i13	M1	i33	J12	i53	L8
i14	J8	i34	J12	i54	L8
i15	J8	i35	J12	i55	L8
i16	J8	i36	J12	i56	J22
i17	K5	i37	J18	i57	J22
i18	k5	i38	J18	i58	J22
i19	K5	i39	J18	i59	J22
i20	K5	i40	J18	I60	J22

Table 11

Selected center for post the blood packs by BCCs and MTs

MT &BCCs	L6	L8	M1	M2	M3	M4
J2			•
J4			•
J8	•
J12			•
J20			•
J22		•
K3			•
K5			•
K6		•

Table 12

Selected center for post the blood samples by BCPCs

MT &BCCs	M1	M2	M3	M4
L6	•
L8	•

5 Conclusion

In response to the growing need of having an efficient blood transfusion system, the blood collection and distribution network design problem was modeled as a bi-objective MILP formulation. The major contribution was to develop a novel hybrid approach based on stochastic programming, MCGP and robust optimization to simultaneously model two types of uncertainties by including stochastic scenarios for total blood donations and polyhedral uncertainty sets for demands. This model makes optimal decisions regarding the number and locations of MTs, BCCs and BCPCs in a multi-period planning horizon. Moreover, the model assigns donors to the steed up facilities in each period. The application of the proposed methodology was demonstrated using a case study in EAPI.

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Cities	City codes	Population	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13	C14	C15	C16	C17	C18	C19	C20	C21	Tabriz	Ardabil	Oromieh	Zanjan
Marand	C1	240000	0	73	171	286	257	257	190	184	74	66	114	129	176	201	227	221	180	112	214	252	233	73	311	208	369
Jolfa	C2	55000	73	0	126	353	235	249	182	256	146	138	186	201	249	273	299	293	279	211	286	324	299	138	376	217	434
Varzaqan	C3	49000	171	126	0	153	124	107	56	116	110	165	150	152	200	224	250	244	190	122	197	235	216	91	209	237	354
Khodaafrin	C4	34000	286	353	153	0	46	124	153	215	219	332	251	276	358	381	411	398	280	252	285	313	346	219	268	404	498
kaleybar	C5	92000	257	235	124	46	0	88	67	125	183	255	223	238	286	310	263	330	272	208	202	321	301	173	223	317	410
Horand	C6	4000	257	249	107	124	88	0	67	127	183	255	223	238	296	310	336	330	276	208	204	361	297	169	219	315	407
Ahar	C7	155000	190	182	56	153	67	67	0	60	116	188	156	166	219	243	229	263	205	141	137	254	238	112	163	258	350
Heris	C8	71000	184	256	116	215	125	127	60	0	110	182	150	165	213	237	263	257	263	76	77	189	183	91	169	237	322
Tabriz	C9	1652000	74	146	110	219	183	183	116	110	0	72	40	55	103	127	153	147	133	65	140	178	159	0	228	150	305
Shabestar	C10	127000	66	138	165	332	255	255	188	182	72	0	122	127	175	199	225	219	205	137	214	250	233	72	290	180	368
Osku	C11	77000	114	186	150	251	223	223	156	150	40	122	0	195	243	102	128	122	173	105	180	218	191	36	248	127	326
Azarshahr	C12	104000	129	201	152	276	238	238	166	165	55	127	195	0	48	72	98	92	188	120	195	233	187	58	290	124	348
Ajabshir	C13	86000	176	249	200	358	286	296	219	213	103	175	243	48	0	34	50	44	144	168	243	219	147	98	310	163	300
Bonab	C14	132000	201	273	224	381	310	310	243	237	127	199	102	72	34	0	26	20	120	192	267	195	125	121	353	187	277
Malekan	C15	105000	227	299	250	411	263	336	229	263	153	225	128	98	50	26	0	47	286	218	293	331	120	145	333	165	279
Margheh	C16	245000	221	293	244	398	330	330	263	257	147	219	122	92	44	20	47	0	100	212	287	175	108	139	323	204	261
Hashtrud	C17	68000	180	279	190	280	272	276	205	263	133	205	173	188	144	120	286	100	0	68	143	75	61	119	225	264	197
Bostanabad	C18	101000	112	211	122	252	208	208	141	76	65	137	105	120	168	192	218	212	68	0	75	113	114	58	261	204	250
Sarab	C19	138000	214	286	197	285	202	204	137	77	140	214	180	195	243	267	293	287	143	75	0	188	183	128	87	273	319
Miyaneh	C20	200000	252	324	235	313	321	361	254	189	178	250	218	233	219	195	331	175	75	113	188	0	86	165	153	310	133
charoymaq	C21	36000	233	299	216	346	301	297	238	183	159	233	191	187	147	125	120	108	61	114	183	86	0	193	234	300	177

DG	RL	DG	RL	DG	RL
i1	J2	i21	J12	i41	L6
i2	K3	i22	J12	i42	L6
i3	K3	i23	L6	i43	L6
i4	K3	i24	L6	i44	L6
i5	J4	i25	L6	i45	J20
i6	J4	i26	L6	i46	J20
i7	K3	i27	L6	i47	J20
i8	J4	i28	L6	i48	J20
i9	M1	i29	L6	i49	J20
i10	M1	i30	L6	i50	K6
i11	M1	i31	J12	i51	K6
i12	J8	i32	J12	i52	L8
i13	M1	i33	J12	i53	L8
i14	J8	i34	J12	i54	L8
i15	J8	i35	J12	i55	L8
i16	J8	i36	J12	i56	J22
i17	K5	i37	J18	i57	J22
i18	k5	i38	J18	i58	J22
i19	K5	i39	J18	i59	J22
i20	K5	i40	J18	I60	J22

DG	RL	DG	RL	DG	RL
i1	J2	i21	J12	i41	L6
i2	K3	i22	J12	i42	L6
i3	K3	i23	L6	i43	L6
i4	K3	i24	L6	i44	L6
i5	J4	i25	L6	i45	J20
i6	J4	i26	L6	i46	J20
i7	K3	i27	L6	i47	J20
i8	J4	i28	L6	i48	J20
i9	M1	i29	L6	i49	J20
i10	M1	i30	L6	i50	K6
i11	M1	i31	J12	i51	K6
i12	J8	i32	J12	i52	L8
i13	M1	i33	J12	i53	L8
i14	J8	i34	J12	i54	L8
i15	J8	i35	J12	i55	L8
i16	J8	i36	J12	i56	J22
i17	K5	i37	J18	i57	J22
i18	k5	i38	J18	i58	J22
i19	K5	i39	J18	i59	J22
i20	K5	i40	J18	I60	J22

Robust stochastic multi-choice goal programming for blood collection and distribution problem with real application

Abstract

Keywords

1 Introduction

2.1 Goal programming approach

Table 7 Dimensions of the proposed mathematical model |I| × |J| × |K| × |L| × |M| × |T| × |S| Binary variables Continuous variables60 × 23 × 16 × 8 ×4 × 4 ×3 11792 93323

References

Table 7
Dimensions of the proposed mathematical model

|I| × |J| × |K| × |L| × |M| × |T| × |S| Binary variables Continuous variables60 × 23 × 16 × 8 ×4 × 4 ×3 11792 93323

DG	RL	DG	RL	DG	RL
i1	J2	i21	J12	i41	L6
i2	K3	i22	J12	i42	L6
i3	K3	i23	L6	i43	L6
i4	K3	i24	L6	i44	L6
i5	J4	i25	L6	i45	J20
i6	J4	i26	L6	i46	J20
i7	K3	i27	L6	i47	J20
i8	J4	i28	L6	i48	J20
i9	M1	i29	L6	i49	J20
i10	M1	i30	L6	i50	K6
i11	M1	i31	J12	i51	K6
i12	J8	i32	J12	i52	L8
i13	M1	i33	J12	i53	L8
i14	J8	i34	J12	i54	L8
i15	J8	i35	J12	i55	L8
i16	J8	i36	J12	i56	J22
i17	K5	i37	J18	i57	J22
i18	k5	i38	J18	i58	J22
i19	K5	i39	J18	i59	J22
i20	K5	i40	J18	I60	J22