Modeling a production-inventory-routing problem of blood products using heuristic solution methods

3.1 Research contributions

Although many studies have been carried out on blood supply chain management duo to the attractiveness and importance of research in this field, however, most studies have used the numerical and non-analytical simulation approaches because of the complex nature of problem; so that the transfusion between hospitals and the possibility of supplying demand with matched groups are not investigated within the optimization approach at all. Also, most studies on the management of production and distribution of blood products have provided production policy for the blood center (the place where blood products are produced) and the issue of routing it to hospitals is under consideration. This study covers studies of the application of research to operations in the blood supply chain. This research includes the application of operations research in the blood supply chain. In addition to production management, inventory management, and distribution planning, this study considers inter-hospital transfers to cope with demand uncertainty, as well as offers an innovative problem-solving approach to a case study of Blood transmission services. Research contributions are:

Providing a robust and flexible mathematical model for the integrated production-routing inventory of blood products

3.2 Problem statement: Production-routing-inventory

The problem is modeled as a three-level supply chain, including a blood center and a set of hospitals and patients in a limited time horizon. The blood center is constrained by the production (or replenishment) of each period and should decide on the amount of blood products at the start of each period. Also, a limited-capacity vehicle is available, which provides a delivery route of blood products from the blood center to a subset of hospitals at the start of each period. However, since hospitals have different demand scenarios in each period, there is a possibility of transfusion between hospitals and also direct transfusion from the blood center to hospitals (in general, transfusion) after occurrence of any of any scenario during the period. Other assumptions are as follows.

The blood banks and hospitals inventory is known for any age group at the start of the first period.

Next, after the definition of sets and indices, parameters and variables, the problem presented in certainty condition is modeled; then a brief explanation of the robust method is given and finally, the robust problem is developed.

Notation

Sets and Indices

V: set of hospitals and blood centers i, j∈ { 0, 1, 2, . . , n }

V′: Set of hospitals i, j∈ { 1, 2, . . , n }

T: set of periods t∈ { 1, 2, . . , τ }

G: set age group g, gg∈ { 1, 2, 3, . . , H }bsv

S: set of scenarios s, s ∧ ′ ∈ {1, 2, . . Ω}

Parameters

h _i : Storage cost at the ith hospital and the blood center

PC: Production cost of each unit of blood products

C _ij : Transportation cost of a vehicle between nodes i and j

b _ij : Transfusion cost between node i and j

Ω: Weighted penalty for trade-off between solution robustness and model robustness (shortage cost)

BB ^t : Maximum amount of production at the blood center in the period t

d i t : Demand of the ith hospital in the period t

d i t , s Realization of d i t under the scenario s

I i o , g : Inventory at the start of the first period of the age group g in the ith hospital and blood center

Q: vehicle capacity

C _i : Storage capacity at the ith hospital

ML: Maximum waste amount

λ: Weight assigned to the objective function variance

p _s : sth scenario probability

Variables

I i t , g : Level of inventory in the blood center and the ith hospital from the age group g at the end of period t

I i t , g , s : Realization of I i t , g : under the scenario s

X ij t 1, If node i is met after node j in period t, 0, otherwise

q i t , g : Amount of blood products delivered to hospital i from the age group g in period t

w ij t , g : Amount of blood transmitted from i to j from the age group g in period t

w ij t , g , s : Realization w ij t , g : under the scenario s

δ i t , s : Realization of the demand for blood products (shortage) in hospitals under the scenario s

d i t , g : Amount of blood product demand in hospitals that is provided by age group g in period t

d i t , g , s : Realization of d i t , g : under the scenario s

vv i t Auxiliary variable for removing sub tours

l i t , g : Auxiliary variable for applying FIFO policy

l i t , g , s : Realization of l i t , g : under the scenario s

Mathematical model of the first problem in certainty conditions

min ∑ i ∈ V ∑ t ∈ T ∑ g ∈ G h i I i , t g + ∑ i ∈ V ∑ j ∈ V ∑ t ∈ T c ij x ij t + ∑ i ∈ V ∑ j ∈ V ′ ∑ t ∈ T ∑ g ∈ G b ij w ij t , g(1)

I o t , g = I o t - 1 , g - 1 - ∑ i ∈ V ′ q i t , g - ∑ i ∈ V ′ w o , i t , g g = 2 , 3 , . . H , t ∈ T(2)

I o t , g = F t - ∑ i ∈ V ′ q i t , g - ∑ i ∈ V ′ w o , i t , g g = 1 , t ∈ T(3)

I i t , g = I i t - 1 , g - 1 + q i t , g + ∑ i ∈ V w j , i t , g ∑ i ∈ V ′ - w i , j t , g + d i t , g i ∈ V ′ , g = 2 , 3 , . . H , t ∈ T(4)

I i t , g = q i t , g + ∑ i ∈ V w j , i t , g - ∑ i ∈ V ′ w i , j t , g + d i t , g g = 1 , i ∈ V ′ , t ∈ T(5)

∑ t ∈ T ∑ i ∈ V I i t , g ≤ ML g = H(6)

F t ≤ BB t t ∈ T(7)

d i t = ∑ g ∈ G d i t , g i ∈ V ′ t ∈ T(8)

d i t , g ≤ C i l i t , g i ∈ V ′ t ∈ T g ∈ G(9)

l i t , g - 1 ≤ C i l i t , g i ∈ V ′ t ∈ T g = 2 , 3 , . . . , H(10)

C i ( 1 - l i t , g - 1 ) ≥ ∑ gg = g H ( I i t - 1 , gg - 1 + q i t , gg + ∑ j ∈ V w j , i t , gg - ∑ j ∈ V w i , j t , gg ) - d i t i ∈ V ′ , t ∈ T , g = 2 , 3 , . . . , H(11)

∑ g ∈ G I i t , g ≤ C i i ∈ V ′ , t ∈ T(12)

q i t , g ≤ C i ∑ j ∈ V ′ x ij t t ∈ T , i ∈ V ′ , g ∈ G(13)

∑ g ∈ G q i t , g ≤ C i - ∑ g = 1 H - 1 I i t - 1 , g i ∈ V ′ , t ∈ T(14)

∑ g ∈ G ∑ i ∈ V ′ q i t , g ≤ Q t ∈ T(15)

∑ j ∈ V x ij t = ∑ j ∈ V x ji t t ∈ T , i ∈ V(16)

v v i t - v v j t + Qx ij t ≤ Q - ∑ g ∈ G q j t , g t ∈ T , i , j ∈ V ′(17)

∑ g ∈ G q i t , g ≤ v v i t ≤ Q i ∈ V ′(18)

∑ i ∈ V x io t ≤ 1 ∈ T(19)

x i , j t ∈ { 0 , 1 } i , j ∈ V i ≠ j , t ∈ T I i t , g ≥ 0 t ∈ T , g ∈ G , i ∈ V , j ∈ V(20)

w ij t , g ≥ 0 t ∈ T , g ∈ G , i ∈ V , j ∈ V ′(21)

d i t , g ≥ 0 t ∈ T , g ∈ G , i ∈ V ′(22)

v v i t ≥ 0 t ∈ T , i ∈ V ′(23)

F t ≥ 0 t ∈ T(24)

q i t , g ≥ 0 t ∈ T , g ∈ G , i ∈ V ′(25)

The objective function (1) minimizes inventory, routing, transfusion, and production costs. The constraint (2) updates the blood level of the blood center for the upper two years old age group, while constraint (3) indicates the level of blood center for a one-day blood product according to the amount of production. The constraints (4) and (5) determine the level of hospital inventory for upper two years old age group and upper one-year-old age group age group, respectively. The constraint (6) limits the number of waste. The constraint (7) indicates the production constraint. The constraint (8) indicates that demand can be provided from any age group. The constraints (9) to (11) are used to enforce FIFO policy for use in hospitals. The constraint (9) shows that if demand is provided from an age group g, its associated auxiliary variable ( I i t , g ) , and constraint (10) states that the auxiliary variable for the g-1 group equals one if the auxiliary variable associated with the group g is one; namely the age group g- 1 is used to supply demand if the age group g is not enough to supply demand. Finally, constraint (11) is defined to prevent all auxiliary variables become one, and states that if the available inventory from the upper-g age group is sufficient for the supply of demand, then the other age group g-1 is used to supply demand (the right side of the constraint being positive will result in the zeroing the auxiliary variable related to the g-1 age group). The constraint (12) indicates the capacity of hospitals to hold blood products. The constraint (13) states that in case of meet the hospital by the vehicle, the blood product will be delivered and constraint (14) limits the delivery amount to the hospital’s available capacity.

The constraint (15) indicates vehicle capacity constraints. The constraint (16) is the tour formation constraint and Constraints (17) and (18) are the tour removing constraints. The constraint (19) states that a vehicle is available. The constraints (21) to (25) show the type of variables.

The network considered for this problem includes a blood center and several hospitals. The function of this network is as follows: firstly, a large, limited-capacity vehicle visits a set of hospitals through routing and supplies some blood products to them. Since the blood center and hospitals deal with different supply and demand scenarios, there is the shortage probability. Hence, there is the possibility of transfusion between hospitals and direct sending from the blood center to hospitals (generally, transfusion) and estimating demand with a matched blood type (shortly replacement) to prevent shortage. Figure 1 demonstrates a schematic view of the second problem. More clearly, when a hospital suffers from a lack of a blood type, it can do several measures: 1) to allow for occurrence of shortage, 2) to acquire blood from a matched blood group in the same hospital, 3) to acquire blood from the same blood group in the blood center or another hospital; or 4) to use the matched blood type available in the blood center or other hospital to supply demand. Also, the blood center can also respond to the lack of the matched blood type. Other proposed model assumptions are as follows:

OPEN IN VIEWER

Fig. 1

The schematic view of the second problem: inventory-routing with transfusion.

Table 1 shows the matched blood groups in terms of the Red Blood Cell product. For example, the O^- blood group (blood type 2) can supply blood to all of blood groups and can only receive blood from its blood group. Plasma and whole blood also have their own capabilities. There is no matching problem about platelets, and all blood group owners can donate platelets to each other. The only exception is for the use of this product in lower aged patients, that matching with ABO blood group should be considered in order to receive platelets.

The indices, sets, parameters, and variables related to the second model are presented in the following section. It should be noted that the additional and existing items provided in the first model are no longer expressed.

Notation

Sets and Indices

R: blood group set r, u∈ { 1, 2, . . , 8 }

parameters

F^t,r: Degree of replenishment of the blood center from the blood group r in period t

F^t,r,s: Realization of F^t,r under the scenario s

d i r , t : Demand for hospital i from the blood group r in period t

d i r , t , s : Realization of d i r , t under the scenario s

I i o , r : Inventory of the blood group r in the ith hospital and the blood center at the start of the first period

MS^r,u: Matching matrix of blood groups, r blood donator and u blood receptor group

Variables

I i t , r : inventory level in the blood center and ith hospital from the blood group r at the end of period t

I i t , r , s : Realization of I i t , r under the scenario s

q i t , r : amount of blood product delivered to hospital i from the blood type r in period t

w ij t , r : amount of blood product transferred from i to j for the blood group r in period t

w ij t , r , s : Realization of w ij t , r under the scenario s

δ i t , r , s : Realization of blood product demand (shortage) in the hospital I under the scenario s

d i t , r , u : amount of red blood cell demand from the blood type r, which is supplied by the matched blood type u in hospital i and in period t

d i t , r , u , s : Realization of d i t , r , u under the scenario s

The mathematical model of the second problem in certainty conditions

min ∑ t ∈ T ∑ i ∈ V ∑ J ∈ V C ij x ij t + ∑ t ∈ T ∑ r ∈ R h o I i t , r + ∑ t ∈ T ∑ i ∈ V ′ ∑ r ∈ R h i I i t , r + ∑ t ∈ T ∑ i ∈ V ∑ j ∈ V ′ ∑ r ∈ R b ij w ij t , r + ∑ t ∈ T ∑ i ∈ V ′ ∑ r ∈ R ∑ u ∈ R CB r , u d i t , r , u(26)

I o t , r = I o t - 1 , r + F t , r - ∑ i ∈ V ′ q i t , r - ∑ i ∈ V ′ w o , i t , r t ∈ T , r ∈ R(27)

I i t , r = I i t - 1 + q i t , r + ∑ i ∈ V w j , i t , r - ∑ j ∈ V ′ w ij t , r - ∑ u ∈ R MS r , u d i t , r t ∈ T , r ∈ R , i ∈ V ′(28)

d i t , r = ∑ u ∈ R MS u , r d i t , u , r t ∈ T , r ∈ R , i ∈ V ′(29)

∑ r ∈ R I i t , r ≤ C i t ∈ T , i ∈ V ′(30)

q i t , r ≤ C i ∑ j ∈ V x ij t t ∈ T , r ∈ R , i ∈ V ′(31)

∑ r ∈ R q i t , r ≤ C i - ∑ r ∈ R I i t - 1 , r t ∈ T , i ∈ V ′(32)

∑ i ∈ V ′ ∑ r ∈ R q i t , r ≤ Q t ∈ T(33)

∑ i ∈ V x ij t = ∑ j ∈ V x ji t t ∈ T , i ∈ V(34)

v v i t - v v j t + Qx ij t ≤ Q - ∑ r ∈ R q j t , r t ∈ T , i , j ∈ V ′(35)

∑ r ∈ R q i t , r ≤ v v i t ≤ Q t ∈ T , i ∈ V ′(36)

∑ i ∈ V x io t ≤ 1 t ∈ T(37)

x i , j t ∈ { 0 , 1 } i , j ∈ V i ≠ j , t ∈ T I i t , r ≥ 0 t ∈ T , r ∈ R , i ∈ V(38)

q i t , r ≥ 0 t ∈ T , r ∈ R , i ∈ V ′(39)

w ij t , r ≥ 0 t ∈ T , r ∈ R , i ∈ V , j ∈ V ′(40)

d i t , u , r ≥ 0 t ∈ T , r , u ∈ R , i ∈ V ′(41)

v v i t ≥ 0 t ∈ T , i ∈ V ′(42)

The objective function (26) is to minimize the cost of routing, the cost of storage in blood center, the cost of storage in hospitals, the cost of transfusions between hospitals, and the direct transfer from the blood center to hospitals and replacement costs. The constraint (27) indicates that the blood center’s inventory of any blood group in each period is equal to the blood center’s inventory of that blood group at the end of the previous period plus the amount of replenishment of that blood group, minus the amount sent from that blood group to hospitals via routing, minus the amount sent from that blood group directly to the hospitals. The constraint (28) states that the level of inventory of each hospital for each blood type is equal to the level of inventory of that blood group at the end of the previous period plus the amount received from that blood group via routing, plus the sum of the amounts received from that blood group from the blood center (direct delivery) and other hospitals, minus the sum of the amounts sent from that blood group to other hospitals, minus the sum of the amounts used to estimate the demand for a matched blood type (e.g. the blood group AB^- can be divided into two blood types namely AB- and AB+. So, the coefficient d i t , 8 , 7 and d i t , 8 , 8 are one in the inventory constraint of this blood group, and for the rest namely d i t , 8 , u , u ∈ { 1 , 2 , 3 , 4 , 5 , 6 } are zero. The constraint 3–30 indicates that the demand for each blood type can be estimated by matched blood groups. For example, the second blood group can only receive blood from itself, that is, d i 2 , t = d i t , 2 , 2 . The constraint (30) states that hospital inventory levels should not exceed the capacity of hospitals at the end of the period. The constraints (31) and (32) state that if a vehicle visits a hospital, the delivery value should not exceed the capacity of the hospital. The constraint (33) indicates the capacity limit of the vehicle. The constraint (34) is used to form a tour, while the constraints (35) and (36) are limiting the elimination of the substrate. The constraint (37) also states that a vehicle is available. Finally, the constraints (38) to (42) indicate the type of variables.

In order to model scenario-oriented problems where objective function and problem constraints are measured under different conditions, methods should be used to answer the solution for all the different robust response scenarios (close to the optimal answer). Robust method is one of the ways in which an answer can be achieved by finding the most suitable performance in all scenarios. See Mulvey [22] and Yu & Li [30] for more information in this regard.

4.1 The heuristic algorithm

The proposed solution method starts with an initial solution (i.e., an initial value problem) and improves the solution found iteratively by two local searches called “inserting the best” and “removing the worst” and also a shocking step to prevent the algorithm from being trapped. It is optimally localized.

4.2 Developing the initial solution

Developing the initial solution involves two steps; in the first stage, the paths are formed, and then the values of other variables are determined according to the fixed paths. In order to form the route, 75% of the hospitals are selected randomly in each period, and then, they are included in that route using the cheapest Insertion rule. It is worth noting that the 2-opt operator should be used to improve the formed routes as long as the improvement is possible. For the second stage, the heuristic algorithm uses a mathematical model known as the flow network problem. See Coelho & Laporte [4] and Coelho et al. [4] for further information. The flow grid model for the first and second problems is studied as follows.

•the flow network model for the first problem

Min PCO + ∑ s ∈ s p s Z s + λ ∑ s ∈ s p s ( ( Z s - ∑ s ′ ∈ s p s ′ Z s ′ ) + 2 θ s ) + ω ∑ s ∈ s ∑ i ∈ V ∑ t ∈ T ∑ r ∈ R p s δ i t , r , s(78)

According to the constraints (7), (15), (24) – (25), (47) – (61) and

q i t , g ≤ C i r i t t ∈ T , i ∈ V ′ , g ∈ G(79)

Where r i t is the binary parameter in relation to the absence or presence of hospital i in the vehicle route in period t, as determined in the previous step. The constraint (79) ensures that delivery is only made to hospital visited by the vehicle.

•The flow network model for the second problem

Min ∑ s ∈ s p s Z s + λ ∑ s ∈ s p s ( ( Z s - ∑ s ′ ∈ s p s ′ Z s ′ ) + 2 θ s ) + ω ∑ s ∈ s ∑ i ∈ V ∑ t ∈ T ∑ r ∈ R p s δ i t , r , s(80)

Considering the constraints (33), (39), (67) – (77) and

q i t , r ≤ C i r i t t ∈ T , r ∈ R , i ∈ V ′ , s ∈ S(81)

Construction (83) ensures that delivery is only made to hospital, visited by the vehicle.

As mentioned earlier, two local searches are used to improve the answers. Local search adds the first local search (inserting the best) of unvisited hospitals to the route, that inserting them decrease the total cost more than the insertion of the other hospital. To determine these hospitals as well as the values of other variables attributes such as the amount of production, the amount delivered by the vehicle, the amount of transfer between hospitals, etc., the following model is used, which is derived from the idea et al. (2012). cited in the work by Archetti

•the best insertion model for the first problem

Min ∑ i ∈ V ′ ∑ t ∈ T b i t v i t + PCO + ∑ s ∈ s p s Z s + λ ∑ s ∈ s p s ( ( Z s - ∑ s ′ ∈ s p s ′ Z s ′ ) + 2 θ s ) + ω ∑ s ∈ s ∑ i ∈ V ∑ t ∈ T ∑ r ∈ R p s δ i t , r , s(82)

According to the constraints (7), (15), (24) – (25), (47), (61) and

q i t , r ≤ C i ( r i t + v i t ) t ∈ T , r ∈ R , i ∈ V ′ , s ∈ S(83)

v i t ≤ 1 - r i t i ∈ V ′ , t ∈ T(84) v i t ∈ { 0 , 1 } i ∈ V ′ , t ∈ T

Where v i t is the binary variable related to the insertion or not insertion of the hospital i in route at the t period. b i t denotes the cost of inserting hospital i in period t and it is calculated as b i t = c ij + c jk - c ik (see Fig. 2 from left to right). The constraint (83) shows that delivery will only be made to hospitals located on the route and the inserted hospitals. The constraint (84) also indicates that the hospital, inserted in the route, should not be in the route by itself.

OPEN IN VIEWER

Fig. 2

How to calculate hospital insertion and elimination cost.

•The best insertion model for the second problem

Min ∑ s ∈ s p s Z s + λ ∑ s ∈ s p s ( ( Z s - ∑ s ′ ∈ s p s ′ Z s ′ ) + 2 θ s ) + ω ∑ s ∈ s ∑ i ∈ V ∑ t ∈ T ∑ r ∈ R p s δ i t , r , s + ∑ i ∈ V ′ ∑ t ∈ T b i t v i t(85)

Considering the constraints (33), (39), (67) – (77) and

q i t , r ≤ C i ( r i t + v i t ) t ∈ T , r ∈ R , i ∈ V ′ , s ∈ S(86)

v i t ≤ 1 - r i t i ∈ V ′ , t ∈ T(87) v i t ∈ { 0 , 1 } i ∈ V ′ , t ∈ T

The second local search seeks to remove the visited hospitals, which, in turn it would lead to the value of objective function being further reduced. According to a previous local search, a mathematical model is used to determine these hospitals and their corresponding variable sets.

The worst elimination model for the first problem

Min ∑ i ∈ V ′ ∑ t ∈ T a i t u i t + PCO + ∑ s ∈ s p s Z s + λ ∑ s ∈ s p s ( ( Z s - ∑ s ′ ∈ s p s ′ Z s ′ ) + 2 θ s ) + ω ∑ s ∈ s ∑ i ∈ V ∑ t ∈ T ∑ r ∈ R p s δ i t , r , s(88)

According to the constraints (7), (15), (24) – (25), (47) – (61) and

q i t , r ≤ C i ( r i t - u i t ) t ∈ T , r ∈ R , i ∈ V ′ , s ∈ S(89)

u i t ≤ r i t i ∈ V ′ , t ∈ T(90) u i t ∈ { 0 , 1 } i ∈ V ′ , t ∈ T

Where u i t is the binary variables related to the elimination or not elimination of hospital i from route at t period. a i t is the reduction in the cost of transportation for the elimination of hospital i from route at t period, and is calculated as a i t = c ik - c jk - c ij (see Fig. 2 from right to left). The constraint (89) states that delivery will only be made to hospitals on the route and to non-removed hospitals. The constraint (90) also states that the hospital will be removed from the route that is currently on the route.

The worst elimination model for the second problem

Min ∑ s ∈ s p s Z s + λ ∑ s ∈ s p s ( ( Z s - ∑ s ′ ∈ s p s ′ Z s ′ ) + 2 θ s ) + ω ∑ s ∈ s ∑ i ∈ V ∑ t ∈ T ∑ r ∈ R p s δ i t , r , s + ∑ i ∈ V ′ ∑ t ∈ T a i t u i t(91)

Considering the constraints (33), (39), (67) – (76) and

q i t , r ≤ C i ( r i t - u i t ) t ∈ T , r ∈ R , i ∈ V ′ , s ∈ S(92)

u i t ≤ r i t i ∈ V ′ , t ∈ T(93) u i t ∈ { 0 , 1 } i ∈ V ′ , t ∈

There is a possibility of falling into a local optimal solution if the algorithm is always improving the solution. Therefore, a shocking stage is incorporated into the solution methodology, i.e., like random walk, to avoid this and to expand the search space. At this stage, a period is randomly selected and all hospitals visited at that period are eliminated. To determine the values of other variables, the flow network model (like determining the initial response) is used.

4.5 Adaptive Large Neighborhood Search Algorithm

The Adaptive Large Neighborhood Search Algorithm (ALNS) starts with an initial response, and then addresses searching for neighboring answers in a loop. The search phase in the ALNS algorithm is divided into several sections. In each replication of each section, an operator is selected from several possible operators to move to the neighboring answer. The operator selection is performed by the roulette wheel based on the weight of the operators (the weight of all operators is 1 in the first section). There are three modes in the operation of the operators: 1) operator obtains the best answer. In this case, the operator’s range is increased byσ₁ (the range of all operators is zero at first); 2) the operator has a better answer than the current answer. In this case, the domain increases by σ₂; 3) the result obtained by the operator has not improved that the operator’s domain increases or does not change by σ₃, depending on whether the answer is accepted or rejected according to the acceptance criterion. So that σ₁ ≥ σ₂ ≥ σ₃ ≥ 0. The acceptance criterion mechanism is in this way that a random number is created between zero and one; if the generated number of is lower that ΔS (94) phrase, the new answer is accepted; otherwise, it will be rejected. S _n in this case denotes the value of the objective function of the new answer, and S _f denotes the value of the objective function of the previous answers. In fact, in early iterations where the temperature is too high, it is more likely to accept worse solutions. With a gradual decrease in temperature, there will be less chance of accepting worse solutions in the final iterations. Each section also requires that the range of operators to be set at zero and the weight of the operators is calculated using Equation (95).

Δ S = 100 * ( S n - S f ) / S f(94)

w i = { w i , if o ij = 0 ( 1 - η ) w i + η π v i o ij , if o ij ≠ 0(95)

Where W _i is the weight of the operator i, o _ij is the times of using the operator i in the last section j; η ∈ [0, 1] is the response factor and v _i  ≥ 1 is the normalization factor (this factor depends on the operator’s computational time). π _i is the total range of the operator i in the last section. It should be noted that the weighted acceptance criterion is used in this study. The idea of this acceptance criterion is that the temperature is so high in the initial iterations so that it is more likely to accept worse solutions. With a gradual decrease in temperature, there will be less chance of accepting worse solutions in final iterations.

The results obtained by the heuristic algorithm and ALNS are compared in this section. This comparison is provided in Table 4. As you can see, the heuristic algorithm outperforms the ALNS algorithm in terms of gap and time to solution.

As mentioned earlier, ω leads to a trade-off between the solution robustness and the model robustness. Generally, lower values of Ω lead to a lower solution robustness (closer to the optimal answer) and higher model robustness (farther than the feasible answer). When Ω increases, the solution robustness increases, while the model robustness decreases (closer to the feasible answer and farther away from the optimal answer). Figure 4 indicates the balance between the solution robustness (total cost) and the model robustness (under realization of demand or shortage) for 70 hospitals. As one can see, with a rise of Ω from 10,000 to 100,000 Tomans, the total cost increases to 818699891.5; so that the number of shortages is reduced to zero. A risk-averse decision maker chooses high values of Ω to completely avoid shortages; however, the risk-taker chooses low values of Ω to reduce their costs. In fact, any decision-maker can choose some of the Ω according to its degree of risk-appetite. The increase in λ causes the final answer to be more sensitive to changes in input data (Mulvey et al., 1975); so according to Lalmazloumian et al. [18], λ is considered equal to one in this study.

OPEN IN VIEWER

Fig. 4

The trade-off between the model robustness and the solution robustness.

In this section, a comparison is made between the total cost, the number of shortages and the number of losses in two modes of transfusion and lack of transfusion between hospitals for 70 hospitals. For this purpose, the shortage cost is set at 70,000 Tomans. As shown in Figs. 5 and 6, the cost and number of shortage are far less than that in the transfusion mode than lack of transfusion. The results also show that the amount of waste is the same in the two cases and does not exceed the maximum planned loss. In fact, the transfusion has not helped to reduce waste and the application of FIFO policy has been investigated in two cases for consumption and production optimal policy leading to a lack of loss in periods other than the first period [17].

OPEN IN VIEWER

Fig. 5

Comparison of the number of shortage in the two modes of transfusion and lack of transfusion between hospitals.

OPEN IN VIEWER

Fig. 6

Comparison of total cost in two modes of transfusion and lack of transfusion between hospitals.

The comparison of the results obtained with the heuristic algorithm and the ALNS algorithm for the second model is presented in Table 5. As it is clear, the heuristic algorithm outperforms ALNS algorithm in all cases. It has only a few weaknesses in terms of the gap in examples with 10 hospitals.

Based on the trade-off between the solution robustness (total cost) and the model robustness (under realization of demand or shortage) for 70 hospitals, it can be seen that with an increase of Ω from 0 to 50,000 Tomans, the total cost increases to 5042401719, so that the number of shortages is reduced to zero. The number of shortages from each blood group also reaches zero as shown in Fig. 7. Moreover, Figure 8 shows the Number of shortages from each blood type for shortage cost variations. A risk-averse decision maker chooses high values of Ω to completely avoid shortages; however, a risk-taker chooses low values of Ω to reduce his costs.

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Fig. 7

Trade-off between the solution robustness and the model robustness.

As mentioned earlier, three low, medium and high demand and supply scenarios with probabilities of 0.3, 0.5 and 0.2 (basic assumption) are considered, which allows for changing conditions, such as, changing the probability of scenarios. Furthermore, sensitivity analysis is conducted in this section to detect changes in results (total cost, shortage, vehicle value, direct delivery, transfer, replacement, and inventory level) in four modes of: 1) both transfusion and replacement, 2) only replacement, 3) only transfusion, and 4) neither replacement nor transfusion. To this end, replacements, transfusion, and both are eliminated from the model, respectively. It should be noted that only transfusion between hospitals are eliminated and direct transfusions from the blood center to hospitals are not eliminated. Also, Ω is adjusted at 40,000 Tomans for these comparisons. Also, four assumptions I, II, III and IV are considered for scenarios probabilities. In assumption I, all scenarios have the same probability of occurrence) p₁ = p₂ = p₃ = 1/3. In assumptions II, III and IV, the highest probability (0.8) is assigned to the first, second and third scenarios, respectively and the remaining (two) scenarios have a probability of 0.1. The results of this sensitivity analysis are presented in Table 6.

As it is clear from the Table 6, the total cost in the first case where both replacement and transfusion are allowed is lower than other states in the mentioned assumptions. In general, in the absence of transfusion between hospitals, the total cost is higher than the one in which the transfusion is allowed. In fact, it can be concluded that transfusion helps reducing cost mostly through decreasing transportation costs than reducing replacements costs. Also, the total cost in assumption II (allocation of higher probability to the first scenario) is lower than all assumptions.

Regarding the usefulness of the transfusion and replacement in reducing the shortage, the transfusion is more important as is clear from the results. In fact, the replacement value in the basic assumptions I, II and III is very low and is not proposed by the model. However, when the probability of the third scenario increases (assumption VI), the replacement value is also increased and it is suggested that a replacement be made to reduce the shortage. That is, when the probability of high demand increases, it is suggested that a replacement be made to reduce the shortage.

At the shortage cost of 40,000 Tomans, the shortage amount for basic assumptions I, II is less than 100 units, while it is very high under the VI and III assumptions. Also, the shortage cost for the VI assumption is increased to 100,000 Tomans, and it is observed that in order to reduce the shortage, the direct delivery, transfer, replacement and inventory levels of the blood center should increase to 8761, 8756.621, 946, and 6203 units. This is while the amount of vehicle and hospital inventory levels should be reduced to 4809 and 12313 units, respectively. Also in this case, the total cost will be 52,612,330 Tomans and the shortage will be 9 units. Figure 9 shows the replacement levels for matched blood groups for this assumption and shows the shortage cost of 100,000 units. As it is clear, most of replacements is related to the blood type O ⁺ (donator) and B ⁺ (receptor), followed by the blood group A+ (donator) and AB ⁺ (receptor).

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Fig. 8

Number of shortages from each blood type for shortage cost variations.

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Fig. 9

Replacement levels for matched blood groups.