South Africa faces a critical shortage of blood donors, leading to substantial deficits in the national blood supply. Blood donations are vital for the treatment of life-threatening conditions, making it crucial to develop eficient models for the management of blood stocks. This paper presents a mathematical model to optimize blood donation and ensure suficient supply to meet fluctuating demands. The model captures the complex interactions within the blood banking system, focusing on minimizing costs, reducing waste, and eficiently distributing blood units. Specifically, it addresses daily supply challenges by minimizing the need for emergency imports and reducing blood wastage due to expiration while meeting all demand requirements. The core objective is to minimize blood wastage and reduce the reliance on imported blood banks during emergencies. The proposed objective function incorporates variables such as emergency importation and expiration rates, and robust optimization techniques are applied to identify optimal solutions while satisfying operational constraints. Symbiotic Organism Search (SOS), Genetic Algorithm (GA), and Particle Swarm Optimization (PSO) methods are utilized for optimization. Among these, SOS demonstrated superior performance, achieving the lowest levels of importation and wastage. However, the algorithms were unable to significantly reduce supply levels due to the accumulation of excess stock from the previous day, which carried over into the next day. This paper provides valuable information on blood supply management and highlights the potential for optimization techniques to improve eficiency and sustainability in blood banking. expiration, Blood assignment problems HC@AIxIA 2024: 3rd AIxIA Workshop on Artificial Intelligence For Healthcare ∗Corresponding author. †These authors contributed equally.
Blood is an essential fluid that delivers oxygen and nutrients to cells while removing carbon dioxide
and waste products. The primary constituents of blood consist of red blood cells, responsible for
oxygen and carbon dioxide transport; white blood cells, crucial for combating pathogens and aiding
in immune response; platelets, which facilitate clot formation to prevent blood loss from injuries; and
plasma, the fluid component transporting blood cells and platelets throughout the body [
A blood bank is a center where blood collected from blood donation is stored and preserved for later use in blood transfusion. Key participants and entities within the blood supply chain process include blood donors, blood banks, hospitals, and patients. Figure 1 outlines the general interactions between each participant. In addition, it highlights the flow of blood from donation to utilization, emphasizing the crucial roles played by each entity.
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The operations in a blood bank include the collection of blood from donors, processing the blood, testing its health and specific properties, separating blood units based on blood type and other factors, and finally, storing these operations. As illustrated in Figure 1, the process of blood collection and distribution begins with the testing and screening phase, where blood collected by the South African National Blood Service (SANBS) undergoes rigorous testing to ensure its safety and suitability for transfusion. This includes screening for diseases such as HIV and other potential contaminants. Following successful screening, the blood collected is transferred to designated blood banks for further processing. Here, blood is subjected to processing techniques to preserve its quality and extend its shelf life. Once processed, blood is classified according to its blood type and subjected to additional screenings to confirm its safety for transfusion. Blood is quickly discarded to avoid potential harm if abnormalities or contaminants are detected during this stage. However, if the blood passes all necessary screenings and is deemed safe, it is sent to various hospitals and medical facilities according to their demands and needs. This ensures that hospitals have a steady blood supply to meet the demands of patients who require transfusions. The model derived later will focus only on the phases after donor acceptance and blood collection. It will not consider the initial step of donor evaluation or the risk of donor rejection, as these factors can vary over time and significantly impact the inflow of blood units to a hospital.
Alternatively, hospitals can transfer or export surplus blood to other facilities facing emergencies
or experiencing lower demand. This collaborative approach helps optimize the distribution of blood
resources, ensuring that they are used eficiently where they are most needed. Any surplus blood that
remains after transfusion or exportation is stored in designated storage facilities. These blood reserves
serve as a crucial backup to ensure that an adequate blood supply is always available, particularly
during increased demand or emergencies. However, it is essential to note that despite these measures,
blood units have a finite shelf life. If blood expires before it can be utilized, it is discarded to maintain
the integrity and safety of the blood supply. Hospitals can import additional blood units from other
facilities when they face a sudden surge in blood demand. This emergency measure helps to address
immediate needs and ensures that patients receive timely and life-saving transfusions. These blood
products use first-in-first-out (FIFO) and last-in-first-out (LIFO) methods to reach hospitals [
The demand for blood has increased worldwide, while there are low levels of blood donations. This means that the demand for blood exceeds the blood supply in hospitals. This poses a threat to patients in need of blood. Blood is being wasted through expiration when a specific type of blood is not in demand. There is insuficient blood stored for emergency events such as car accidents.
This section presents a brief overview of preliminary studies and related works to further emphasise the current study’s relevance.
According to Charpin et al. [
Blood compatibility is one of the most important factors in blood transfusions. Additionally, natural
blood grouping restricts transfusion options due to blood compatibility. Karl Landsteiner’s 1901
discovery identified the human blood system, the ABO system, comprising four primary groups. In 1940,
Reid et al. [
The shelf life of blood depends on the type of blood product and the temperature conditions. Whole
blood lasts 30 days, red blood cells 24 days, plasma 12 months, and platelets 5 days. This study focuses
on whole blood cells; thus, we will use 30 days for the expiration period in accordance with the study
conducted in [
A model that describes the inflow and outflow processes of blood units is necessary to enhance the
blood supply chain. In 1976, a planning model was developed by Cumming et al. [
The study by Dufourq et al. [
and − To enhance the mathematical model introduced in prior research, which omitted considerations such as blood expiration and emergency demand, this study aims to refine it. Thus, the expanded model incorporates factors including the rate of blood expiration, the volume of blood expiring per unit time, the quantity of blood imported from other blood banks for each blood group, and the volume of blood exported to other blood banks for each blood group. The mathematical model maintains eight diferential equations, each corresponding to a distinct blood type: +, −, +, −, +, −, +, . Each equation denotes the rate of change of the total blood units for the respective blood type, determined by subtracting the total available blood for transfusion from the total units transfused, adjusted for expired blood and accounting for both imported and exported blood units for emergency purposes at other hospitals.
Figure 2 shows the interactions between blood types +, −, +, −, +, −, +, and −, with
simplified as . The flowcharts depict how each type evolves through expiration, importation, exportation, and donations. Each blood type is color-coded, representing the system of diferential equations governing changes in blood units. Inputs are added, and outputs are subtracted from the equations.
This study addresses the removal of expired units after transfusions or transfers and importation during emergencies. Importation increases blood availability, while exportation decreases it. The model represented by Equation 1, 2, 3, 4, 5, 6, 7, and 8 serves as a mathematical representation of the problem, accounting for the complexities of diferent blood types. 3.1. Mathematical Model − = − − ( − − + − + + − − + − + + − − + − + + − − + − +) + −−
1 − − −, + = + − ( + + + + + + + + + + +) + + − 2 + − +, − = − − ( − − + − + + − − + − +) + − − 3 − − −, + = + − ( + + + + +) + + − 4 + − +, − = − − ( − − + − + + − − + − +) + − − 5 − − −, + = + − ( + + + + +) + − − 6 + − +, − = − − ( − − + − +) + − − 7 − − −, + = + − + + + + − 8 + − +. (1) (2) (3) (4) (5) (6) (7) (8) For representing an element of +, −, +, −, +, −, +, and −, where is denoted as , and for as an element of {1, 2, 3, 4, 5, 6, 7, 8}, the explanation of each term in this model is as follows: denotes the amount of blood available for donation, while represents the rate of change of the blood type over time. indicates a source of blood from donations, and the terms refer to various difusion rates between diferent blood types. The term represents an external blood input from other hospitals, often in emergencies. stands for the expiration rate of blood units for each blood type, and accounts for the degradation or removal process due to expiration, proportional to the concentration of . Finally, represents the exportation of blood to other hospitals to meet their emergency demands. The number of blood units of a particular blood type imported ( ) and exported ( ) on any given day will depend on the number of emergencies in the hospital. The blood units indicated by will be imported into the hospital to address emergencies, while those represented by will be exported to other hospitals to meet their emergency needs. Additionally, the volume of blood transfused during emergencies is critical in determining the net inflow and outflow of blood units within the healthcare system.
The equation =1 =1 = min (∑ () + ∑ () ) , (9) subject to
denotes the formulation of the model objective function and aims to minimize the amount of blood imported and expired for each blood type. It will be tested with the dataset, and its results will be compared to those reported in previous studies.
Equation 10 calculates the remaining amount of blood after it expires 30 days post-issuance, rendering it unsuitable for hospital use due to health reasons: subject to > 30 . Where () determines the amount of blood that remains after it has been issued.
If the demand over the period after 30 days is less than the supply before the 30-day expiration, the expired amount will be the diference between this supply and the demand, as the blood will surpass the 30-day expiration period. This can be mathematically represented as follows: subject to When > 30 , if then, where 1 ≤ ≤ 365 . 3.2.2. Importation () = ∑ (), ∞ =31 () = () − (), () ≥ (), 1 ≤ ≤ 365.
=30 ( ∑ ()) < ( − 30), () = ( − 30) − ( ∑ ()) ,
=30 Hospitals frequently encounter situations where there is an urgent need for blood due to emergencies. In such cases, hospitals can initially import blood from other compatible blood types and then resort to importing blood from other hospitals when needed. Equation 12 calculates the amount of blood imported for each type.
() = +() + −() + +() + −()+ +() + −() + +() + −()
Blood banks and hospitals can import blood from other compatible blood types when there is a shortage of a particular blood type, which can be represented as:
If then
() > (), () = () − (). represents a distinct value derived from the blood data set, facilitating individual manipulation in subsequent calculations. (10) (11) (12) (13) (14)
In this research, Metaheuristic Algorithms such as Symbiotic Organism Search (SOS) [
The SOS and GA algorithms have demonstrated superior performance compared to other algorithms examined in each study. The algorithm incorporates three phases inspired by real-world biological interactions: mutualism, commensalism, and parasitism phases.
Mutualism Phase: An organism is selected to pair with organism . Together, these organisms engage in mutation to enhance the survival probabilities of organisms within the ecosystem. The ofspring solutions new and new are computed based on the mutualistic symbiosis between the parent organisms and . The calculations are defined as: (15) (16) (17) (18) new = + rand(0, 1) ∗ ( best − Mutualism_Vector ∗ BF1), new = + rand(0, 1) ∗ ( best − Mutualism_Vector ∗ BF2),
Mutualism_Vector = +
. 2
Commensalism Phase: Randomly selecting two organisms, and , from the ecosystem, we modify organism with the assistance of organism . The resulting child solution, derived from this modification through commensal symbiosis between organisms and , is expressed as: new = + rand(−1, 1) ∗ ( best − ).
Parasitism Phase: Organism is randomly selected, and a Parasite Vector is created by duplicating and modifying some dimensions. Then, another organism is chosen as the host and is replaced by the parasite vector, which usually has a better fitness value than . However, if has a higher fitness, it becomes immune to the Parasite Vector, preventing it from surviving in the ecosystem.
This algorithm utilizes recombination and mutation to generate new chromosomes, akin to biological reproduction. The mutation alters genes within the chromosome. GA aims to evolve the optimal chromosome for solving a given problem. The algorithm comprises three main components: natural selection, mutation, and crossover.
Natural Selection: In nature, individuals with superior survival traits survive for longer periods. Consequently, over time, the population becomes dominated by genes from these superior individuals, while genes from inferior individuals diminish. Species with high survival rates thrive, whereas those with low survival rates perish. This is the theory of natural selection.
Crossover: During the crossover operation, two individuals combine genetic material to create diverse ofspring. The parent strings yield children strings based on a chosen crossover point. With a crossover probability , 100 ⋅ % of the population undergoes crossover, while 100 ⋅ (1 − )%remains unchanged. Common methods include single-point and double-point crossovers.
Mutation: Mutation introduces random variations into the genetic search process, thereby preventing the population from becoming stuck in local optima. It enhances diversity within the population by operating at the bit level: during reproduction, each bit in the ofspring has a small probability of mutation, typically denoted as mutation probability .
The PSO algorithm creates global memory for the whole population of particles by recording the
bestever position and the corresponding fitness value. This is computed on every iteration of the algorithm.
In a PSO system, particles traverse a multi-dimensional search space by ”flying” around until they reach
a relatively stable state or until computational constraints are met [
The global best position found by the swarm from the beginning of the search up to the current iteration is represented by the term is calculated using the equation:
The study uses real-world blood datasets to implement state-of-the-art metaheuristic algorithms that include SOS, GA, and PSO. Data sourced from the Enugu National Blood Transfusion Service in Nigeria, spanning from 2010 to 2018, was adapted from the Nigerian Enugu blood bank’s records over a decade (2009-2019). These datasets detail monthly distributions of blood units across various blood types. Ethical concerns have been carefully considered throughout the process of collecting data. The confidentiality and anonymity of the individuals in the data set have been strictly maintained.
In this section, we describe a series of experiments conducted to evaluate the practicality of the proposed mathematical model and the eficiency of the GA, SOS, and PSO optimization algorithms. The experiments were performed on a computing platform equipped with an Intel Core i3 CPU running at 1.20 GHz, 8 GB of RAM, and the Windows 11 operating system. All three algorithms were implemented using Python.
Diferent population sizes of 50, 100, and 150 were tested during the experiments. These population
sizes were chosen to maintain consistency with a study in [
+1 = +1 + . =
, { , = 1, 2, .., , = 1, 2, .., }.
selection of parameters for the SOS, PSO, and GA algorithms is consistent with the implementation
used in study [
The discrepancy between demand and supply must be zero for an optimal solution, but no such solution was found due to the excess supply. This occurred because leftover blood units from the previous day were carried over, increasing the total supply. As a result, the algorithms couldn’t meet the required optimal supply. Additionally, import values should be minimal, but this condition wasn’t satisfied due to high import quantities. However, the expiration values, which should approach zero, were met. All supply, import, and expiration conditions must be satisfied for a valid solution.
The SOS algorithm achieved the lowest importation levels compared to GA and PSO, as shown in Tables 3 and 5, which aligns with our goal to minimize imports. Additionally, from Tables 3, 4, and 5, the PSO algorithm consistently minimized blood expiration, with values very close to zero across all population sizes. In contrast, other algorithms still had blood units expiring after 30 days. The diferent population sizes investigated show trends in monthly blood volume (imported versus expired) for a population of 50 monitored over a108 months(2010–2018). − − + + − − + + − − + + − −
The comparison of the SOS, GA, and PSO algorithms in managing blood supply, import, and expiration across all months (Tables 3, 4, and 5) reveals distinct performance patterns. For a population size of 50, the SOS algorithm delivers the highest supply for −, with minimal import and near-zero expiration, − + − + − + −
+, making it the most eficient. The GA algorithm shows good supply for +, lower import for −, but struggles with expiration rates for +, −, and −. The PSO algorithm minimises import and expiration, particularly for +, though zero importation is impractical. For a population size of 100, SOS ofers a high supply for most blood types with low imports and expirations but struggles slightly with +. GA performs well for −, with low import for −, though expiration for
needs improvement. PSO achieves high supply for + and −, while minimizing import and expiration rates, making it the most efective at maintaining supply with minimal waste. At a population size of 150, SOS maintains a high supply for − with low import and expiration rates. GA provides a strong supply for + and −, but shows less eficient management for expiration, especially for + and
−. PSO delivers the highest supply for −, maintaining minimal import and expiration across all blood types. rates.
In summary, SOS demonstrates the most consistent performance in supply management, while PSO excels in minimizing import and expiration. GA shows room for improvement in managing expiration
When the population is initially set to 50, Figures 4, 5, and 6 reveal that imported blood volume exhibits high variability with occasional spikes, while expired volume remains consistently low. Notably, the PSO model shows the highest peak in imported blood, followed by GA, with SOS exhibiting lower peaks. Despite the algorithmic diferences, trends indicate that imported blood volume fluctuates significantly over time, whereas PSO maintains a zero expired volume, indicating SOS’s eficiency in this scenario.
As the population increases to 100, a correlation between importation and expiration is anticipated. In Figure 7, SOS shows significant fluctuations in importation and more minor variations in expired blood. Conversely, Figure 8 depicts both volumes experiencing frequent volatility, yet the expired volume remains low. The PSO model in Figure 9 illustrates more significant import variability while the expired volume remains near zero. Overall, GA proves to be the most eficient at this population size.
With a population size of 150, blood imports display similar fluctuations to those seen in populations of 50 and 100, but with slightly lower peaks. In Figure 10, the SOS algorithm shows that expired blood remains low relative to imports, suggesting a negative correlation between expired units and population size. Figure 11 shows more frequent, less pronounced peaks in imports for GA, while expired units occur more frequently than in SOS, indicating less eficient usage. The PSO model in Figure 12 reflects a slight decrease in imports compared to smaller populations, with expired units remaining low. In this case, SOS again emerges as the most eficient.
Overall, across varying population sizes, the SOS algorithm consistently demonstrates superior eficiency in managing blood imports and minimizing expired units.
The computation times for the SOS, GA, and PSO algorithms at diferent population sizes ( ∈ {50, 100, 150}) reveal distinct trends. Additionally, Figure 13 shows that SOS is always higher compared to GA and PSO, and there is a positive relationship between population size and computational time. Its time significantly increases with larger populations, indicating its sensitivity to dataset size, which suggests it may not be the most eficient for higher populations. The GA algorithm also shows a rise in computation time with increasing population size, but this increase is more moderate than that of SOS. While GA requires more time than PSO, it remains faster than SOS, making it a better balance between accuracy and eficiency. In contrast, the PSO algorithm consistently exhibits the lowest computation times at each population size, scaling eficiently with a relatively linear increase in time. Thus, PSO is highly suited for managing larger populations while maintaining computational eficiency. In summary, all algorithms show increased computation times as population size grows. However, SOS has the most MA SOS GA PSO MA SOS GA PSO MA SOS GA
PSO substantial rise, GA exhibits moderate scaling, and PSO is the most scalable and eficient for larger datasets.
Additionally, diferent iteration counts of 1000, 1500, 2000, and 3000 were investigated to analyze how increasing the number of iterations would impact the results. Most prior studies utilized 1000 iterations as a standard, providing a baseline for comparison. The selection of these four specific iteration counts was deliberate, following a systematic progression that allows for a thorough examination of the efects of increased iterations on convergence and solution quality. Starting with 1000 iterations, a commonly accepted threshold in the literature, ensures consistency with existing studies. The increments of 500 allow for a gradual exploration of the efects of increased computational efort on the results. The results for iteration 1000 are represented by Table 3 which was used initially before changing the iteration sizes.
Increasing the number of iterations from 1000 to 1500, 2000, and 3000 significantly improved the results across the algorithms used (SOS, GA, and PSO) in terms of importation and expiration metrics. Both the SOS and PSO algorithms exhibited a marked reduction in importation values, achieving nearzero expiration rates, particularly at higher iterations, which aligns with the goal of minimizing non-zero importation and eliminating expiration. In contrast, the GA algorithm showed less sensitivity to iteration increases, maintaining low importation levels but failing to reach zero expiration consistently. Overall, the findings suggest that higher iterations enhance the performance of most algorithms, particularly SOS and PSO, in optimizing blood supply management. Table 6 and 8 show that the SOS algorithm outperformed the other algorithm as it has the smallest values for both importation and expiration. But for Table 7, the PSO algorithm has the smallest values for both factors.
The analysis of blood unit trends across diferent algorithms and iterations (Figures 14–22) suggests that the SOS algorithm may be the most efective in balancing blood imports and minimizing expirations over time. At iterations 1500, 2000, and 3000, SOS (Figures 14, 17, and 20) shows relatively stable import rates with consistently low expiration levels, indicating an eficient management of blood inventory. In contrast, the GA (Figures 15, 18, and 21) exhibits high and frequent import peaks across all iterations, which could lead to unnecessary over-importation without a proportional decrease in expirations. Similarly, the PSO algorithm (Figures 16, 19, and 22) maintains high import levels but with fewer expirations, indicating limited success in minimizing imports. Overall, the SOS algorithm emerges as the most suitable for minimizing both blood imports and expirations as iterations increase, demonstrating more excellent stability and eficiency in blood unit management compared to GA and PSO.
These results are quite similar to those obtained in the study by Ezugwu et al. [
To enhance their efectiveness, it is essential to introduce variables that account for the remaining blood from the previous day, as carrying over the remainder did not yield satisfactory results. Future work should focus on minimizing the remainder to reduce or eliminate carryover. Additionally, factors such as seasonal variations, patient demographics by supplying blood based on the demographics of the patient population, and the investigation of other algorithms should also be considered.