Scenario: In a hospital laboratory there is a blood banking lab, which 
is responsible for providing different blood types to patients in need. 
The blood banking lab is basically in charge of ordering the different 
blood types and storing them until they are needed. Disregarding certain 
protein factors, the basic blood types are A, B, AB, and O.

Problem: The most significant problem facing the blood banking lab is 
how much to order of each type of blood. If the lab orders too few of 
a certain blood type, then there is a blood shortage and people in need 
of blood will go with out. Also, they have to deal with increased shipping 
costs as a result more frequent ordering. If the lab orders too many of 
a certain blood type, then they have to deal with carrying costs, such 
as money tied up in the inventory and storage of the unused blood. Also, 
they have to deal with the money loss that may come if the blood expires 
and has to be disposed of.

Observe: The data that needs to be collected includes the demand for each 
blood type, the current ordering status of each blood type, the amount of 
blood product disposed of due to expiration, carrying costs, shipping costs, 
and any shortages in blood products. It should be determined if it is more 
efficient to store excess blood (even with the knowledge that some blood 
products will expire) or to order more frequently with current shipping costs.

Mathematical Model: Each blood type is a separate product; therefore a 
mathematical model would need to be created for each blood type. The mathematical 
model for each product’s problem would involve data for the cost of storing 
excess blood (carrying costs including money tied up in product inventory 
and lost cost for expired product) and data from ordering small orders 
frequently. This data would then be graphed using cost against quantity.

Verification and Prediction of Model: The graph would then be interpreted 
using calculus to determine the optimum quantity and which method (ordering 
in excess or in small quantities) is most efficient for each blood type.

Suitable Alternatives: Based on the model, alternatives will be offered that 
best meet the objectives of the blood bank.

Present Results and Conclusion of Study: Display results found from the model 
to the blood bank laboratory and advise them of the most efficient ordering 
practice found in study.

Implement and Evaluate Recommendations: If the blood bank accepts the conclusion, 
aid in setting up the new ordering system which the study found to be most efficient. 
Then, monitor the new system to ensure that it is in fact following the model and 
enabling the blood bank to meet its objectives.