Introduction
Hospitals are normally mandated with the responsibility of ensuring that there is a good blood inventory system for optimum patient care. However, this is normally not an easy task for hospitals because they handle commodities that have a pre-determined shelf-life. This fact has prompted classical optimum inventory management systems to try to establish a model that greatly reduces under-stocking or over-stocking costs (Miller 1). Hospital inventories have been carefully analyzed in the past to establish optimum stock levels and many researchers have agreed that holding and ordering costs are not very suitable here (Tokay 2).
Inventory management is normally desirable in instances where it is difficult to determine the demand for blood within any given region (Montgomery 4). As mentioned in this study, blood has a limited shelf life and it is a crucial commodity for any country that intends to provide excellent healthcare services to its people. However, blood is not obtained from any source; it can only be sourced from a healthy person. Its use is also strongly felt among injured or ill patients who may need it for survival or healing. Nonetheless, blood cannot be given to any person. It can only be used by people from one blood group (Norwitz 27). From this understanding, a national blood bank is a very important health resource for any country because it is a strong determinant of a nation’s health and wealth. Ataturk affirms that,
“It is here that stocks of blood are maintained in healthy conditions to meet the demand. By the very nature of this perishable commodity, the policy matters regarding blood bank inventory management are average inventory level to be maintained for each group, average age of blood at the time of transfusion and the average amount of blood that perished” (Atamturk 57).
Blood demand normally varies if hospitals or patients are far from the supply chain, but more importantly, there is a highly flexible shift in the orders registered upstream. This observation has been termed the “Bullwhip effect” (Nemhauser 32). In solving the problems arising from this effect, experts have come up with several strategies including dividing the inventory control problem into two parts, which include determining the order quantity and determining the reorder point. The latter problem refers to the minimal amount of blood that triggers the replenishment of bloodstock. The division of the inventory model into two parts stems from a famous researcher named Wilson, whose work birthed the stochastic models (Schrijver 54).
Various models were developed to solve the above inventory problem. However, in detail, three models have been advanced as remedies to the above problem and they include the Newsboy model, Base stock model, and the (Q, r) model (Moon 825). The first model is usually too simplistic because it only considers a single replenishment and its only problem is determining the correct order quantity (Simchi-Levi 56). The base stock model is designed to consider the replenishment of stocks, one at a time, even though there is a ransom demand (Nemirovski 78). The (Q, r) model considers two parts of the inventory management model, ‘Q’ and ‘r’. ‘R’ represents the random demand in the inventory model (which later determines the quantity (Q) which is to be ordered). This quantity is normally supplied after a lead time (1). Since the Newsboy model is simplistic, it will not be used in this paper because it is not used in instances where there are multiple replenishments required. Here, the last two models will be more beneficial to this paper.
This paper focuses on the stochastic demand for optimum blood inventory management in Thailand. Several models can be used in this context including the (Q, r) model, News Vendor model, and the Base stock model but this paper focuses on the base stock model as the ideal model (after which, physical stimulation will be applied to determine the validity of the model). The base stock model will be used for various purposes in determining the optimum blood inventory management in Thailand. However, to understand its appropriateness in this study, a conceptual analysis needs to be done to ascertain its usefulness.
Conceptual Analysis
The base stock model operates in a continuous time-frame and makes several assumptions including the fact that blood demand occurs one at a time; backorders occur when the blood demand is not supplied from the stock; the lead times for replenishments are normally known and fixed; replenishments normally occur one at a time, and products are normally analyzed individually (Wincel 98). The base stock model has been equated to many types of models but the most notable model is the Japanese Kansas system because both models use one order quantity.
The base stock model is based on several principles. First, the model ensures that there is no chance of the blood inventory management model experiencing any stock-outs because it establishes a safety stock level that acts as a buffer stock (Zipkin 3). Secondly, the model is based on the premise that to achieve a given fill rate (the fraction of demand that will be filled from stock), the safety stock will be an increasing function of the mean and standard deviation of demand. Finally, the base stock model is based on the premise that the base stock level (in multistage production) is very similar to the above-mentioned Japanese Kanban model (Nemirovski 75). This paper assumes a Poisson distribution of blood from the national blood bank of Thailand. This variable will be used to establish the reorder point, order quantity, and the safety stock to be established so that any chance of experiencing a total shortage of blood is eliminated.
Application of Base Stock Model on the Blood Inventory Management System
When using the base stock model on the inventory management system, a replenishment lead time will apply to the desired stock level of ‘r’ (Ben-Tal 4). If there is a surge in the demand for blood, the units are obtained from the stock ‘r’. For instance, if one unit of blood is sourced from the stock, the resultant stock level would be ‘r-1’. Here, an order will be placed on one unit of blood to restore the initial level of bloodstock which was ‘r’ (Gallego 51). The blood is bound to be available after a given lead time. Here, the healthcare institution which sourced blood from the country’s main blood bank is bound to replenish the main blood bank with the same number of units that it ordered. The time when this replenishment is bound to happen may vary depending on the technicalities of implementing the replenishment exercise. However, on average, this may range from zero to four days (Copacino 12).
Countries that have used this system, have always noted that blood replenishments sometimes fail to match the blood type ordered from the blood bank but other countries have noted that the probability of such an event occurring is between the ranges of 0.05% to 5% (Montgomery 4). However, for this study, we will ignore the probability of the blood inventory system failing to be replenished by the same blood type.
The base stock inventory model has in the past been studied by many experts and they have noted that the model operates an optimum blood inventory management model, subject to a few assumptions. This fact defines the ‘R-policy’ because ‘r’ is the main variable in this framework and the entire model is also based on the same variable (Handfield 34). Other inventory models (which have the same characteristic as the above model), have their lead time characterized and explained by the queuing theory. Several authors have shown the mathematical connotations of the base-stock inventory model and several other authors have exhibited how the model can be used in the blood inventory management system.
The base stock model is mainly used in this context because the time for ordering and reordering is divided into periods of equal length (Bertsimas 3). In a given instance, several events may occur and they include a receipt of replenishment order, a receipt of inventory, and a receipt of random inventory. Here, it should be noted that the lead time is the fixed number of the period order is received.
In the base stock model, if an order is received, the blood bank officer is bound to verify that the stock demanded is indeed available. After verifying the existence of the stock, he or she is bound to assign the healthcare institution with the oldest possible blood units (Gallego 651). Immediately, a replacement order should be made to reimburse the ordered number of units. Often, the hospital that ordered the blood should be the same institution to reimburse the blood bank. If the blood is replenished immediately, there will be a lead time of zero but if the blood is replenished after a long time (say, if the hospital fails to find the right donors); there will be a positive lead time. Using the base stock model, the desired lead time of not more than four days is recommended but the model encourages the replenishment of blood supplies within zero to two days as opposed to three to four days (Montgomery 4). An exponential distribution is normally established to define the appropriate lead time.
The number of units that may be demanded from the national blood bank within a given week may vary, but the figure is probably a discreet random variable (Federgruen 18). Here, the probability that the national blood bank will have an ‘x’ or less demand is normally given by the cumulative Poisson distribution which Ayers explains as:
“P(l, x) = S / x! and the probability that the lead time is less than y days is g(m, y) = One – e -y. The average lead time will be 1/m. Define p=1/m and let P (x, r) denote the cumulative Poisson distribution up to and including x with parameter r. Let bibe the probability of having I units of blood as on-hand inventory. Then it can be shown that bi = RI i!, i = 0, 1,2,….,R The average on-hand inventory is: I = R – . The fraction of time the system will be out of stock is given by: g = RR e -r / and the service level will then be (1 – g)” (Ayers 11).
Assuming there is a prevailing demand of six units every week and the mean lead time is two days, the average lead time will be 2/7 weeks and the intensity factor r = 1/m will be 1.71.
“If a safety stock of five units is maintained, the on-hand inventory will be – I = 600 – 1.71(0.976/00.993) which is equal to 4.32, or it is rounded off to the nearest whole number, it will be four units. The service level will be (1-g) which becomes (1-0.0002) = 99%. This is only an illustration and the working of the model depends on the values of and R. Taking l = 12 for the O+ group, the intensity factor was found to be r = 3.48” (Bienstock 21).
In computing the remaining values of the ideal safety stock level, several measures will be calculated including the average on-hand stock, beginning stock, and the service factor (Golany 248). Considering the above figures, it is correct to note that the on-hand inventory increases with an increase in the stock registered at the beginning of the computation period.
To determine the inventory level, the following formula is used:
“s minus demand over l + one period” but it is crucial to note that, before the blood demand for a given period is met, the inventory level (plus on-order) equals S (Fleischmann 17). It is also crucial to note that all on-order demand for a prior period is normally met before the following period. However, during the supply of blood for sequential orders, the orders will be supplied at the same time. For instance, if there is a supply to be made for orders realized between the second to the fourth periods, the orders will not be completed until the order for the fourth period is met. Here, there is very little possibility that any demand (for all periods in question) will not be met.
In determining the expected on-hand inventory and backorders, a sharp similarity is drawn between the base stock model and the Newsvendor model. The similarity arises from the fact that the order quantity (s) and the demand distribution follow the same pattern as that realized in the base stock model because, for computation purposes, all the models follow demand over l + one period (Ehrhart 121). The on-hand delivery which is realized at the end of one period can be determined as the expected left-over inventory Q = s. Equally, the expected backorder for a given period can be evaluated as the expected lost units, with Q being equal to the order quantity (Axsater 2). Here, we can see that, in the base stock model, the inventory that remains after a given period is used for the following period as opposed to being salvaged.
From the above analysis, we can establish that the base stock model recommends a short lead time as the ideal measure of optimal inventory control. However, we can also establish that the model is most appropriate in a situation of uncertain demand (Iyengar 257). However, the baseload model needs to have valid results before it is properly relied on as the National blood inventory model.
Physical Simulation
Physical simulation of the baseload model will be aimed at ensuring the baseload model projects valid results (Scarf 7). Physical simulation is crucial in this setup because it affirms that optimum inventory levels including the re-order level (among other statistics discussed in earlier sections of this study) are realistic values. If these figures are not effectively controlled, high holding costs and backorders are likely to be realized because it is difficult to maintain excessive inventory and it is also risky to maintain inadequate inventory (Iglehart 45).
In distributing the demand for blood, it is crucial to maintain a Poisson distribution, where a ‘stat-fit’ software is used to ensure that accurate demand quantities are maintained. At the end of every phase of the inventory model, the total number of excessive inventory and backorders are registered (Ozbay 65). Comprehensively, the physical simulation tool is used to include all the assumptions which are evident in the base stock model, and therefore, this purpose equates the base stock model to the physical simulation tool. At every point of the simulation, the backorder and inventory numbers are calculated.
Conclusion
This paper affirms that governments are normally mandated with the responsibility of ensuring that there is a good blood inventory system for optimum patient care. However, this is normally not an easy task for healthcare departments because they handle commodities that have a pre-determined shelf-life. The base stock model is used in this paper because it is appropriate in situations where there is no deterministic demand. The base stock model assumes that the holding costs are bound to increase as the replenishment lead time also increases. From this understanding, we have seen that the base stock model works best when the lead time is short. Physical simulation is effectively used to validate the results of the base stock model because research studies have affirmed that a physical simulation is a useful tool in modeling stochastic systems. This attribute can also be equated to its ability to model organizational supply systems as well. Moreover, it has been used to validate the results of other similar mathematical models (Montgomery 4).
Comprehensively, we have established that to manage Thailand’s blood bank, an attitude of “act according to the situation” is appropriate. This observation is supported by the fact that it is crucial to maintain a crucial balance between the supply and demand of blood from the national blood bank. The stock base model has been adopted as a viable model that manages Thailand’s blood bank, but the cost variables have been excluded from this analysis because of the nature of the operations (voluntary nature of blood donations and its importance in maintaining good health among the country’s population). Since the national blood bank mainly operates through the withdrawal and replenishment of blood supplies, the base stock policy is a viable model for the blood inventory management process.
Works Cited
Atamturk, Zhang. Two-Stage Robust Network Flow and Design under Demand Uncertainty. California: University of California at Berkeley, 2004. Print.
Axsater, Zipkin. Logistics of Production and Inventory, Handbooks in Operations Research and Management Science. Amsterdam: Elsevier (North-Holland), 1993. Print.
Ayers, James. Supply Chain Project Management. New York: St. Lucie Press, 2004. Print.
Ben-Tal, Goryashko. “Adjusting robust solutions of uncertain linear programs”. Mathematical Programming 99.2 (2004): 351 – 376. Print.
Bertsimas, Sim. “Price of robustness,” Operations Research 52 (2003): 35 – 53. Print.
Bienstock, Ozbay. Computing Robust Basestock. Columbia: Columbia University, 2005. Print.
Copacino, William. Supply Chain Management The Basics And Beyond. New York: The St. Lucie Press/ APICS Series on Resource Management, 1995. Print.
Ehrhart, Richard. “Policies for a dynamic inventory model with stochastic leadtimes,” Operations Research 32 (1984): 121 – 132. Print.
Federgruen, Alex. Logistics of Production and Inventory, Handbooks in Operations Research and Management Science. Amsterdam: Elsevier (North-Holland), 1993. Print.
Fleischmann, Bloemhof-Ruwaard. “Quantitative Models for Reverse Logistics: A Review.” European Journal of Operational Research 103 (2010): 1 – 17. Print.
Gallego, Griffins. “Distribution free procedures for some inventory models,” Journal of Operations Research Society 45 (1994): 651 – 658. Print.
Gallego, Ramachandran. A Note on the Inventory Management of High Risk Items. London: Macmillian, 2005. Print.
Golany, Nemirovski. “Retailer-Supplier Flexible Commitments Contracts: A Robust Optimization Approach.” MSOM 7 (2005): 248-271. Print.
Handfield, Robert. Introduction to Supply Chain Management. Prentice-Hall, Inc, 1999. Print.
Iglehart, Dan. Dynamic Programming and Stationary Analysis in Inventory Problems, Multistage Inventory Models and Techniques. California: Stanford University, 2000. Print.
Iyengar, George. “Robust Dynamic Programming,” Math. of OR 30 (2005): 257 – 280. Print.
Miller, Star. Inventory Control, Theory and Practice. London: Prentice-Hall, 1974. Print.
Montgomery, Johnson. Operations Research in Production Planning and Inventory Control. London: John Wiley, 1974. Print.
Moon, Griffins. “The distribution free newsboy problem: Review and extensions,” Journal of Operations Research Society 44 (1993): 825 – 834. Print.
Nemhauser, Wolsey. Integer and Combinatorial Optimization. New York: Wiley, 1988. Print.
Nemirovski, Ben-Tal. “Robust convex optimization,” Mathematics of Operations Research 23 (1998): 769 – 805. Print.
Nemirovski, Ben-Tal. “Robust solutions of linear programming problems contaminated with uncertain data,” Mathematical Programming Series 88 (2000): 411 -424. Print.
Norwitz, Duncan. “Opportunity costs and elementary inventory theory in the hospital service,” Opl.Res.Qty 24 (1973): 27-34.
Ozbay, Nathaniel. Robust Inventory Problems. Columbia: Columbia University, 2006. Print.
Scarf, Hardley. A Min-Max Solution Of An Inventory Problem, Studies In The Mathematical Theory Of Inventory And Production. California: Stanford University Press, 1958. Print.
Schrijver, Aggrey. Geometric Algorithms and Combinatorial Optimization. New York: Springer-Verlag, 1993. Print.
Simchi-Levi, Ryan. “Minimax analysis for _nite horizon inventory models,” IIE Transactions 33 (2001): 861 – 874. Print.
Toktay, Wein. “Inventory Management of Remanufacturable Products,” Management Science 46.11 (2000): 1412 – 1426. Print.
Wincel, Jeffrey. Lean Supply Chain Management: A Handbook For Strategic Procurement. New York: Productivity Press, 2004. Print.
Zipkin, Peter. Foundations of Inventory Management. Boston: McGraw Hill, 2000. Print.