Optimization of stochastic inventory model for MEDICINES AT hospital
Importance of medication goes on increasing with the population and receiving proper medication at the right place at the right time is mandatory for good health. Medicine shortages not only lead to financial losses to pharmaceutical companies but also affect majorly to patients. This motivates the need for proper management of inventories related to medicines and also to ensure proper supply to the hospitals. With this paper I tried to optimize the inventory system by proper and continuous review on production and also for determining optimal solutions for inventory lot size, lead time and distribution.
Research in this field of inventory management at hospitals is very crucial as it is directly related to patient’s health. Careful management of pharmaceutical expresses country’s concern about its citizens. Many healthcare industries face issues regarding management of pharmaceutical which ultimately put negative effect on patient’s health either due to lack of supply or late supply. There is a term, PSC (pharmaceutical supply chain), which consists of three major players- producers, purchasers and pharmaceutical providers. Producers consists of pharmaceutical companies, purchasers consist of grouped purchasing organisation, pharmaceutical wholesalers etc and pharmaceutical providers include hospitals. This chain (PSC) holds very high responsibility ensuring that the right medication reach at the right place at the right time to resolve problems related to any kind of disease. Product perishability is another crucial issue related to medicine as each and every medicine comes with an expiry date and it can’t be use after that date which again demands for proper and timely supply of pharmaceuticals. However, 100% availability at the right time all over world is somewhat difficult but to reach as close as possible to tackle this issue of medicine management must be our aim. In order to achieve this aim by considering PSC, one need to decide order quantities, purchasing dates and inventory level which one can carry without any loss that incur due to excess or under inventory. To form proper inventory management policies operation research plays a very important role as it provides wide range of methodologies that can help hospitals and other health care centres to significantly improve their operations.
Notations and Assumptions:
M- Number of products controlled in supply chain decision variables
Qi- Order quantity for ith product where i= 1,2,3,…………..M.
L- Lead time (in days)
n- Total no of lots of M products delivered by pharmaceutical company to hospital in one production cycle.
Di- Average demand for ith product per year.
di- Expiry rate for ith product.
hi- Holding cost per unit per year.
AI- Order cost per order for the ith product.
pi- Purchasing cost per unit for the ith product.
si- Selling price per unit for the ith product.
ri- Reorder point for the ith product.
Xi- Lead time demand for the ith product and it follows normal distribution with finite mean of DiL and standard deviation of ?i root (L) where ?i is the standard deviation for demand per unit time for the ith product.
E(Xi-ri)- Expected demand shortage for the ith product at the end of cycle.
?i- Fraction of demand that stock is unable to meet, so 1- ?i is the service level.
F- Fixed transportation cost for all products per delivery.
fi- Storage space for ith product.
W- Total space available for M products.
Hospital uses a continuous review inventory policy for all products and the order quantity Qi for the ith product is placed when its inventory level falls to a reorder point ri.
Reorder point is given by: ri = DiL + ki ?i root (L) where ki is the safety factor for product i and satisfies Pr(Xi>ri)=qi and qi is the allowable stockout probability for i during the lead time.
The quantity of the ith raw material required for production in each production cycle is instantaneous.
All expired pharmaceutical products held in inventory by the pharmaceutical company are a constant fraction of the accumulated inventory.
Inventory Model For The Hospital
In this model an OR model is formulated to identify the optimal inventory lot size, lead time and total number of deliveries in a production cycle by minimizing expected total cost incurred by the hospital.
Suppose for product i , hospital places an order of Qi units.
Average cycle time = Qi/Di
Expected order cost = AIDi/Qi
Lead time per unit time = DiR(L)/Qi
Note: Order cost involves cost of preparing invoices, telephone bill & travel expenses.
Before arrival of procurement quantity, the expected net inventory level is equivalent to safety stock only which is given by:
ssi = ri-DiL= ki ?i root (L)
After arrival of procurement Qi , expected net inventory level becomes
= Qi+ ki ?i root (L)
Hence expected on hand inventory = (Max on hand + Min on hand)/2
= (Qi+ ki ?i root (L) + ki ?i root (L))/2
= Qi/2 + ki ?i root (L)
Note: Trade credit finance – In real life, business via share marketing , trade credit finance or permissible payment delays can improve sales in health care industries. Many pharmaceutical companies offer interest free credit period to promote market competition and improve health policy & public health. Before the end of trade credit period hospital can sell goods and accumulate revenue and earn interest. However higher interest is charged if the payment is not settled by the end of trade credit period, it makes economic sense to delay the payments to the end of permissible delay period given by company and here in this model, there is an assumption that pharmaceutical company has given trade credit period for all kinds of products.
So, let tc be the common credit period for all products and hi the holding cost per unit time excluding interest charges.
Expected inventory = Qi/2 + ki ?i root (L)
Expected cycle stock = Qi/2
Holding cost per unit time for cycle stock = hiQi/2
Safety stock carried throughout cycle = ki ?i root (L)
Total cost of safety stock = holding cost + interest
= (hi+piIC) ki ?i root (L)
Where Ic= Interest charge paid per unit(cuurency) in stock to the bank for all products per year.
Id= Common deposit interest rate for all products per year.
So, the interest earned by the hospital per unit time for product i is:
siIdDi/Qi ?0tcDitdt = Di2tc2siId/2Qi
The expected shortage cost is completely backlogged in the previous cycle and is cleared at the starting of current cycle.
Interest earned by hospital = sitcIdDi/Qi*E(Xi-ri) perunit time during credit period
Unsold units at the end of permissible delay period = Qi-Ditc . The company will charge interest for this unit. The hospital has a loan from the bank for unpaid purchase costs for unsold items at an interest of Ic.
Opportunity interest cost per unit time for unsold units of product I is given by
piIcDitC/Qi?tcQi/Di(Qi-Dit)dt = (Qi-Dit)2piIc/2Qi
Since pharmaceutical products are perishable, so by taking it into account we consider di as the proportional probability for expiry of product and so the total amount of expired product for i is diQi.
Expiry cost = Cdi(L) i.e.it is a function of lead time and Cdi(L)= Zi+Z0L where 0<=Z<=1 and Z is some constant.
Transportation cost = FDi/Qi & Labour cost = viDiSo, the expected total cost per unit time for product i is given by:
ETChi(Qi,L)=Di/Qi (Ai+F+R(L))+ (Qi-Dit)2piIc/2Qi + hiQi/2 +
(hi+piIc) ki ?i root (L)+ Ci(diCdi(L)+vi) – Di2tc2siId/2Qi
The expected total cost per unit time for total M products will be then :ETCh(Q,L)=?i=1M ETChi(Q,L)= ?i=1M Di/Qi (Ai+F+R(L))+ (Qi-Dit)2piIc/2Qi + hiQi/2 +
(hi+piIc) ki ?i root (L)+ Ci(diCdi(L)+vi) – Di2tc2siId/2Qi
-sitcIdDi/Qi*E(Xi-ri) where Q = Q1 Q2 Q3 …….QM
Now, this is the total expected cost and by differentiating it wrt to lead time , lot size etc we can get the optimum values.
Management of pharmaceutical inventory in hospitals has become a troublesome task as at the same time hospital need to think economically as well as for the requirement of the patient. Here I have given an OR model with the aim of achieving optimal value for inventory by keeping which hospital can secure itself from the loss related to money as well as potential loss by providing proper supply at the right time. This model improves operational, health policy and strategic decision making for inventory management. It can be used to maintain medical inventory without overstocking or expiration. This approach can be used in hospitals to provide better medical/healthcare services to the patients at a minimum inventory cost.
1-The Plunkett Research Group, 2010, Health Care Trends, http://www. plunkettresearch.com/Industries/HealthCare/HealthCareTrends/tabid/294/ Default.aspx (accessed 10.05.2010).
2- A. Almarsdóttir, J. Traulsen, Cost-containment as part of pharmaceutical policy, Pharm. World Sci. 27 (2005) 144–148. 3-R.B HYPERLINK “http://refhub.elsevier.com/S2211-6923(13)00015-5/sbref4” h . Handfield, E.L. Nichols, Introduction to Supply Chain Management, Prentice Hall, New Jersey, 1999.4-L.R. Burns, Wharton School Colleagues, The Health Care Value Chain Producers, Purchasers, and Providers, Jossey-Bass, San Francisco, 2002.5-I. Karaesmen, A. Scheller-Wolf, B. Deniz, Managing perishable and ageing inventories: review and future research directions, in: K. Kempf, P. Keskinocak, P. Uzsoy (Eds.), Planning Production and Inventories in the Extended Enterprise, Springer, Berlin, Germany, 2011, pp. 393–436.6-O. Aptel, H. Pourjalali, Improving activities and decreasing of cost of logistic in hospitals. A comparison of US and French hospitals, Int. J. Account. 36 (2001) 65–90.7-R.B. Handfield, E.L. Nichols, Introduction to Supply Chain Management, Prentice Hall, New Jersey, 1999.