Analysis of an order-up-to-level policy for perishables with random issuing

Abstract

Much of the research in perishable items inventory management has focused on the first-in-first-out issue process. However, motivated by the technical characteristics of the blood unit issue process, we model an order-up-to-level policy under periodic review setting with random issue of items from inventory. We provide empirical evidence in support of the random issuing assumption using real data on serial numbers of blood units issued from a blood bank. For general demand distribution we derive exact expressions for per period expected shortage, expected wastage and expected cost as functions of the policy parameters R (order-up-to-level) and T (review period). Since the exact model becomes computationally burdensome with increase in the number of periods of life of the perishable item an approximate model for the random issuing process is developed. The accuracy of the approximation is affirmed using simulation analysis. A gradient search-based heuristic is provided to identify the optimum policy parameters for the approximate model. A real life application of the model is demonstrated in determining the optimum frequency and order-up-to-level for blood collection at a blood bank.

Keywords:

• inventory
• perishable items
• (R, T) policy
• random issuing policy

Acknowledgements

The authors would like to thank Hasmukh Gajjar, Deepak Iyengar, Arun Kumar, two anonymous referees and the seminar participants at Indian Institute of Management Indore for helpful comments and suggestions.

Notes

1 Human blood can be categorized into eight major blood types: A+, B+, O+, AB+, A−, B−, O− and AB−.

2 The eight blood types represent only one major blood classification system known as the ABO and Rh system. There are 32 such classification systems. Blood of same ABO and Rh type may not be always compatible. There can always be some antigens in the donor’s blood which can react with the antibodies in the recipient’s blood, which can be fatal in certain cases.

3 In extreme cases the patient’s blood may not be compatible with most of the blood units in the inventory.

4 See Bakalar (2013) and Aubron et al (2013) for further details.

5 Some papers in this list have considered both the backorders case as well as the lost sales case.

6 Platelets generally do not contain any antigens and hence do not usually carry a blood type. Therefore platelets derived from a given blood type can be generally given to patients of other blood types.

7 We use terms period and cycle interchangeably throughout the text. Also the time (or duration) between two consecutive reviews is nothing but the length of the cycle or period.

8 The terms outdating, wastage, perishing, discarding are used interchangeably throughout the text.

9 One of the equations is redundant.

10 Note that complexity is the function of the life of the item in periods (which determines the number of batches of different ages) and not in days or weeks.

11 This allows us to differentiate the various components of the cost function.

12 We have ignored the holding cost for further analysis since it is negligible in case of blood products. However, if required, the holding cost can be included in the model using the expression

13 Normal and Poisson random variables for demand are commonly used in modelling of perishables since they are ‘infinitely divisible’ and the resulting models are ‘relatively tractable’ (Chiu (1995a).

14

15 Obviously it does not make any sense to increase R beyond d_max since it will only increase wastage without decreasing shortage (proof is omitted).

16 is used instead of R to indicate that the optimization is for a modified cost function.

17 Since the discarded units will also include those units whose lifetime is not over this assumption will lead to overestimation of wastage.

18 Since

19 For the purpose of the given blood bank even the T (time between two consecutive reviews) was assumed to be an integer variable. This reduced the search space in the heuristic considerably since both m and T are assumed to be integer valued. The permitted values of T thus were 1, 2, 3, 5, 6, 10, 15 and 30 days.

20 In case of large blood banks that collect blood themselves through external donation camps the model can help predict the aggregate collection quantity to be collected in the camp but not blood-group wise. This is because the group of the collected unit can only be determined after the blood units have been brought back to the blood bank.