Supply Chain Management of a Blood Banking System. with Cost and Risk Minimization

Transcription

1 Supply Chain Network Operations Management of a Blood Banking System with Cost and Risk Minimization Anna Nagurney Amir H. Masoumi Min Yu Isenberg School of Management University of Massachusetts Amherst, Massachusetts nd Annual POMS Conference Reno, Nevada April 29 - May 2, 2011

2 Acknowledgments This research was supported by the John F. Smith Memorial Fund at the University of Massachusetts Amherst. This support is gratefully acknowledged. The authors acknowledge Mr. Len Walker, the Director of Business Development for the American Red Cross Blood Services in the greater Boston area, and Dr. Jorge Rios, the Medical Director for the American Red Cross Northeast Division Blood Services, for sharing valuable information on the subject, and enlightening thoughts on the model.

3 Outline Background and Motivation Some of the Relevant Literature The Supply Chain Network Model of a Blood Banking System Numerical Examples The Algorithm and Explicit Formulae Summary and Conclusions

4 Background Literature Model Examples Summary

5 Background and Motivation Blood service operations are a key component of the healthcare system all over the world. Over 39,000 donations are needed everyday in the United States, alone, and the blood supply is frequently reported to be just 2 days away from running out (American Red Cross).

6 Background and Motivation Of 1,700 hospitals participating in a survey in 2007, a total of 492 reported cancellations of elective surgeries on one or more days due to blood shortages. Hospitals with as many days of surgical delays as 50 or even 120 have been observed. In 2006, the national estimate for the number of units of blood components outdated by blood centers and hospitals was 1,276,000 out of 15,688,000 units (Whitaker et al. (2007)).

7 Background Literature Model Examples Summary Background and Motivation The hospital cost of a unit of red blood cells in the US had a 6.4% increase from 2005 to In the US, this criticality has become more of an issue in the Northeastern and Southwestern states since this cost is meaningfully higher compared to that of the Southeastern and Central states.

8 Background and Motivation Hospitals were responsible for approximately 90% of the outdates, where this volume of medical waste imposes discarding costs to the already financially-stressed hospitals (The New York Times (2010)).

9 Background and Motivation The health care facilities in the United States are second only to the food industry in producing waste, generating more than 6,600 tons per day, and more than 4 billion pounds annually (Fox News (2011)).

10 Some of the Relevant Literature Nahmias, S. (1982) Perishable inventory theory: A review. Operations Research 30(4), Prastacos, G. P. (1984) Blood inventory management: An overview of theory and practice. Management Science 30 (7), Nagurney, A., Aronson, J. (1989) A general dynamic spatial price network equilibrium model with gains and losses. Networks 19(7), Pierskalla, W. P. (2004) Supply chain management of blood banks. In: Brandeau, M. L., Sanfort, F., Pierskalla, W. P., Editors, Operations Research and Health Care: A Handbook of Methods and Applications. Kluwer Academic Publishers, Boston, Massachusetts,

11 Sahin, G., Sural, H., Meral, S. (2007) Locational analysis for regionalization of Turkish Red Crescent blood services. Computers and Operations Research 34, Haijema, R. (2008) Solving Large Structured Markov Decision Problems for Perishable - Inventory Management and Traffic Control, PhD thesis. Tinbergen Institute, The Netherlands. Nagurney, A., Qiang, Q. (2009) Fragile Networks: Identifying Vulnerabilities and Synergies in an Uncertain World. John Wiley & Sons, Hoboken, New Jersey. Mustafee, N., Katsaliaki, K., Brailsford, S.C. (2009) Facilitating the analysis of a UK national blood service supply chain using distributed simulation. Simulation 85(2),

12 Nagurney, A., Nagurney, L.S. (2010) Sustainable supply chain network design: A multicriteria perspective. International Journal of Sustainable Engineering 3, Nagurney, A. (2010) Optimal supply chain network design and redesign at minimal total cost and with demand satisfaction. International Journal of Production Economics 128, Karaesmen, I.Z., Scheller-Wolf, A., Deniz B. (2011) Managing perishable and aging inventories: Review and future research directions. In: Kempf, K. G., Kskinocak, P., Uzsoy, P., Editors, Planning Production and Inventories in the Extended Enterprise. Springer, Berlin, Germany, Nagurney, A., Yu, M., Qiang, Q. (2011) Supply chain network design for critical needs with outsourcing. Papers in Regional Science 90(1),

13 Regionalized Blood Supply Chain Network Model We developed a generalized network optimization model for the complex supply chain of human blood, which is a life-saving, perishable product. More specifically, we developed a multicriteria system-optimization framework for a regionalized blood supply chain network.

14 Regionalized Blood Supply Chain Network Model We assume a network topology where the top level (origin) corresponds to the organization;.i.e., the regional division management of the American Red Cross. The bottom level (destination) nodes correspond to the demand sites - typically the hospitals and the other surgical medical centers. The paths joining the origin node to the destination nodes represent sequences of supply chain network activities that ensure that the blood is collected, tested, processed, and, ultimately, delivered to the demand sites.

15 Components of a Regionalized Blood Banking System ARC Regional Division Management (Top Tier), Blood Collection Blood Collection Sites (Tier 2), denoted by: C 1, C 2,..., C nc, Shipment of Collected Blood Blood Centers (Tier 3), denoted by: B 1, B 2,..., B nb, Testing and Processing Component Labs (Tier 4), denoted by: P 1, P 2,..., P np, Storage Storage Facilities (Tier 5), denoted by: S 1, S 2,..., S ns, Shipment Distribution Centers (Tier 6), denoted by: D 1, D 2,..., D nd, Distribution Demand Points (Tier 7), denoted by: R 1, R 2,..., R nr

16 Supply Chain Network Topology for a Regionalized Blood Bank 1 ARC Regional Division C 1 C 2 C 3 C nc Blood Collection Sites B 1 B nb Blood Centers P 1 P np Component Labs S 1 S ns Storage Facilities D 1 D 2 D nd Distribution Centers R 1 R 2 R 3 R nr Demand Points Graph G = [N, L], where N denotes the set of nodes and L the set of links.

17 Regionalized Blood Supply Chain Network Model The formalism is that of multicriteria optimization, where the organization seeks to determine the optimal levels of blood processed on each supply chain network link subject to the minimization of the total cost associated with its various activities of blood collection, shipment, processing and testing, storage, and distribution, in addition to the total discarding cost as well as the minimization of the total supply risk, subject to the uncertain demand being satisfied as closely as possible at the demand sites.

18 Notation c a : the unit operational cost on link a. ĉ a : the total operational cost on link a. f a : the flow of whole blood/red blood cell on link a. p: a path in the network joining the origin node to a destination node representing the activities and their sequence. w k : the pair of origin/destination (O/D) nodes (1, R k ). P wk : the set of paths, which represent alternative associated possible supply chain network processes, joining (1, R k ). P: the set of all paths joining node 1 to the demand nodes. n p : the number of paths from the origin to the demand markets. x p : the nonnegative flow of the blood on path p. d k : the uncertain demand for blood at demand location k. v k : the projected demand for blood at demand location k.

19 Formulation Total Operational Cost on Link a ĉ a (f a ) = f a c a (f a ), a L, (1) assumed to be convex and continuously differentiable. Let P k be the probability distribution function of d k, that is, P k (D k ) = P k (d k D k ) = D k 0 F k (t)d(t). Therefore, Shortage and Surplus of Blood at Demand Point R k k max{0, d k v k }, k = 1,..., n R, (2) + k max{0, v k d k }, k = 1,..., n R, (3)

20 Expected Values of Shortage and Surplus E( k ) = (t v k )F k (t)d(t), k = 1,..., n R, (4) v k vk E( + k ) = (v k t)f k (t)d(t), k = 1,..., n R. (5) 0 Expected Total Penalty at Demand Point k E(λ k k + λ+ k + k ) = λ k E( k ) + λ+ k E( + k ), (6) where λ k is a large penalty associated with the shortage of a unit of blood, and λ + k is the incurred cost of a unit of surplus blood.

21 Formulation Arc Multiplier, and Waste/Loss on a link Let α a correspond to the percentage of loss over link a, and f a denote the final flow on that link. Thus, f a = α a f a, a L. (7) Therefore, the waste/loss on link a, denoted by w a, is equal to: w a = f a f a = (1 α a )f a, a L. (8)

22 Total Discarding Cost function ẑ a = ẑ a (f a ), a L. (9) Non-negativity of Flows x p 0, p P, (10) Path Multiplier, and Projected Demand µ p a p α a, p P, (11) where µ p is the multiplier corresponding to the loss on path p. Thus, the projected demand at R k is equal to: v k p P wk x p µ p, k = 1,..., n R. (12)

23 Relation between Link and Path Flows δ ap α a, if {a < a} Ø, a α ap <a δ ap, if {a < a} = Ø, (13) where {a < a} denotes the set of the links preceding link a in path p. Also, δ ap is defined as equal to 1 if link a is contained in path p; otherwise, it is equal to zero. Therefore, f a = p P x p α ap, a L. (14)

24 Cost Objective Function Minimization of Total Costs Minimize ĉ a (f a ) + ẑ a (f a ) + a L a L n R k=1 subject to: constraints (10), (12), and (14). ( λ k E( k ) + λ+ k E( + k )) (15)

25 Supply Side Risk One of the most significant challenges for the ARC is to capture the risk associated with different activities in the blood supply chain network. Unlike the demand which can be projected, albeit with some uncertainty involved, the amount of donated blood at the collection sites has been observed to be highly stochastic. Risk Objective Function Minimize a L 1 ˆr a (f a ), (16) where ˆr a = ˆr a (f a ) is the total risk function on link a, and L 1 is the set of blood collection links.

26 The Multicriteria Optimization Formulation θ: the weight associated with the risk objective function, assigned by the decision maker. Multicriteria Optimization Formulation in Terms of Link Flows Minimize ĉ a (f a ) + ẑ a (f a ) + a L a L n R k=1 ( λ k E( k ) + λ+ k E( + k )) + θ a L 1 ˆr a (f a ) (17) subject to: constraints (10), (12), and (14).

27 The Multicriteria Optimization Formulation Multicriteria Optimization Formulation in Terms of Path Flows Minimize (Ĉp (x) + Ẑp(x) ) + n R p P k=1 ( λ k E( k ) + λ+ k E( + k )) + θ p P ˆR p (x) (18) subject to: constraints (10) and (12).

28 The total costs on path p are expressed as: Ĉ p (x) = x p C p (x), p P wk ; k = 1,..., n R, (19a) Ẑ p (x) = x p Z p (x), p P wk ; k = 1,..., n R, (19b) ˆR p (x) = x p R p (x), p P wk ; k = 1,..., n R, (19c) The unit cost functions on path p are, in turn, defined as below: C p (x) a L c a (f a )α ap, p P wk ; k = 1,..., n R, (20a) Z p (x) a L z a (f a )α ap, p P wk ; k = 1,..., n R, (20b) R p (x) a L 1 r a (f a )α ap, p P wk ; k = 1,..., n R. (20c)

29 Formulation: Preliminaries It is proved that the partial derivatives of expected shortage at the demand locations with respect to the path flows are derived from: E( k ) x p = µ p P k x p µ p 1, p P wk ; k = 1,..., n R. p P wk (21) Similarly, for surplus we have: E( + k ) x p = µ p P k x p µ p, p P wk ; k = 1,..., n R. (22) p P wk

30 Formulation: Lemma The following lemma was developed to help us calculate the partial derivatives of the cost functions: Lemma 1 Ĉ p (x) x p Ẑ p (x) x p a L a L ĉ a (f a ) f a α ap, p P wk ; k = 1,..., n R, (23a) ẑ a (f a ) f a α ap, p P wk ; k = 1,..., n R, (23b) ˆR p (x) x p a L 1 ˆr a (f a ) f a α ap, p P wk ; k = 1,..., n R. (23c)

31 Variational Inequality Formulation: Feasible Set and Decision Variables Let K denote the feasible set such that: K {x x R np + }. (24) Our multicriteria optimization problem is characterized, under our assumptions, by a convex objective function and a convex feasible set. We group the path flows into the vector x. Also, the link flows, and the projected demands are grouped into the respective vectors f and v.

32 Variational Inequality Formulation Theorem 1 The vector x is an optimal solution to the multicriteria optimization problem (18), subject to (10) and (12), if and only if it is a solution to the variational inequality problem: determine the vector of optimal path flows x K, such that: n R Ĉp(x ) + Ẑp(x ) + λ + k x p x µ pp k xp µ p p p P wk k=1 p P wk λ k µ p 1 P k xp µ p + θ ˆR ] p (x ) [x p xp ] 0, x K. x p p P wk (25)

33 Variational Inequality Formulation Theorem 1 (cont d) The variational inequality mentioned, in turn, can be rewritten in terms of link flows as: determine the vector of optimal link flows, and the vector of optimal projected demands (f, v ) K 1, such that: + n R k=1 a L 1 [ ĉa (f a ) f a + a L C 1 + ẑ a(f a ) f a [ ĉa (f a ) f a a ) + θ ˆr a(f f a a ) + ẑa(f f a ] [f a f a ] ] [f a f a ] [ λ + k P k(v k ) λ k (1 P k(v k )) ] [v k v k ] 0, (f, v) K 1, where K 1 denotes the feasible set as defined below: K 1 {(f, v) x 0, (10), (12), and (14) hold}. (26)

34 Standard form of Variational Inequality Formulation (25) The variational inequality (25) can be put into standard form as follows: determine X K such that: F (X ) T, X X 0, X K, (27) If we define the feasible set K K, and the column vector X x, and F (X ), such that: F (X ) Ĉp(x) + Ẑp(x) + λ + k x p x µ pp k x p µ p p p P wk λ k µ p 1 P k x p µ p + θ ˆR p (x) ; p P wk ; k = 1,..., n R, x p p P wk then (25) can be reexpressed in standard form (27). (28)

35 Illustrative Numerical Examples

36 Example 1 Organization 1 a C 1 b B 1 c P 1 d S 1 e D 1 f R 1 Demand Point Supply Chain Network Topology for Numerical Examples 1 and 2

37 Example 1 (cont d) The total cost functions on the links were: ĉ a (f a ) = f 2 a + 6f a, ĉ b (f b ) = 2f 2 b + 7f b, c c (f c ) = f 2 c + 11f c, ĉ d (f d ) = 3f 2 d + 11f d, ĉ e (f e ) = f 2 e + 2f e, c f (f f ) = f 2 f + f f. No waste was assumed so that α a = 1 for all links. Hence, all the functions ẑ a were set equal to 0. The total risk cost function on the blood collection link a was: ˆr a = 2fa 2, and the weight associated with the risk objective, θ, was assumed to be 1.

38 Example 1 (cont d) There is only a single path p 1 which was defined as: p 1 = (a, b, c, d, e, f ) with µ p1 = 1. Demand for blood followed a uniform distribution on [0, 5] so that: P 1 (x p1 ) = xp 1 5. The penalties were: λ 1 = 100, λ+ 1 = 0.

39 Example 1: Solution Substitution of the numerical values, and solving the variational inequality (25) yields: x p 1 = 1.48, and the corresponding optimal link flow pattern: f a = fb = f c = fd = f e = ff = Also, the projected demand is equal to: v 1 = x p 1 = 1.48.

40 Example 2 Example 2 had the same data as Example 1 except that now there was a loss associated with the testing and processing link with α c =.8, and ẑ c =.5f 2 c. Solving the variational inequality (25) yields: which, in turn, yields: f a The projected demand was: x p 1 = 1.39, = f b = f c = 1.39, fd = f e = ff = v 1 = x p 1 µ p1 = 1.11.

41 Summary of the Numerical Examples 1 and 2 An Interesting Paradox Comparing the results of Examples 1 and 2 reveals the fact that when perishability is taken into consideration, with α c =.8 and the data above, the organization chooses to produce/ship slightly smaller quantities so as to minimize the discarding cost of the waste, despite the shortage penalty of λ 1.

42 Summary of the Numerical Examples 1 and 2 Nevertheless, an appropriate increase in the unit shortage penalty cost λ 1 results in the organization processing larger volumes of the blood product, even exceeding the optimal flow in Example 1, which makes sense intuitively. The optimal path flow in Example 2 as a function of the link multiplier and the shortage penalty simultaneously can be expressed as: xp 1 = 5α c(λ 1 αc(λ ) 120. (29) + 50) + 65

43 Sensitivity Analysis Results Table 1: Computed Optimal Path Flows xp 1 Objective Function as α c and λ 1 Vary and Optimal Values of the α c λ xp OF xp OF xp OF xp OF xp OF Note: Numbers in red are the paradoxical results.

44 Sensitivity Analysis Results Table 1 (cont d) α c λ xp OF xp OF xp OF xp OF

45 The Solution Algorithm The realization of Euler Method for the solution of the blood bank supply chain management problem governed by variational inequality (25) induces subproblems that can be solved explicitly and in closed form. At iteration τ of the Euler method one computes: X τ+1 = P K (X τ a τ F (X τ )), (30) where P K is the projection on the feasible set K, and F is the function that enters the standard form variational inequality problem (27).

46 Explicit Formulae for the Euler Method Applied to Our Variational Inequality Formulation x τ+1 p = max{0, xp τ +a τ (λ k µ p(1 P k ( xp τ µ p )) λ + k µ pp k ( xp τ µ p ) p P wk p P wk Ĉp(x τ ) x p Ẑp(x τ ) x p θ ˆR p (x τ ) x p )}, p P. (31) The above was applied to calculate the updated product flow during the steps of the Euler Method for our blood banking optimization problem.

47 Example 3 1 ARC Regional Division 1 2 C 1 C 2 Blood Collection Sites B 1 B 2 Blood Centers 7 8 P 1 P 2 Component Labs 9 10 S 1 S 2 Storage Facilities D 1 D 2 Distribution Centers R 1 R 2 R 3 Demand Points Supply Chain Network Topology for Numerical Example 3

48 The demands at these demand points followed uniform probability distribution on the intervals [5,10], [40,50], and [25,40], respectively: µ p x p 5 ( ) p P w1 P 1 µ p x p =, P 2 ( µ p x p 40 p P w2 µ p x p ) =, 5 10 p P w1 p P w2 P 3 ( µ p x p 25 p P w3 µ p x p ) =. 15 p P w3 λ 1 = 2200, λ+ 1 = 50, λ 2 = 3000, λ+ 2 = 60, λ 3 = 3000, λ+ 3 = 50. and θ = 0.7. ˆr 1 (f 1 ) = 2f 2 1, and ˆr 2 (f 2 ) = 1.5f 2 2,

49 Solution Procedure The Euler method (cf. (31)) for the solution of variational inequality (25) was implemented in Matlab. A Microsoft Windows System with a Dell PC at the University of Massachusetts Amherst was used for all the computations. We set the sequence a τ =.1(1, 1 2, 1 2, ), and the convergence tolerance was ɛ = 10 6.

50 Example 3 Results Table 2: Total Cost and Total Discarding Cost Functions and Solution for Numerical Example 3 Link a α a ĉ a(f a) ẑ a(f a) fa f f 1.8f f f 2.7f f f 3.6f f f 4.8f f f 5.6f f f 6.8f f f 7.5f f f 8.8f f f 9.4f f f 10.7f

51 Example 3 Results Table 2 (cont d) Link a α a ĉ a(f a) ẑ a(f a) fa f f 11.3f f f 12.4f f f 13.3f f f 14.4f f f 15.7f f f 16.4f f f 17.5f f f 18.7f f f 19.4f f f 20.5f The computed amounts of projected demand for each of the three demand points were: v 1 = 6.06, v 2 = 44.05, and v 3 =

52 Summary and Conclusions We developed a supply chain network optimization model for the management of the procurement, testing and processing, and distribution of a perishable product that of human blood. Our model: captures perishability of this life-saving product through the use of arc multipliers; it contains discarding costs associated with waste/disposal; it handles uncertainty associated with demand points; it assesses costs associated with shortages/surpluses at the demand points, and it also quantifies the supply-side risk associated with procurement.

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