Two-Stage Stochastic Optimization for the Allocation of Medical Assets in Steady State Combat Operations

Transcription

1 Two-Stage Stochastic Optimization for the Allocation of Medical Assets in Steady State Combat Operations LTC(P) Larry Fulton, Ph.D. Leon Lasdon, Ph.D. Reuben McDaniel, Jr., Ed.D. Barbara Wojcik, Ph.D.

2 + Purpose + Motivation + Literature + Context + Simulation + Optimization + Results + Conclusions + Limitations and Future Work Outline

3 Purpose + Purpose: To investigate algorithms for determining the optimal site locations for medical assets in stability operations Motivation + Iraq base realignment + Afghanistan base realignment + Army Experimentation

4 Literature + Swain, Revelle, and Bergman (1971) + Similarly, Revelle & Hogan (1989) + Marianov and Taborga (2001) + Batta and Mannur (1990) + Aly and White (1978) + Neebe (1988) + Eaton, Sanchez, Ricardo, and Morgan (1986) + Silva and Serra (2008) + Geoffrion and Graves (1974) + Bouma (2005) + Santoso, Ahmed, Goetschalckx, and Shapiro (2005) + Snyder (2006)

5 Simple Overview Evac site hospital Evac site 1.0 Evac site 1.4* time 1.4* time * time 1.0* time hrs hospital Evac site 1.0 hospital

6 Scenario Parameters / Assumptions + OIF-like scenario + 10 possible hospital locations, 5 to be selected possible evacuation locations, 10 to be selected + Feasible locations randomly generated over grid (size of Iraq) + Four areas with casualty clusters (e.g., brigades) + Frequency distribution assigns the casualties occurring within each area based on unit mission and priority (e.g., main effort or supporting effort)

7 Hospital Parameters / Assumptions + Hospitalization sites = subset of evacuation sites + Total beds for all hospital designs by type were: + 1,000 minimal care ward (MCW) beds (200 per hospital), + 1,000 intermediate care ward (ICW) beds (200 per hospital) intensive care unit (ICU) beds (48 per hospital); + Bed types need not be equally distributed among hospitals. + Beds represent those available in 5 x 248-bed combat support hospitals with attached minimal care capability. + Assumption: over the relevant range, hospital bed availability (e.g., percent unoccupied) is 25% per type leaving available a daily average of 250 MCW beds, 250 ICW beds, and 60 ICU beds.

8 Vehicle Parameters / Assumptions + Based on operating characteristics of the platforms and limitations of human performance, a ground ambulance can conduct at most 10 lifts/day and air ambulances 5 lifts/day. + Ground ambulances are assumed to operate uniformly on the interval 45 +/-15 nautical miles per hour (accounting for patient loading, unloading, circuitous routing, etc. ) + Air ambulances are assumed to operate at 95 +/- 20 nautical miles per hour (again, accounting for patient loading, unloading, threat, terrain, etc.).

9 Casualty Generation + Inverse CDF sampling to 1. Determine brigade center {n,o} where casualty-producing event occurred. 2. Determine random location around brigade where casualty-producing event occurred. 3. Determine number of casualties at location. 4. Determine casualty severity. 5. Determine casualty bed type.

10 Casualty Distributions 1. Determine brigade center {n,o} where casualty-producing event occurred. Grid {n,o} % of Casualties P(X<=x) {100,100} 20%.20 {400,100} 35%.55 {100,400} 25%.80 {400,400} 20% 1.00

11 Casualty Locations

12 Casualty Numbers 3. Determine number of casualties at location. Fulton, McMurry, & Kerr (2009). Monte Carlo Simulation of Air Ambulance Requirements.., Military Medicine, In Press # Patients** P(X=x) 1 patient patients patients patients.016 ** The number of patients is subject to a DoE lethality multiplier.

13 Casualty Severity (Path Weights) 4. Determine casualty severity and bed assignment + Sample from OIF ISS data + Sample from ISS fatality distribution (National Trauma Databank, 2007) + Weight path by geometric expectation (inverse of survival probability). + Assign bed category based on ISS scores groupings. Injury Severity Score P(X=x) >24.11 Survival by Injury Severity Score (ISS), 2007 ISS P(Fatal) P(Non-Fatal) Penalty Weight >

14 Design of Experiment Factors Design: III, I=AB2 C, I=BCD Main Effects Clear + A: Lethality multiplier, {1.0, 1.5, 2.0} + B: Maximum casualty distance from center of unit, {50NM, 100NM, 150NM} + C: Percent of ground ambulance use, {.2,.3,.4} + D: Days in which casualty numbers might be experienced, {4, 7, 10}

15 Simulation Runs + Based on an initial simulation run of size n=100, the standard deviation for the time to travel along the network was.629 hours. + All 9 runs with their estimated 1,719 casualties should provide a 95% confidence interval around mean transport time of +/- 2 minutes.

16 Sets + T = scenarios for evaluation with index + I = set of locations where injuries may occur with index + J = set of candidate helicopter evacuation sites with index + K = set of candidate hospital sites with index + L = type of bed requirement for a patient (minimal, intermediate, and intensive care) with + M = type of evacuation asset, ground or air

17 Problem Data p t = probability that scenario t occurs (probability distribution) a it = injury severity weight of most critically-injured patient departing from site i for scenario t, assigned stochastically b ijt = distance between injury location i and evacuation site j for scenario t c ikt = distance between injury location i and hospital location k for scenario t d jkmt = speed of transport from j to k by vehicle m for scenario t, stochastic e it = number of total patients injured at location i for scenario t, stochastic f l = the number of patients with type l bed requirements ecap mt = capacity for patient evacuation by m-type vehicles hcap lt = capacity for hospital acceptance of l-type patients enod jt = evacuation node capacity for each node j hnod kt = hospital node capacity for each node k u = maximum number of hospitalization sites to be occupied v = maximum number of evacuation sites to be occupied wx jt = percent ground ambulances required for each evacuation site

18 Decision Variables + x ijklmt = number of patients traveling from injury location i with bed-type requirement l on vehicle m to hospital site k by a vehicle and crew located at evacuation site j for scenario t + y j = 1 if air evacuation site j is chosen, 0 otherwise + z k = 1 if hospitalization site l is chose, 0 otherwise

19 Constraint 1 j k l m x ijklmt = e it, i, t Synopsis: All patients are evacuated LHS: total number of casualties who are transported RHS: the number of casualties specified as originating at location i in scenario t.

20 i j k l Constraint 2 x ijklmt ecap mt, m, t Synopsis: Patients traveling the network cannot exceed evacuation capacity. LHS: Number of patients traveling the network RHS: Evacuation system capacity by vehicle by scenario Note: RHS may be set large (open) for requirements estimation.

21 i j k m Constraint 3 x ijklmt hcap lt, l, t Synopsis: Patients traveling the network cannot exceed hospitalization capacity. LHS: Number of patients traveling the network RHS: Hospitalization system capacity by bed by scenario Note: RHS may be set large (open) for requirements estimation.

22 Constraints 4 & 5 i k l m i j l m x x ijklmt ijklmt <= <= enod hnod jt kt * y * y j, j, t k, k, t Two constraints allow for node throughput limits by restricting capacity at either evacuation or hospitalization nodes. To allow unconstrained nodes and post-hoc determination of what is needed, both enod and hnod might be set large.

23 xijkl, m= i j k l m Constraint 6 'G',t wx jt * i j k l m x ijklmt When evacuation capacity is large such as in stability operations, solutions will generally select the military's preferred and primary source of evacuation assets, the air ambulance. Weather sometimes limits the ability of aircraft to fly, however. To account for limitations in use of air ambulances, an additional constraint was used. The left hand side insures that the evacuations conducted by ground ambulance are greater than a percentage of total evacuations at each evacuation site j., t

24 Constraints 7 and 8 j y = v, z = j k The total number of all open evacuation sites and hospitals are limited using the following constraints. For this analysis, v represents the available evacuation sites, while u represents the available hospitalization sites. One can see that the decision variables, y and z, are not indexed by scenario. In this way, only a single set of open nodes are chosen which is necessary to find the optimal over all scenarios. k u

25 Constraints 9, 10, and 11 x jikt 0, y j {0,1}, z k {0,1} Finally, the flow variables x must be nonnegative while the others must be binary. If certain nodes must remain open, one can add constraints forcing their associated y or z value to 1.

26 Objective Function Min O p ( a itxijklmt (bijt + = t t j k l m i c ikt )/(d The objective is to minimize the expectation over all scenarios of the total, penaltyweighted time traveled by all patients. O sums the penalty-weighted time for evacuating patients via helicopter or ground ambulance along all paths for all scenarios. If some candidate locations are fixed, one simply does not define variables for these indices. ijkmt ))

27 Solver + GAMs provided the platform for both the simulation and the optimization. The simulation leveraged inverse CDF sampling while the optimization model was solved via the IBM-based OSL solver. + GAMS provides reasonable capability for simulation via pseudo-random number generation, seed assignment, probability functions, and programming flow control. + GAMS also provides an excellent optimization modeling platform.

28 Complexity + Prior to optimization, the generation of the model (the stage 1 simulation runs) took 256MB of memory and 37.7 seconds for initialization and initial simulation runs. + The model contained 1,234 single equations with 1,090 rows and 864,031 columns, 2.6 million single variables (30 discrete), and 6.0 million non-zero elements. + On a Dell XPS 1730 laptop with 3.8 GHZ Intel Core 2 Extreme processor with 4 GB of RAM and twin Raid 0 solid state hard drives, the OLS solver completed 3,400 simplex iterations before proceeding to branch and bound. After 43,225 iterations and 19.3 minutes, the OSL solver produced an integer optimal solution meeting the.01 optimality criterion.

29 Solution

30 Results, Allocation Average Patients Moved: j=evacuation site k=hospitalization site m=vehicle type l=patient type m=g, Avg. Patients Moved by Ground m=a, Avg. Patients Moved by Air k=h1, Avg. Patients Moved to Hosp. 1 k=h2, Avg. Patients Moved to Hosp. 2 k=h4, Avg. Patients Moved to Hosp. 4 k=h7, Avg. Patients Moved to Hosp. 7 k=h9, Avg. Patients Moved to Hosp. 9 P(j=E* m=g) Ground Ambulance Asset Allocation at Site j P(j=E* m=a) Air Ambulance Asset Allocation at Site j j=e1, Evac. Site % 53 20% 12 j=e2, Evac. Site % 41 17% 10 j=e3, Evac. Site % 8 4% 3 j=e4, Evac. Site % 4 2% 1 j=e5, Evac. Site % 23 9% 5 j=e7, Evac. Site % 23 11% 6 j=e12, Evac. Site % 17 8% 5 j=e13, Evac. Site % 37 15% 9 j=e15, Evac. Site % 3 2% 1 j=e20, Evac. Site % 32 12% 7 Totals E(l=MCW column) Avg. Patients Moved to MCW E(l=ICW column) Avg. Patients Moved to ICW E(l=ICU column) Avg. Patients Moved to ICU P(column l=mcw) 16% 84% 32% 27% 10% 16% 15% 100% % Patients Moved to MCW P(column l=icw) 9% 21% 36% 19% 9% 26% 10% 100% % Patients Moved to ICW P(column l=icu) 14% 51% 32% 10% 10% 34% 14% 100% % Patients Moved to ICU MCW Asset Allocation for Hospital k Total MCW beds in scenario ICW Asset Allocation for Hospital k Total ICW beds in scenario ICU Asset Allocation for Hospital k Total ICU beds in scenario

31 Main Effects of DOE Factors Type III Sum of Source Squares df Mean Square F p Model a <.001 Lethality Radius % ground Days Error Total a. R Squared =.823 (Adjusted R Squared =.821)

32 Limitations and Future Work Improvement of distributions Multi-period extension (possibly), e.g., x d,i,t Addition of set items (e.g., C-130, FST) MCO extension by indicing over phase, e.g., x p,d,i,t

33 Questions?