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Articles in Advance, pp. 1–20 ISSN 0092-2102 (print) � ISSN 1526-551X (online) http://dx.doi.org/10.1287/inte.2014.0779

The AMEDD Uses Goal Programming to Optimize Workforce Planning Decisions

Nathaniel D. Bastian Center for Integrated Healthcare Delivery Systems, Department of Industrial and Manufacturing Engineering, Pennsylvania State University, University Park, Pennsylvania 16802; and Center for AMEDD Strategic Studies,

U.S. Army Medical Department Center and School, Fort Sam Houston, Texas 78234, ndbastian@psu.edu

Pat McMurry AMEDD Personnel Proponency Directorate, U.S. Army Medical Department Center and School, Fort Sam Houston,

Texas 78234, pat.m.mcmurry.civ@mail.mil

Lawrence V. Fulton Center for Healthcare Innovation, Education and Research, Rawls College of Business Administration, Texas Tech University,

Lubbock, Texas 79410, larry.fulton@ttu.edu

Paul M. Griffin H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332,

paul.griffin@isye.gatech.edu

Shisheng Cui, Thor Hanson, Sharan Srinivas Department of Industrial and Manufacturing Engineering, Pennsylvania State University, University Park, Pennsylvania 16802

{suc256@psu.edu, tkh138@psu.edu, sus412@psu.edu}

The mission of the Army Medical Department (AMEDD) is to provide medical and healthcare delivery for the U.S. Army. Given the large number of medical specialties in the AMEDD, determining the appropriate number of hires and promotions for each medical specialty is a complex task. The AMEDD Personnel Proponency Direc- torate (APPD) previously used a manual approach to project the number of hires, promotions, and personnel inventory for each medical specialty across the AMEDD to support a 30-year life cycle. As a means of decision support to APPD, we proffer the objective force model (OFM) to optimize AMEDD workforce planning. We also employ a discrete-event simulation model to verify and validate the results.

In this paper, we describe the OFM applied to the Medical Specialist Corps, one of the six officer corps in the AMEDD. The OFM permits better transparency of personnel for senior AMEDD decision makers, whereas effectively projecting the optimal number of officers to meet the demands of the current workforce structure. The OFM provides tremendous value to APPD in terms of time, requiring only seconds to solve rather than months; this enables APPD to conduct quick what-if analyses for decision support, which was impossible to do manually.

Keywords: workforce planning; mixed-integer linear programming; stochastic optimization; goal programming; multiple-criteria decision making; military medicine.

History: This paper was refereed. Published online in Articles in Advance.

The Army Medical Department (AMEDD) is a spe-cial branch of the U.S. Army whose mission is to provide health services for the Army and, as directed, for other agencies, organizations, and military ser- vices. Since the establishment of the AMEDD in 1775, six officer corps (Medical Corps, Dental Corps, Nurse Corps, Veterinary Corps, Medical Specialist Corps, and Medical Service Corps) have been developed to provide the organizational leadership and profes- sional and clinical expertise necessary to accomplish

the broad soldier-support functions implicit to the mission (Department of the Army 2007). Each corps is made up of individually managed career fields and duty titles called areas of concentration (AOCs); the AMEDD includes 100 officer AOCs. The Medical Specialist Corps has the smallest number with four AOCs; the Medical Corps has the most with 41 AOCs.

The AMEDD manages medical officer personnel over a 30-year life cycle. Given the large number of AOCs in the AMEDD, determining the appropriate

number of hires and promotions for each medical spe- cialty is a complex task. The number of authorized medical personnel positions for each AOC varies sig- nificantly depending on the uniqueness of the career field and the needs of the Army. Some AMEDD offi- cers enter the Army and remain in the same AOC throughout their careers. Others start their careers in one AOC but have the option of obtaining additional education to qualify for a more specialized AOC. Finally, some officers enter the Army in one AOC but must obtain additional education and move to a more specialized AOC to stay competitive for promotion.

The rank structure of the AMEDD includes officers in the ranks of second lieutenant through general, but the AMEDD Personnel Proponency Directorate (APPD) is only responsible for managing officers below the rank of general, namely second lieutenant (2LT), first lieutenant (1LT), captain (CPT), major (MAJ), lieutenant colonel (LTC), and colonel (COL). The pay grades for these ranks are O-1 to O-6, respec- tively. APPD is only responsible for managing these officers through the 30th year of commissioned fed- eral service, because officers not selected for pro- motion to general are generally limited to a 30-year career in the military. The AMEDD’s promotion pol- icy is set forth by the Defense Officer Personnel Management Act (DOPMA) of 1980, which provides guidance on promotion selection target percentages (Rostker et al. 1993). For example, the targeted pro- motion rate from LTC to COL is 50 percent based on DOPMA, although this does not apply to physi- cians or dentists. In addition to the challenge of meet- ing federal mandates associated with promotion rates, uncertain officer continuation rates further compli- cate the workforce planning problem for the AMEDD, because an officer may decide to leave at any time (if that officer has no remaining active-duty service obli- gation). Thus, uncertainty caused by attrition makes the officer accession and promotion decisions even more complex.

Although the Army Surgeon General has author- ity over the entire AMEDD, each corps has a corps chief who is responsible for making many decisions impacting the officers in his (her) corps. Some of the key decisions include, how many new officers to recruit and hire into active duty each year within his (her) corps, how many officers to promote to the next

higher rank (grade) each year, and how many offi- cers to train in each career field (or clinical specialty). The APPD seeks to provide workforce planning deci- sion support to the corps chiefs by projecting the number of hires, promotions, and personnel inven- tory needed to support a 30-year life cycle within a corps’ authorized officer positions. A 30-year life cycle allows APPD to assess the availability of an offi- cer throughout the anticipated lifespan. Although the model is rerun year after year to reassess each offi- cer’s availability, the number of hires must necessar- ily be based on the requirements forecast based on attrition and promotion data. This approach, which is used throughout the U.S. Army, was implemented initially by Gass (1991).

Literature Review Military workforce planning models have been used for decades. Bres et al. (1980) developed a goal- programming model for planning officer hires to the U.S. Navy from various commissioning sources. They used transition rates to project the on-board expected flows between starts in successive periods in Marko- vian fashion to which they also added new hires into the system. Gass et al. (1988) developed the Army manpower long-range planning system, which inte- grated a Markov chain and linear goal-programming model to forecast the flow of an initial force (given by grade and years of service) to a future force over a 20- year planning horizon, and to determine the optimal transition rates (continuation, promotion, and skill migration) and accession values to obtain the desired end-state force structure or the rates required to min- imize the deviation from the desired end-state force structure. Silverman et al. (1988) developed a multi- period, multiple-criteria trajectory optimization sys- tem to help manage the enlisted force structure of the U.S. Navy. Their workforce accession planning model employs an interactive augmented weighted Tcheby- cheff method, while examining various recruitment and promotion strategies. Gass (1991) built network- flow goal-programming models to provide the U.S. Army with decision support tools to effectively man- age its workforce. The transition-rate models describe people going from one state to another during their life cycle in the workforce system. Weigel and Wilcox (1993) developed the Army’s enlisted personnel deci- sion support system, which combines a variety of

modeling approaches (goal programming, network models, linear programming, and Markov-type inven- tory projection) with a management information sys- tem to support the analysis of long-term personnel planning decisions.

In addition to these long-term workforce planning models, Corbett (1995) developed a workforce opti- mization model that assists personnel planners in determining yearly officer hires as well as transfers to functional areas as part of the branch detail pro- gram. The model employs a multiyear weighted goal program designed to maximize the Army’s ability to meet forecasted authorization requirements. Yamada (2000) developed the infinite-horizon workforce plan- ning model using convex quadratic programming for managing officer hires, promotions, and separa- tions annually to best meet desired inventory targets. Henry and Ravindran (2005) presented both preemp- tive and nonpreemptive goal-programming models for determining the optimal-hires cohort—the num- ber of new Army officers into each of 15 career branches. Shrimpton and Newman (2005) developed a network-optimization model to designate mid- career level officers into new career fields to meet end-strength requirements and maximize the overall utility of officers.

Cashbaugh et al. (2007) used network-based mathe- matical programming to model the assignment of U.S. Army enlisted personnel in a 96-month planning hori- zon. Kinstler et al. (2008) developed a Markov model using promotion and attrition rates to improve work- force management decisions in the U.S. Navy nurse corps. Hall (2009) used dynamic programming and linear programming techniques to model the opti- mal retirement behavior for an Army officer from any point in his (her) career. He addresses the opti- mal retirement policies for Army officers, incorpo- rating the current retirement system, pay tables, and Army promotion opportunities. Coates et al. (2011) investigated the U.S. Army’s captain retention pro- gram and used a chi-square and odds ratio analy- sis to determine whether the practice of providing financial bonuses to individuals agreeing to continue their service is an effective retention tool. Lesiński et al. (2011) used discrete-event simulation (DES) to model the current flow process that an officer negoti- ates from precommissioning to the first unit of assign- ment. This model assisted with synchronization of the

officer accession and training with the Army force generation process.

Given that some of the officer continuation rates are uncertain parameters, we discuss several tools for addressing optimization problems in the pres- ence of uncertainty. Different algorithms have been developed for stochastic optimization problems, and research has shown that they can be used successfully in many planning applications. The type of data avail- able to the decision maker(s), the assumptions on risk, and the structure and properties of the stochastic opti- mization problem guide which method to use. Our workforce planning problem incorporates stochastic components in the constraints; therefore, we are con- cerned with methods for solving chance-constrained stochastic programming problems that Charnes and Cooper (1959) proposed originally. In general, chance- constrained stochastic programs have two difficulties (Ahmed and Shapiro 2008). One difficulty is accu- rately computing the probabilistic constraints. With- out this difficulty, we could transform the stochastic optimization problems to their respective determinis- tic equivalents and then convert them to general non- linear programs that are solvable with traditional non- linear techniques. Cheon et al. (2006) and Ruszczyński (2002) proposed algorithms for these types of prob- lems. However, such processes are usually difficult to solve in practice and are only successful for special cases. The second difficulty arises when the feasible region is not convex. In this case, which occurs fre- quently in workforce planning, the optimization prob- lem becomes difficult to solve efficiently.

Most chance-constrained stochastic programs are solved using approximation methods. Numerous methods have been developed for problems in which both difficulties exist. Both Nemirovski and Shapiro (2006) and Calafiore and Campi (2005) proposed such solution methods. Kleywegt et al. (2001) introduce a Monte Carlo simulation-based approach to stochastic discrete optimization problems, in which a random sample is generated and the expected-value function is approximated by the corresponding sample average function. The obtained sample average approximation optimization problem is then solved.

A particular subclass within chance-constrained stochastic programs is chance-constrained stochas- tic goal programs, which can be used to solve

multiple-criteria optimization problems under uncer- tainty. This subclass belongs to goal programming, where there are probabilistic rather than determin- istic constraints. Because we cannot usually convert a chance-constrained stochastic goal program to a deterministic equivalent, we typically apply Monte Carlo simulation methods. The general concept is that we can approximate the uncertain constraint func- tions using stochastic simulation, and then solve the problem using the approximated function.

Motivation and Purpose Effective long-term workforce planning and person- nel management of all medical professionals within the AMEDD is a complex problem. Prior to 2010, APPD used a manual approach to project the appro- priate hires and promotion goals for each medi- cal specialty across the six separate corps. McMurry et al. (2010) developed a set of nonlinear mathemat- ical programming models to provide the APPD with a multiple-criteria decision support mechanism for determining optimal hiring and promotion policies.

We extend the work of McMurry et al. (2010) by introducing the objective force model (OFM), a deterministic, mixed-integer linear weighted goal- programming model to optimize AMEDD workforce planning for the Medical Specialist Corps (SP). This linear, multicriteria optimization model is signifi- cantly more difficult in that the constraints are more varied as a result of substitutability. We also intro- duce two stochastic variants of the linear OFM, which incorporate probabilistic components associated with uncertain officer continuation rates (following the completion of grade-based active-duty service obliga- tions); these rates may fluctuate significantly. We use discrete-event simulation to verify and validate the results of the deterministic OFM.

Our improved optimization models allow for bet- ter transparency of AMEDD personnel for both the corps chiefs and the health services human resource planners at APPD, while effectively projecting the workforce skill levels (by grade) required to meet the demands of the current force. Note that we model a 30-year life cycle because of Title 10 U.S. code sec- tion 634, which states that each officer who holds the grade of colonel in the regular Army and is not on the selection list to brigadier general must retire the

first day of the month after the month he (she) com- pletes 30 years of active federal-commissioned ser- vice. Therefore, because we model officer ranks up to and including colonel, 30 years constitutes the maxi- mum life cycle and represents the target steady-state inventory of officers within each specialty, rank, and years of service.

Methodology We first present the formulation for the OFM, a mixed-integer linear weighted goal-programming model, to solve the workforce planning problem for AMEDD’s Medical Specialist Corps, given determin- istic continuation rates. We then briefly describe two solution methods for the stochastic goal programs used to solve the workforce planning problem under uncertainty. Finally, we discuss the discrete-event sim- ulation model performed for OFM verification and validation.

Deterministic Variant of the Objective Force Model AMEDD officers in the SP are hired into the Army at the grade (rank) of either O-1 (2LT), O-2 (1LT), or O-3 (CPT). Unlike the more specialized and diver- sified corps, these officers remain in the same AOC throughout their careers. SP consists of four career AOCs: occupational therapists (OTs), physical ther- apists (PTs), clinical dietitians (CDs), and physician assistants (PAs). We note that promotion decisions are made only for officers at the grade (rank) of O-4 (MAJ), O-5 (LTC), and O-6 (COL). According to APPD, a noninteger solution for promotions is an acceptable simplification. The noninteger structure is appropriate because of the concept of full-time equiv- alent employees, which may be fractional. Although we may not hire a fractional person, we can augment any fractional requirement along the entire 30-year timeline.

We now present a brief description of the deter- ministic OFM sets, parameters, variables, objective function, and goal and hard constraints (Appendix A provides a full description), and we provide some definitions and explanations of the military human resources terminology. The set G represents the grade of the SP officers that is indexed using {1, 2, 3, 4, 5, 6}. The set I represents the officer AOCs within the SP, which is indexed as {1 = OT, 2 = PT, 3 = CD, 4 = PA}.

The set K represents the year of service for an offi- cer, which is {11 210001 30} representing a full officer career. The set F represents the set of goals specified by APPD, which is {11 21 31 41 5}.

Authorizations are officer positions funded by the U.S. Congress to carry out the mission of the U.S. Army. A documented authorization is a funded posi- tion within an organization that identifies a specific specialty and rank required to meet a stated capa- bility. There are two types of documented authoriza- tions. The first is a career field specialty (or AOC) authorization that can be filled only by an officer specifically trained for that job (e.g., physical thera- pist). The second type of documented authorization is an immaterial authorization. Immaterial positions do not require an individual with a specific career spe- cialty. Most immaterial authorizations are executive or leadership positions, such as commanders, directors, and administrators. The last type of authorization provides allowances for officers who are not assigned or contributing to the mission of an organization. This includes officers who are students, in transit between assignments, in long-term hospitalization or pending discharge (e.g., wounded warriors), or removed for disciplinary reasons (e.g., court martial).

In the OFM, the parameter cig reflects the doc- umented authorizations for each AOC and grade, which reflect the requirements for SP officers to sup- port both peace- and wartime healthcare delivery for the Army. The parameter SPIMMg represents the SP immaterial documented authorizations for each grade; these are SP authorizations that SP officers can fill regardless of AOC; AMIMMg represents the AMEDD immaterial documented authorizations for each grade that are AMEDD authorizations that SP officers can fill regardless of AOC; THSg represent the transient, holdee, and student documented authoriza- tions for each grade, which are authorizations that SP officers can fill regardless of AOC. Table 1 displays these data, which APPD provided.

The parameter Cap i

is the maximum allowable number of officers for each AOC (i.e., capacity). According to APPD, this upper bound applies only to OTs, PTs, and CDs. The parameter Floori is the min- imum acceptable number of officers for each AOC; this lower bound applies only to OTs and CDs. The parameter Cap

ig is the maximum allowable number

Area of concentration Documented authorizations OT PT CD PA Total SPIMM AMIMM THS

Total 75 255 122 780 11505 13 44 216 COL 3 6 5 3 28 4 5 2 LTC 9 23 19 28 103 1 15 8 MAJ 20 46 38 149 325 2 15 55 CPT 31 111 28 534 798 6 7 81 1LT 12 19 10 66 179 0 2 70 2LT 0 50 22 0 72 0 0 0 Company grade 43 180 60 600 11049 6 9 151

Table 1: This table shows the medical specialist corps documented num- ber of authorizations (cells) by rank (rows) and area of concentration (columns).

Area of concentration

Number of officers OT PT CD PA

Total (Max) 96 295 154 Total (Min) 93 149 COL (Max) 4 COL (Min) 4 8 7 LTC (Max) 12 LTC (Min) 20

Table 2: This table details both the minimum and maximum number of medical specialist corps officers by rank (rows) and by AOC (columns).

of officers for each AOC and each grade; this upper bound applies only to COL and LTC who are OTs. The parameter Floorig is the minimum acceptable number of officers for each AOC and each grade; this lower bound applies only to COL for OTs, PTs, and CDs and to LTC for CDs. Table 2 displays these data provided by APPD.

Promotion rate is the number of officers selected for promotion divided by the number of officers consid- ered. The number of officers selected is a variable in the OFM bounded by promotion rates usually based on DOPMA objectives ±10 percent when possible. In the OFM, the parameter pfig is the minimum promo- tion rate for each AOC and grade, which is not appli- cable to 2LT in each AOC. The parameter pcig is the maximum promotion rate for each AOC and grade, which is also not applicable to 2LT in each AOC. Table 3 displays these data, which APPD provided.

Note that although solving by hand might appear to be reasonable for some of our smaller models (such as the example of SP Corps OFM we discuss here),

OT PT CD PA DOPMA

Promotion rate 1LT CPT MAJ LTC COL 1LT CPT MAJ LTC COL 1LT CPT MAJ LTC COL 1LT CPT MAJ LTC COL

Max 1 0095 1 008 006 0098 0095 1 008 006 1 0095 1 008 006 1 0095 1 008 006 Min 1 0095 007 006 004 0098 0095 007 0055 003 1 0095 007 006 003 1 0095 006 004 002

Table 3: This table details the minimum and maximum DOPMA promotion rates for medical service corps officers by order statistic (rows), rank (columns), and area of concentration (table split).

Year

Promotion evaluation 2 4 11 17 22

2LT → 1LT 98% 1LT → CPT 95% CPT → MAJ 80% MAJ → LTC 70% LTC → COL 50%

Table 4: This table details the DOPMA promotion standards by rank and year of service.

others are exceedingly large with significant range between the floor and ceiling constraints. Table 4 shows both the scheduled promotion evaluations by year and grade and the DOPMA standard promotion rates, which APPD again targets at ±10 percent.

The continuation rate is the percentage of officers that stay in the Army from one year to the next year, categorized by specialty, rank, and years of service. The rates are based on a five-year average of actual data collected on every officer on active duty for each specialty. In the OFM, the parameter rigk reflects the deterministic continuation rate for each AOC, grade, and year of service, which reflects both those SP offi- cers who are selected for promotion when considered and those SP officers who are considered for promo- tion but not selected. Note that officers not selected for promotion are limited to a set number of years that they may remain on active duty (rank depen- dent) before mandatory separation. These data pro- vided by APPD come from the medical operational data system, which (again) is derived from multiyear averages. Finally, the parameter wf reflects APPD’s weight for each goal in the model.

The model decision variables pig represent the num- ber of SP officers promoted in AOC i at grade g (for MAJ, LTC, and COL only). The model decision vari-

ables aig represent the number of SP officers hired for AOC i at grade g (for 2LT, 1LT, and CPT only). The model decision variables dig represent the actual num- ber of SP officers in the system for each AOC and grade, whereas the model decision variables Invigk represent the projected inventory of SP officers in the system by AOC i, grade g, and year k. In terms of goal deviation variables, pos

f is the positive deviation for

goal f and neg f

is the negative deviation for goal f . The objective function of the deterministic OFM

seeks to minimize the sum of the weighted goal devi- ations. The target for the first goal constraint is for the total number of officers (over each grade and AOC) to equal the total documented authorizations (over each grade and AOC as well as the SP immaterial, AMEDD immaterial, and THS). The target for the sec- ond goal constraint is for the number of COLs (over each AOC) to equal the COL documented authoriza- tions (over each AOC as well as the SP immaterial, AMEDD immaterial, and THS). The target for the third goal constraint is for the number of LTCs (over each AOC) to equal the LTC documented authoriza- tions (over each AOC as well as the SP immaterial, AMEDD immaterial, and THS). The target for the fourth goal constraint is for the number of MAJs (over each AOC) to equal the MAJ documented authoriza- tions (over each AOC as well as the SP immaterial, AMEDD immaterial, and THS). The target for the last goal constraint is for the number of company grade (sum of 2LT, 1LT, and CPT) officers (over each AOC) to equal the company grade documented authoriza- tions (over each AOC as well as the SP immaterial, AMEDD immaterial, and THS).

The hard constraints, as we define in Appendix A, force inventory controls, promotion controls (floors and ceilings by AOC), and transition controls. These constraints were developed based on known pro- motion restrictions, transition data, and (primarily)

decision-maker input. Some constraints apply to all AOCs; however, others are AOC specific. All con- straints were developed in conjunction with APPD. For example, multiple constraints provided promo- tion floors and ceilings by year, AOC, and grade. These constraints were necessary to achieve decision- maker personnel requirements. In addition, inventory constraints were necessary to ensure proper rollover from one period to another by AOC and grade. Addi- tional constraints ensured that promotions were con- sidered only during those years and by grade when feasible.

Stochastic Variants of the Objective Force Model In solution method #1, we use a scenario-based Monte Carlo simulation approach to approximate the objec- tive value and the optimal solution of a stochas- tic goal program. We generate S scenarios, where each scenario corresponds to one realization of the deterministic optimization problem. We use the sam- ple average across the S scenarios to approximate the optimal objective value and optimal solution. In solution method #2, we leverage the sample aver- age approximation (SAA) method (Rubinstein and Shapiro 1990) because prior samples are easily gener- ated. Appendix B provides additional details.

In the stochastic variant of the OFM, we model officer continuation rates as random variables from the normal distribution based on historical multiyear averages captured by APPD. We use both stochas- tic method #1 and stochastic method #2 to solve the stochastic variant of the mixed-integer linear weighted goal-programming model. For stochastic method #1, we solved the deterministic OFM for each scenario with stochastically generated continuation rates, where the final objective value and optimal solution is the average over the scenarios.

Appendix C contains the full description of the model formulation for stochastic method #2. The key differences between the deterministic and stochastic method #2 model formulations are as follows. First, we include an additional set S representing the sce- nario under consideration. Second, the data parame- ter for SP officer continuation (stochastic) rate rigks is now computed over each AOC, grade, year, and sce- nario. Third, the decision variables digs and Invigks are now computed over each scenario. Fourth, there are

now positive and negative goal deviations for each goal and scenario, pos

fs and neg

fs , respectively. Fifth,

the objective function now seeks to minimize the sam- ple average of the sum of the weighted goal devia- tions over the scenarios. Finally, the number of goal and hard constraints are increased as a result of exe- cution over each scenario.

Discrete-Event Simulation Model To verify and validate the deterministic OFM pre- sented previously, we developed a DES model that processes SP officer hires (i.e., number of arrivals) for each AOC through the 30-year life cycle. At the beginning of each year, the DES model probabilis- tically determines if the arrival will be transferred to another AOC (not applicable for SP officers), pro- moted to a higher grade, or continue for another year of service. At the end of each year, for each AOC, the arrival’s current grade (O1–O6) is incremented. At the end of each DES model replication, the model generates 30 years of data for each AOC, including replication number, life cycle year, and number of each grade. The DES model uses SP officer promotion rates and number of hires (entity arrival information) determined by the deterministic OFM. The determin- istic SP officer continuation rates are also used as an empirical distribution in the simulation. The DES model is replicated 999 times using common random number streams for variance reduction.

Results and Discussion We discuss the results of the deterministic and stochas- tic mixed-integer linear weighted goal-programming models to solve the SP workforce planning problem and provide decision support to APPD. We discuss how the results of the discrete-event simulation verify and validate the results of the deterministic variant of the OFM.

Deterministic Objective Force Model Results Upon formulating and solving the deterministic OFM using Microsoft Excel with the OpenSolver, we obtained an objective function value of 0, indicating that all the specified goal constraints were satisfied. That is, the documented number of SP officer autho- rizations and the officer quantities determined by the model (for all the grades) were satisfied. Only two of

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COL 2 3 2 1 1 1 1 2

LTC 6 5 4 5 4 2 4 2 1

MAJ 112111210777

CPT 9 17 22 28 17 19 18 5 2

LT 16 15 18 20 6

Obj force 20 20 19 19 19 18 17 16 15 15 13 13 10 10 10 9 8 8 7 7 6 6 3 2 1 1 1 1 0 0

Army medical specialist corps Physical therapist

Figure 1: (Color online) This chart is the typical OFM (what the AMEDD needs, depicted as a line by year) and the officer inventory by grade (depicted as stacked bars). The vertical bars of the histogram represent the actual inventory in terms of primary year group (cohort based on years of service) and rank. The black line represents the results of the OFM projections. Vertical bars that extend above the OFM line suggest that the AOC is over- strength for that particular year group, and vertical bars that fall below the OFM line suggest that the AOC is under-strength for that particular year group.

the goal deviation variables were positive. The model had 60 constraints and 27 variables, and solved in 0.53 seconds.

The optimal numbers of hires determined by the model were as follows: eight 1LT hires for OTs, 20 2LT hires for PTs, 10 2LT hires for CDs, 85 1LT hires for PAs, and five CPT hires for PAs. The goal for the aver- age promotion rate (across all ranks) for each AOC was approximately 74 percent based on the DOPMA promotion standards (see Table 4); the optimal solu- tion achieved an average promotion rate of 70 per- cent. On average, 95 percent of the CPTs, 76 percent of the MAJs, 55 percent of the LTCs, and 36 percent of the COLs were promoted across all AOCs. Table 5

displays the promotion percentage and the number of officers promoted by officer AOC and rank (CPT through COL).

Similarly, on average, 69 percent of CPTs, 75 per- cent of MAJs, and 81 percent of LTCs enter the rank (across all AOCs) and remain for consideration for promotion to the next higher rank. Table 6 shows offi- cer continuation percentages for each AOC and rank (CPT, MAJ, LTC).

We first solved the OFM with equal decision-maker weights (i.e., the documented authorizations at all the grades are given equal weights). We then per- formed a sensitivity analysis to see if the adjustment of weights had any impact on the authorizations. We

CPT MAJ LTC COL

OT 95% 83% 60% 43% 7.6 3.9 1.8 0.5

PT 95% 70% 55% 39% 18.4 10.2 5.1 1.5

CD 95% 89% 65% 41% 9.5 5.2 3 1.1

PA 95% 61% 40% 20% 85.8 37.1 7 1.2

Table 5: This table provides both the percentage and number promoted (cells) by area of concentration (rows) and rank (columns).

CPT (%) MAJ (%) LTC (%)

OT 62 75 71 PT 79 91 75 CD 62 88 91 PA 71 47 87

Table 6: This table details the percent of medical specialist corps offi- cers remaining entering promotion consideration for the next-higher grade (cells) by area of concentration (rows) and rank (columns).

found that, irrespective of the weights, all the goals were achieved; however, the determined officer quan- tity for each grade depends on the weights that were chosen.

In addition to estimating the number of officer hires and promotions, APPD uses the OFM as a workforce management decision support tool for optimizing AOC grade structure and to assist in workforce-reduction-program decisions. In particular, APPD uses the OFM results to generate histograms that visually represent the current workforce inven- tory (partitioned by the fiscal year the officer entered active duty) plotted against the 30-year life cycle inventory as projected by OFM. For example, Figure 1 displays a generated histogram for the PTs.

In Figure 1, the vertical bars of the histogram rep- resent the actual inventory in terms of primary year group (cohort based on years of service) and rank. The black line represents the results of the OFM pro- jections. Vertical bars that extend above the OFM line suggest that the AOC is over-strength (i.e., above the targeted number of officers) for that particular year group, and vertical bars that fall below the OFM line suggest that the AOC is under-strength

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50 scenarios

100 scenarios

Figure 3: (Color online) This figure depicts the objective value function (y -axis) as a function of the continuation rate standard deviation (x-axis), given the number of scenario runs (separate lines).

(i.e., below the targeted number of officers) for that particular year group. APPD uses these results to provide the senior AMEDD leadership a quick ref- erence to identify specific year groups that could require management focus and key personnel deci- sions. APPD periodically updates the histograms as personnel numbers change during the year and when- ever a new OFM is produced (usually annually).

(10, 0.005) (10, 0.01) (10, 0.05) (10, 0.1) (50, 0.005) (50, 0.01) (50, 0.05) (50, 0.1) (100, 0.005) (100, 0.01) (100, 0.05) (100, 0.1)

N um

be r

of o

ff ic

er s

hi re

(Number of scenarios, continuation rate SD)

OT-1LT

PT-2LT

CD-2LT

PA-1LT

PA-CPT

Figure 4: (Color online) This figure depicts the number of medical specialist corps officers hired (y -axis) by the number of scenarios and by the continuation rate standard deviation (x-axis) for various areas of concentration and ranks (separate bars).

(10, 0.005) (10, 0.01) (10, 0.05) (10, 0.1) (50, 0.005) (50, 0.01) (50, 0.05) (50, 0.1) (100, 0.005) (100, 0.01) (100, 0.05) (100, 0.1)

N um

be r

of M

A J

pr om

ot ed

(Number of scenarios, continuation rate SD)

Figure 5: (Color online) This figure depicts the number of majors promoted (y -axis), given the number of sce- narios and continuation rate standard deviation (x-axis) for various areas of concentration.

(10, 0.005) (10, 0.01) (10, 0.1) (50, 0.005) (50, 0.01) (50, 0.05) (50, 0.1) (100, 0.005) (100, 0.01) (100, 0.05) (100, 0.1)

N um

be r

of L

T C

p ro

m ot

(Number of scenarios, continuation rate SD)

Figure 6: (Color online) This figure depicts the number of lieutenant colonels promoted (y -axis), given the num- ber of scenarios and the continuation rate standard deviation (x-axis) for various areas of concentration.

Stochastic Method #1: Objective Force Model Results We performed the analysis by varying the number of scenarios and the stochastic continuation rate stan- dard deviation (SD). We varied the number of scenar- ios from 10 to 100 and the continuation rate SD from 0.005 to 0.1. For each setting (i.e., number of scenarios and continuation rate SD), we determined the number of SP officers hired, the number of SP officers pro- moted, and the objective function values. Table 7 con- tains the results of the stochastic method #1 that the OFM solved using Microsoft Excel with OpenSolver.

The average numbers of MAJs promoted were 3.77, 10.26, 5.33, and 37.93 in OT, PT, CD, and PA, respec- tively. The average numbers of LTCs promoted were 1.73, 5.17, 3.04, and 7.27 for OT, PT, CD, and PA, respectively. The average numbers of COLs promoted were 0.54, 1.52, 1.12, and 1.31 in OT, PT, CD, and PA respectively. In Figure 2, we see that the compu- tational time increased with the number of scenar- ios run.

In most of the cases, we observed that the com- putational time was higher when the continuation rate SD was lower. The objective function value also appeared to depend on the continuation rate SD (see Figure 3). The value of the objective function increased as the continuation rate SD increased, irre- spective of the scenarios. This might be because when the uncertainty level increases, achieving all the tar- gets becomes difficult.

The number of SP officers hired remained relatively flat for all the settings (see Figure 3). On average, 8.36 officers of rank 1LT for OTs, 20.58 officers of rank 2LT for PTs, 9.90 officers of rank 2LT for CDs, 75.12 offi- cers of rank 1LT for PAs, and 20.11 officers of rank CPT for PAs were hired. The number of SP officers promoted also remained the same for all the settings. We observed that on average, the number of officers for PAs promoted was higher than for any other AOC (see Figures 4–7).

Stochastic Method #2: Objective Force Model Results Upon formulating and solving the stochastic method #2 OFM, we often did not achieve an integer solution because of the large number of additional constraints from each scenario. We therefore relaxed the integer

constraints for number of SP officers hired. We per- formed 100 scenarios and 10 iterations, and then cal- culated the estimated expected values and variances for each decision variable. We assumed the probabil- ity density function of the officer continuation rates as a normal distribution with mean equal to the solution of the deterministic optimization model and standard deviation of 0.005.

Upon solving the model using the General Alge- braic Modeling System (GAMS) 23.9.3 with IBM ILOG CPLEX 12.4.0.1, we obtained an average objective function value of 0.58. The total solution time was 160.3 seconds. There were 66,432 constraints and 75,449 variables in total per iteration. Table 8 shows the results for the number of estimated average SP officer hires and promotions. The optimal numbers of estimated average hires determined by the model were as follows: 8.44 1LT hires for OT, 19.81 2LT hires for PTs, 9.51 2LT hires for CDs, 89.44 1LT hires for PAs, and no CPT hires for PAs. Compared to the deterministic OFM results, the number of hires for the former 1LT-OT, 2LT-PT, and 2LT-CD was nearly identical. However, we see a clear trade-off for the latter 1LT-PA and CPT-PA. This stochastic model solu- tion increased the number of 1LT hires for PAs by eliminating CPT hires for the same AOC. In terms of the estimated average number of SP officer pro- motions, these results were nearly identical to those determined by the deterministic variant of the opti- mization model.

Compared with the results of stochastic method #1 OFM (using the same condition—100 scenarios with standard deviation of 0.005), we see that the main dif- ferences are the number of 1LTs and CPTs hired for PAs. The model can induce infeasibility problems if the number of CPT hires for PAs is strictly positive, which is hidden in the stochastic method #1 because the number of constraints in stochastic method #1 is much less than in stochastic method #2. Thus, when we use the first method to solve a stochastic optimiza- tion problem, we must be aware of the possible infea- sibility issue because of stochastic parameters. Aside from this difference, the other results were similar, providing verification for stochastic method #1.

From Table 9, we can see that the variances of each decision variable (number promoted, number hired) are not very large, which means that the actual

Number of SP officers hired Number of SP officers promoted

Number of Continuation Comp OT PT CD PA PA OT OT OT PT PT PT CD CD CD PA PA PA Objective scenarios SD time (sec) 1LT 2LT 2LT 1LT CPT MAJ LTC COL MAJ LTC COL MAJ LTC COL MAJ LTC COL function

10 00005 8016 8017 19084 9089 81090 9006 3072 1074 0054 10020 5012 1051 5037 2099 1012 37015 7006 1024 0062 0001 8078 8011 19083 9060 82078 8080 3075 1076 0054 10009 5006 1051 5036 3005 1022 37036 7011 1024 1052 0005 7007 8031 21015 9081 71095 26080 3072 1071 0054 10013 4098 1053 5037 3001 1013 36033 7047 1028 3030 001 6076 8067 22089 10004 66035 36015 3089 1072 0055 10024 5024 1054 5009 3008 1016 40010 7050 1047 18008

50 00005 41092 8013 19080 9089 80031 11029 3076 1075 0054 10016 5012 1051 5031 3001 1014 37034 7007 1024 0074 0001 38002 8018 19078 9076 81077 9042 3075 1073 0054 10015 5014 1051 5034 3003 1020 37012 7008 1025 1071 0005 41049 8048 20023 9075 68090 28093 3066 1067 0055 10025 5022 1051 5036 3008 1014 37084 7032 1031 6015 001 43042 8095 21060 10025 60082 39095 3092 1070 0054 10078 5001 1055 5035 3008 0087 39092 7064 1049 14012

100 00005 233084 8015 19081 9087 81018 10008 3074 1075 0054 10015 5010 1051 5036 3000 1013 37024 7007 1024 0063 0001 244080 8019 19081 9071 81044 10005 3071 1074 0054 10015 5009 1051 5038 3005 1022 37026 7008 1024 1069 0005 236090 8029 20035 9086 72090 23023 3076 1074 0054 10026 5014 1051 5034 3009 1014 38018 7022 1029 5048 001 107093 8071 21082 10034 71012 27054 3085 1069 0055 10052 5086 1057 5034 3000 0095 39035 7058 1040 16047

Table 7: This table depicts the medical specialist corps officer continuation rates, computation time, hiring num- bers, promotion numbers, and objective function of the optimization (separate cell values) by the number of scenarios (row values), area of concentration, and rank (column values).

(10, 0.005) (10, 0.01) (10, 0.05) (10, 0.1) (50, 0.005) (50, 0.01) (50, 0.05) (50, 0.1) (100, 0.005) (100, 0.01) (100, 0.05) (100, 0.1)

N um

be r

of C

O L

p ro

m ot

(Number of scenarios, continuation rate SD)

Figure 7: (Color online) This figure depicts the number of colonels promoted (y -axis), given the number of sce- narios and the continuation rate standard deviation (x-axis) for various areas of concentration.

2LT 1LT CPT

OT Mean 0 804435 0 variance 0 000368 0

PT Mean 1908086 0 0 variance 000498 0 0

CD Mean 905065 0 0 variance 002056 0 0

PA Mean 0 8904375 0 variance 0 0 0

Table 8: This table shows the estimated mean and variance of medical specialist corps officer hires (cells) by area of concentration (rows) and rank (columns).

MAJ LTC COL

OT Mean 305991 106366 005282 variance 000178 000085 000017

PT Mean 1003909 503402 105332 variance 000468 000844 00003

CD Mean 407525 3002 101294 variance 005233 000022 000027

PA Mean 360932 702497 103927 variance 003778 000972 002425

Table 9: This table shows the estimated mean and variance of medical specialist corps officer promotions (cells) by area of concentration (rows) and rank (columns).

AOC 2LT 1LT CPT MAJ LTC COL Total

AOC average OT 806527 4501211 2304194 1102202 40002 9204154 PT 20.3804 4003904 1220926 7604414 2602923 801241 2940555 CD 10 2002653 5801712 3808719 2001291 607768 1540214 PA 1770867 5430512 1860128 4404675 80967 9600941

AOC standard deviation OT 100405 606646 906917 804417 503506 2101086 PT 0.6084 204412 1007675 1907641 1103142 609472 3304218 CD 0 005072 606033 1303928 1004438 601126 2509942 PA 30715 2203338 290749 1606983 800492 5007138

AOC 95% confidence (half-width) OT 0 0006452 0041328 0060099 0052347 003318 1030896 PT 0.03773 0015138 006677 1022559 007016 004308 2007251 CD 0 0003145 0040947 0083049 0064762 0037905 1061191 PA 0 0023037 1038493 1084475 1003547 0049914 3014479

AOC inventory data from deterministic OFM OT 0 806667 4502563 2306091 1104679 4 93 PT 20.3954 4006908 1230328 7507375 2603997 8 2940551 CD 10 2005 5800988 3804012 20 7 154 PA 0 1770846 5440234 1870124 4502453 9 9630449

Table 10: This table shows the results of the discrete-event simulation and the deterministic OFM by area of concentration (rows) and rank (columns). The first three table splits are the average, standard deviation, and confidence interval for the simulation. The last split represents the deterministic model results.

expected values are not far from the estimated means. After comparing the stochastic model results with those of the deterministic OFM, we conclude that these results are both reasonable and useful to APPD.

We also implemented the stochastic method #2 opti- mization model in Microsoft Excel using VBA and the OpenSolver to provide APPD with a user-friendly decision support tool.

Discrete-Event Simulation Model Results As a means to verify and validate the deterministic OFM, we used the output of the discrete-event simu- lation (coded in MedModel 2011) to create lower- and upper-bound limits to the SP officer total inventory values (summed over the 30-year officer life cycle by grade) for each AOC. We then checked whether the optimal values generated by the deterministic opti- mization model fell within these AOC total inventory limits. Based on the 999 DES model replications, we calculated the average and standard deviation of the total inventory values for each AOC and grade. Using these values, we computed the 95 percent confidence

interval half widths to create the lower- and upper- bound AOC limits. We extracted the optimal values generated by the OFM. Table 10 shows these results.

We then compared the optimization values to the 95 percent confidence upper- and lower- 30-year AOC total inventory limits. From Table 11, we can see that all the optimal values computed from the deter- ministic OFM fell within the simulation-based upper and lower bounds, with the exception of two values (PT → 1LT and CD → 1LT); however, these two val- ues (depicted in bold) were both just slightly greater than its respective upper-bound limit.

These DES model results validate and verify the results of our deterministic, mixed-integer linear weighted goal-programming model for optimal work- force planning of the Medical Specialist Corps. In future work, we aim to apply this method to the stochastic variants of the optimization model by gen- erating statistical quality control charts.

Concluding Remarks In this paper, we described the objective force model currently in use by APPD for long-term workforce

Bounds AOC 2LT 1LT CPT MAJ LTC COL Total

Upper OT 8072 45053 24002 11074 4033 93072 Opt value 8067 45026 23061 11047 4 93 Lower 8059 44071 22082 1007 3067 91011 Upper PT 20042 40054 123059 77067 26099 8055 296063 Opt value 2004 40069 123033 75074 2604 8 294055 Lower 20034 40024 122026 75022 25059 7069 292048 Upper CD 10 2003 58058 3907 20078 7016 155083 Opt value 10 2005 5801 3804 20 7 154 Lower 10 20023 57076 38004 19048 604 15206 Upper PA 17801 54409 187097 4505 9047 964009 Opt value 177085 544023 187012 45025 9 963045 Lower 177064 542013 184028 43043 8047 95708

Table 11: This table provides the mean estimate as well as the 95% con- fidence interval for the 30-year area of concentration total inventory lim- its (cells) by the parameter estimate (rows), area of concentration (table dividers), and rank.

planning of medical professionals in the AMEDD, and we specifically discussed the OFM and its stochas- tic variants applied to the Medical Specialist Corps. We investigated several methods of estimating the optimal number of hires and promotions for the AMEDD’s Medical Specialist Corps and found high degrees of consistency among models. We described the deterministic optimization OFM and stochastic optimization OFM variants, and then further com- pared deterministic results with those from DES, find- ing a high level of congruency. The expert workforce management decision support for estimating the cor- rect number of hires and promotions necessary to achieve desired personnel structure under uncertainty is of significance.

The OFM provides tremendous value to APPD; it requires only seconds to solve rather than months. This enables APPD to produce quick turnaround analysis in a transparent way and provides decision support that was almost impossible using a manual process. One of the greatest benefits of the OFM is that APPD uses the optimization results to provide senior AMEDD leaders with a quick reference to iden- tify specific-year groups that could require specific management focus as well as key personnel decisions in terms of hiring, promoting, or firing. As the U.S. Army continues to reduce the size of its workforce structure, the use of the OFM will become vital for the AMEDD in its future medical professional reduc- tion program. In future work, we plan to model the

stochasticity of changing workforce structure autho- rization levels to provide decision support for down- sizing decisions.

In terms of personnel strategy and policy, the impact from OFM-supported decisions includes recruiting goals for the Army’s recruiting command, classroom capacity for specialty training, promotion requirements, and force-reduction objectives.

Appendix A. Deterministic Variant of the Objective Force Model

Objective Force Model Sets G—Index for officer grade with g ∈ G. I—Index for officer career specialty (AOC) with i ∈ I. K—Index for an officer’s year in service with k ∈ K. F —Index for each goal with f ∈ F .

Objective Force Model Parameters cig —Documented authorizations for AOC i in

grade g. rigk—Continuation (deterministic) rate for AOC i in

grade g in year k. Cap

i —Maximum allowable officer quantity of AOC i.

Floori—Minimum acceptable officer quantity for AOC i.

Capig —Maximum allowable officer quantity of AOC i at grade g.

Floorig —Minimum acceptable officer quantity for AOC i at grade g.

pfig —Minimum promotion rate for AOC i at grade g.

pcig —Maximum promotion rate for AOC i at grade g.

SPIMMg —Medical Specialist Corps immaterial authorizations for grade g.

AMIMMg —AMEDD immaterial authorizations for grade g.

THSg —Transients, holdees, and students (THS) authorizations for grade g.

wf —Decision-maker weight for goal f .

Objective Force Model Decision and Goal Deviation Variables pig —Number of officers promoted in AOC i at grade g,

∀ g = 41 51 6. aig —Number of officers accessed (hired) for AOC i at

grade g, ∀ g = 11 21 3. dig —Officer quantity for AOC i in grade g.

Invigk—Projected inventory of AOC i in grade g in year k. posf —Positive deviation for goal f . neg

f —Negative deviation for goal f .

Objective Force Model Formulation

min Z = ∑

wf 4posf +negf 50 (A1)

subject to

∑

dig −

(

∑

4cig +SPIMMg +AMIMMg

+THSg5 )

−posf =1 +negf =1 = 00 (A2)

∑

dig −

(

∑

cig +SPIMMg +AMIMMg +THSg

)

−posf =2 +negf =2 = 0 ∀g = 60 (A3)

∑

dig −

(

∑

cig +SPIMMg +AMIMMg +THSg

)

−pos f =3 +negf =3 = 0 ∀g = 50 (A4)

∑

dig −

(

∑

cig +SPIMMg +AMIMMg +THSg

)

−posf =4 +negf =4 = 0 ∀g = 40 (A5)

3 ∑

g=1

∑

dig −

( 3 ∑

g=1

∑

4cig +SPIMMg +AMIMMg

+THSg5 )

−pos f =5 +negf =5 = 00 (A6)

The objective function in (A1) of the deterministic OFM seeks to minimize the sum of the weighted goal deviations. The target for the first goal constraint in constraint (A2) is for the total number of officers over each grade and AOC to equal the total documented authorizations (over each grade and AOC and the SP immaterial, AMEDD immaterial, and THS). The target for the second goal constraint in (A3) is for the total number of COLs over each AOC to equal the COL documented authorizations over each AOC and the SP immaterial, AMEDD immaterial, and THS. The target for the third goal constraint in (A4) is for the total number of LTCs over each AOC to equal the LTC documented autho- rizations over each AOC and the SP immaterial, AMEDD immaterial, and THS. The target for the fourth goal con- straint in (A5) is for the total number of MAJs over each AOC to equal the MAJ documented authorizations over each AOC and the SP immaterial, AMEDD immaterial, and THS. The target for the last goal constraint in (A6) is for the total number of company grade (sum of 2LT, 1LT, and CPT) officers over each AOC to equal the company grade docu- mented authorizations over each AOC and the SP immate- rial, AMEDD immaterial, and THS.

∑

di=11g ≤ Capi=10 (A7)

∑

di=31g ≥ Floori=30 (A8)

∑

di=11g ≥ Floori=10 (A9)

∑

di=31g ≤ Capi=30 (A10)

∑

di=21g ≤ Capi=20 (A11)

∑

dig ≥ ∑

cig ∀ i ∈ I0 (A12)

3 ∑

g=1

dig ≥ 3 ∑

g=1

cig ∀ i ∈ I0 (A13)

di=31g=5 ≥ Floori=31g=50 (A14)

di=31g=6 ≥ Floori=31g=60 (A15)

di=11g=6 ≤ Capi=11g=60 (A16)

di=21g=6 ≥ Floori=21g=60 (A17)

di=11g=5 ≤ Capi=11g=50 (A18)

di=11g=6 ≥ Floori=11g=60 (A19)

dig ≥ cig ∀ i ∈ I1g = 41 51 60 (A20)

Constraint (A7) is used to place a maximum allowable officer quantity for OTs, whereas constraint (A8) is used to place a minimum acceptable officer quantity for CDs. Con- straint (A9) is used to place a minimum allowable officer quantity for OTs, whereas constraint (A10) is used to place a maximum allowable officer quantity for CDs. Constraint (A11) is used to place a maximum allowable officer quantity for PTs. Constraints (A12) ensure that the officer quantity must meet or exceed the total documented authorizations for each AOC i. Constraints (A13) ensure that the total offi- cer quantity for company grade (sum of 2LT, 1LT, and CPT) officers must meet or exceed the total documented autho- rizations for company grade (sum of 2LT, 1LT, and CPT) officers for each AOC i. Constraint (A14) ensures that the LTC officer quantity for CDs must meet or exceed the min- imum acceptable LTC officer quantity for CDs. Constraint (A15) ensures that the COL officer quantity for CDs must meet or exceed the minimum acceptable COL officer quan- tity for CDs. Constraint (A16) ensures that the COL officer quantity for OTs must be less than or equal to the maximum allowable COL officer quantity for OTs. Constraint (A17) ensures that the COL officer quantity for PTs must meet or exceed the minimum acceptable COL officer quantity for PTs. Constraint (A18) ensures that the LTC officer quantity for OTs must be less than or equal to the maximum allow- able LTC officer quantity for OTs. Constraint (A19) ensures that the COL officer quantity for OTs must meet or exceed the minimum acceptable COL officer quantity for OTs. Con- straints (A20) ensure that the MAJ, LTC, and COL officer quantity must meet or exceed MAJ, LTC, and COL docu- mented authorizations for each AOC i.

pi1g=4 ≥ pfi1g=4Invi1g=31k=10 ∀ i ∈ I0 (A21)

pi1g=5 ≥ pfi1g=5Invi1g=41k=16 ∀ i ∈ I0 (A22)

pi1g=6 ≥ pfi1g=6Invi1g=51k=21 ∀ i ∈ I0 (A23)

pi1g=4 ≤ pci1g=4Invi1g=31k=10 ∀ i ∈ I0 (A24)

pi1g=5 ≤ pci1g=5Invi1g=41k=16 ∀ i ∈ I0 (A25)

pi1g=6 ≤ pci1g=6Invi1g=51k=21 ∀ i ∈ I0 (A26)

Constraints (A21), (A22), and (A23) ensure that the num- ber of resultant field grade (MAJ, LTC, COL) promotions must be greater than or equal to the minimum number of promotions (i.e., the product of the minimum promotion rate and pool) for each respective field grade, for each AOC i. Constraints (A24), (A25), and (A26) ensure that the num- ber of resultant field grade (MAJ, LTC, COL) promotions must be less than or equal to the maximum number of pro- motions (i.e., the product of the maximum promotion rate and pool) for each respective field grade for each AOC i.

dig = ∑

Invigk ∀i ∈I1 g ∈G0 (A27)

Invi1g=11k=1 =ai1g=1ri1g=11k=1 ∀i ∈I0 (A28)

Invi=11g=11k=3 =Invi=11g=11k=2ri=11g=11k=30 (A29)

Invi=11g=21k=3

=4Invi=11g=21k=2 +ai=11g=25ri=11g=21k=30 (A30)

Invig1k=3 =Invig1k=2rig1k=3 ∀g ∈G1 i = 213140 (A31)

Invi=11g=21k=2 =Invi1g=11k=1ri1g=21k=20 (A32)

Invi1g=21k=2 =44Invi1g=11k=1pfi1g=25+ai1g=25ri1g=21k=2

∀i = 213140 (A33)

Invigk =Invigk−1rigk ∀k = 5−10112−16118−21123−301

g ∈G1 i ∈I0 (A34)

Invi1g=11k=2 =Invi1g=11k=141−pfi1g=25ri1g=11k=2

∀i ∈I0 (A35)

Invi1g=31k=4 =44Invi1g=21k=3pfi1g=35+ai1g=35ri1g=31k=4

∀i ∈I0 (A36)

Invi1g=21k=4 =Invi1g=21k=341−pfi1g=35ri1g=21k=3

∀i ∈I0 (A37)

Invi1g=41k=11 =pi1g=4 ∀i ∈I0 (A38)

Invi1g=31k=11 =4Invi1g=31k=10 −pi1g=45ri1g=31k=11

∀i ∈I0 (A39)

Invi1g=51k=17 =pi1g=5 ∀i ∈I0 (A40)

Invi1g=41k=17 =4Invi1g=41k=16 −pi1g=55ri1g=41k=17

∀i ∈I0 (A41)

Invi1g=61k=22 =pi1g=6 ∀i ∈I0 (A42)

Invi1g=51k=22 =4Invi1g=51k=21 −pi1g=65ri1g=51k=22

∀i ∈I0 (A43)

pig ≥ 0 ∀i ∈I1 g = 415163 aig ≥ 0 and integer

∀i ∈I1 g = 112133 posf 1 negf ≥ 0 ∀f ∈F 0 (A44)

Constraints (A27) assign the officer quantity as the total projected inventory (over all years) for each AOC i and grade g. Constraints (A28) assign the inventory for year k = 1 and grade g = 1 for each AOC i. Constraint (A29) assigns the inventory for year k = 3, grade g = 1, and i = 1, whereas constraint (A30) assigns the inventory for year k = 3, grade g = 2, and i = 1. Constraints (A31) assign the inventory for k = 3 for all grades and i = 21 31 4. Constraint (A32) assigns the inventory for i = 1, g = 2, and k = 2, whereas constraints (A33) assign the inventory for g = 2, k = 2, and i = 21 31 4. Constraints (A34) assign the inventory for all years that are not year k = 11 3 or a promotion year, for each grade g and each AOC i. Constraints (A35) assign the inventory for year k = 2 and grade g = 1 for each AOC i. Constraints (A36) assign the inventory for year k = 4 and grade g = 3 for each AOC i. Constraints (A37) assign the inventory for year k = 4 and grade g = 2 for each AOC i. Constraints (A38) assign the inventory for year k = 11 and grade g = 4 for each AOC i. Constraints (A39) assign the inventory for year k = 11 and grade g = 3 for each AOC i. Constraints (A40) assign the inventory for year k = 17 and grade g = 5 for each AOC i. Constraints (A41) assign the inventory for year k = 17 and grade g = 4 for each AOC i. Constraints (A42) assign the inventory for year k = 22 and grade g = 6 for each AOC i. Constraints (A43) assign the inventory for year k = 22 and grade g = 5 for each AOC i. Constraint (A44) represents the nonnegativity and integer constraints for the decision and deviational variables.

Appendix B. Technical Details for Stochastic Optimization Methods In stochastic method #1, we use a scenario-based Monte Carlo simulation approach to approximate the optimal objective value and the optimal solution of a stochastic goal program:

min x∈X

E6f 4x1�570

We generate S scenarios and each scenario corresponds to one realization of the random vector �. We have S deter- ministic optimization problems with forms of

min x∈X

f 4x1�i5 where i = 110001S0

Let zi = minx∈X f 4x1�i5 and xi = arg minx∈X f 4x1�i5. We use z̄ = 41/S5

∑S i zi and x̄ = 41/S5

∑S i xi to be the

approximated optimal objective value and the approxi- mated optimal solution, respectively.

In stochastic method #2, we formulate the stochastic opti- mization model using the following form:

min x∈X

E6f 4x1�570

We leverage the sample average approximation (SAA) method (Rubinstein and Shapiro 1990) because prior sam- ples are easily generated. Suppose �110001�N are N inde- pendent samples from a probability distribution; we then obtain an estimator:

f̂N 4x5 = 1 N

N ∑

j=1

f 4x1�j50

Because f̂N 4x5 is an unbiased estimator of f 4x5, we instead minimize the SAA to obtain an estimate of the opti- mal value:

min x∈X

f̂N 4x5 = 1 N

N ∑

j=1

f 4x1�j50

Let z∗ = minx∈X f 4x5 and ẑN = minx∈X f̂N 4x5. We then con- struct statistical lower and upper bounds suggested by Mak et al. (1999), who show that E6ẑN 7 ≤ z

∗. To estimate E6ẑN 7, we iterate the SAA program M times to obtain M optimal solutions and compute the average of the solutions:

vN1M = 1 M

M ∑

j=1

ẑ j N 1

which is an unbiased estimate of E6ẑN 7. This is a lower bound on z∗. We compute the estimate of variance of the previous estimator as follows:

S2vN1M = 1

M4M − 15

M ∑

j=1

4ẑ j N − vN1M5

For the upper bound, we estimate f 4x̂5 at each of the M solutions (x110001xM ) obtained before by N ′ (which is inde- pendent of N and usually very large) independent samples. Because z∗ is the optimal value,

f̂N ′ 4x̂5 = 1 N ′

N ′ ∑

j=1

f j N ′ 4x̂5 ≥ z

∗1

and f̂N ′ 4x5 is an upper bound. An estimate of the variance of the previous estimator is given by:

S2fN ′x̂ = 1

N ′4N ′ − 15

N ′ ∑

j=1

4f j N ′ 4x̂5 − f̂N ′ 4x̂55

Further, we set the stopping rule according to the value of the estimate of f 4x̂5 − z∗ and the variance. If either gap is not small enough, we increase the sample size and repeat the process.

Appendix C. Stochastic Method #2: Variant of the Objective Force Model

Stochastic Method #2 OFM Sets G—Index for officer grade with g ∈ G. I—Index for officer career specialty (AOC) with i ∈ I. K—Index for an officer’s year in service with k ∈ K. F —Index for each goal with f ∈ F . S—Index for the scenario under consideration with s ∈ S.

Stochastic Method #2 OFM Parameters cig —Documented authorizations for AOC i in

grade g. rigks—Continuation (stochastic) rate for AOC i in

grade g in year k in scenario s. Cap

i —Maximum allowable officer quantity of AOC i.

Floori—Minimum acceptable officer quantity for AOC i.

Capig —Maximum allowable officer quantity of AOC i at grade g.

Floorig —Minimum acceptable officer quantity for AOC i at grade g.

pfig —Minimum promotion rate for AOC i at grade g.

pcig —Maximum promotion rate for AOC i at grade g.

SPIMMg —Medical Specialist Corps immaterial authorizations for grade g.

AMIMMg—AMEDD immaterial authorizations for grade g.

THSg—Transient, holdee, and student (THS) authorizations for grade g.

wf —Decision-maker weight for goal f .

Stochastic Method #2 OFM Decision and Goal Deviation Variables pig —Number of officers promoted in AOC i at grade g1

∀g = 41 51 6. aig —Number of officers accessed (hired) for AOC i at

grade g1 ∀g = 11 21 3. digs—Number of officers for AOC i in grade g in scenario

s. Invigks—Projected inventory of AOC i in grade g in year k

in scenario s. posfs—Positive deviation for goal f in scenario s. neg

fs —Negative deviation for goal f in scenario s.

Stochastic Method #2 OFM Formulation

min Z = ∑

1 S

∑

wf 4posf +negf 50 (B1)

subject to

∑

digs −

(

∑

4cig +SPIMMg +AMIMMg

+THSg5 )

−posf =11s +negf =11s = 0 ∀s ∈S0 (B2)

∑

digs −

(

∑

cig +SPIMMg +AMIMMg +THSg

)

−pos f =21s +negf =21s = 0 ∀g = 61 s ∈S0 (B3)

∑

digs −

(

∑

cig +SPIMMg +AMIMMg +THSg

)

−pos f =31s +negf =31s = 0 ∀g = 51 s ∈S0 (B4)

∑

digs −

(

∑

cig +SPIMMg +AMIMMg +THSg

)

−posf =41s +negf =41s = 0 ∀g = 41 s ∈S0 (B5)

3 ∑

g=1

∑

digs −

( 3 ∑

g=1

∑

4cig +SPIMMg +AMIMMg

+THSg5 )

−posf =51s +negf =51s = 01 ∀s ∈S0 (B6)

∑

di=11gs ≤Capi=1 ∀s ∈S0 (B7)

∑

di=31gs ≥Floori=3 ∀s ∈S0 (B8)

∑

di=11gs ≥Floori=1 ∀s ∈S0 (B9)

∑

di=31gs ≤Capi=3 ∀s ∈S0 (B10)

∑

di=21gs ≤Capi=2 ∀s ∈S0 (B11)

∑

digs ≥ ∑

cig ∀i ∈I1 s ∈S0 (B12)

3 ∑

g=1

digs ≥ 3 ∑

g=1

cig ∀i ∈I1 s ∈S0 (B13)

di=31g=51s ≥Floori=31g=5 ∀s ∈S0 (B14)

di=31g=61s ≥Floori=31g=6 ∀s ∈S0 (B15)

di=11g=61s ≤Capi=11g=6 ∀s ∈S0 (B16)

di=21g=61s ≥Floori=21g=6 ∀s ∈S0 (B17)

di=11g=51s ≤Capi=11g=5 ∀s ∈S0 (B18)

di=11g=61s ≥Floori=11g=6 ∀s ∈S0 (B19)

digs ≥cig ∀i ∈I1g = 415161s ∈S (B20)

pi1g=4 ≥pfi1g=4Invi1g=31k=101s ∀i ∈I1 s ∈S0 (B21)

pi1g=5 ≥pfi1g=5Invi1g=41k=161s ∀i ∈I1 s ∈S0 (B22)

pi1g=6 ≥pfi1g=6Invi1g=51k=211s ∀i ∈I1 s ∈S0 (B23)

pi1g=4 ≤pci1g=4Invi1g=31k=101s ∀i ∈I1 s ∈S0 (B24)

pi1g=5 ≤pci1g=5Invi1g=41k=161s ∀i ∈I1 s ∈S0 (B25)

pi1g=6 ≤pci1g=6Invi1g=51k=211s ∀i ∈I1 s ∈S0 (B26)

digs = ∑

Invigks ∀i ∈I1 g ∈G1 s ∈S0 (B27)

Invi1g=11k=11s =ai1g=1ri1g=11k=11s

∀i ∈I1 s ∈S0 (B28)

Invi=11g=11k=31s =Invi=11g=11k=21sri=11g=11k=31s

∀s ∈S0 (B29)

Invi=11g=21k=31s =4Invi=11g=21k=21s +ai=11g=25

·ri=11g=21k=31s ∀s ∈S0 (B30)

Invig1k=31s =Invig1k=21srig1k=31s

∀g ∈G1 i = 213141 s ∈S0 (B31)

Invi=11g=21k=21s =Invi1g=11k=11sri1g=21k=21s

∀s ∈S0 (B32)

Invi1g=21k=21s =44Invi1g=11k=11spfi1g=25+ai1g=25

·ri1g=21k=21s ∀i = 213141 s ∈S0 (B33)

Invigks =Invigk−11srigks∀k = 5−10112−16118−21123−301

g ∈G1 i ∈I1 s ∈S0 (B34)

Invi1g=11k=21s =Invi1g=11k=11s41−pfi1g=25

·ri1g=11k=21s ∀i ∈I1 s ∈S0 (B35)

Invi1g=31k=41s =44Invi1g=21k=31spfi1g=35+ai1g=35

·ri1g=31k=41s ∀i ∈I1 s ∈S0 (B36)

Invi1g=21k=41s =Invi1g=21k=31s41−pfi1g=35

·ri1g=21k=31s ∀i ∈I1 s ∈S0 (B37)

Invi1g=41k=111s =pi1g=4 ∀i ∈I1 s ∈S0 (B38)

Invi1g=31k=111s =4Invi1g=31k=101s −pi1g=45

·ri1g=31k=111s ∀i ∈I1 s ∈S0 (B39)

Invi1g=51k=171s =pi1g=5 ∀i ∈I1 s ∈S0 (B40)

Invi1g=41k=171s =4Invi1g=41k=161s −pi1g=55

·ri1g=41k=171s ∀i ∈I1 s ∈S0 (B41)

Invi1g=61k=221s =pi1g=6 ∀i ∈I1 s ∈S0 (B42)

Invi1g=51k=221s =4Invi1g=51k=211s −pi1g=65

·ri1g=51k=221s ∀i ∈I1 s ∈S0 (B43)

pig ≥ 0 ∀i ∈I1 g = 415163 aig ≥ 0 and integer

∀i ∈I1 g = 112133 pos fs 1 neg

fs ≥ 0

∀f ∈F 1 s ∈S0 (B44)

Acknowledgments Any opinions, findings, and conclusions or recommenda- tions expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation, Pennsylvania State University, Texas Tech Uni- versity, Georgia Institute of Technology, or United States Army. We would like to thank Michael O’Connor for his assistance at the Center for AMEDD Strategic Studies. This material is based upon work supported by the National

Science Foundation [Grant DGE1255832] and the Seth Bon- der Foundation/INFORMS Bonder Scholarship for Applied Operations Research.

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Verification Letter Paul J. Goymerac, Colonel, U.S. Army, Director, AMEDD

Personnel Proponency Directorate, U.S. Army Medical Department Center & School, Fort Sam Houston, Texas 78234-6100, writes:

“This is to verify that the Army Medical Department Per- sonnel Proponent Directorate has adopted and continues to use the modeling process described by Nathan Bastian in his manuscript “The AMEDD Uses Goal Programming to Optimize Workforce Planning Decisions.” The Objective Force Models are a key factor in the officer personnel recom- mendations my directorate provides to the Army Medical Department leadership to support decisions impacting the number of officers brought on to active duty, the number trained in the various specialties, and the number promoted to a higher rank each year.”

Nathaniel D. Bastian is a NSF graduate research fellow and doctoral candidate of industrial engineering and opera- tions research in the Harold and Inge Marcus Department of Industrial and Manufacturing Engineering at the Pennsyl- vania State University. He is a Captain in the Medical Ser- vice Corps branch of the U.S. Army Medical Department, where he serves as a healthcare operations research analyst. He earned his M.Eng. in industrial engineering from Penn State, M.S. in econometrics and operations research from Maastricht University, and B.S. in engineering management with honors from the U.S. Military Academy at West Point. His research interests include resource allocation optimiza- tion under uncertainty, statistical learning for performance improvement, cost-effectiveness and econometric modeling, prescriptive and predictive analytics, and multiple criteria decision-making with applications in healthcare delivery, military operations, and logistics.

Pat McMurry is an operations research analyst with the U.S. Army Medical Department’s Personnel Proponent Directorate. He earned his B.S. in civil engineering from

New Mexico State University and a M.S.E. in operations research and industrial engineering from the University of Texas at Austin. Pat has spent the last 18 years working in various modeling and simulation positions in the Army Medical Department, with the last six specifically focused on personnel optimization models.

Lawrence V. Fulton is an assistant professor of quanti- tative methods and health organization management in the Rawls College of Business at Texas Tech University and holds a secondary appointment with the Texas Tech Univer- sity Health Sciences Center. His research primarily involves the application of advanced quantitative methods to sup- port decision making within the Department of Defense and medical sector. Larry is also a retired military officer.

Paul M. Griffin is a professor in the H. Milton Stewart School of Industrial and Systems Engineering at the Geor- gia Institute of Technology, where he serves as the Joseph C. Mello Chair. His research and teaching interests are in health and supply chain systems. In particular, his current research activities have focused on cost-effectiveness mod- eling of public health interventions, health logistics, health access, and economic modeling.

Shisheng Cui is a doctoral candidate and research assis- tant in the Harold and Inge Marcus Department of Indus- trial and Manufacturing Engineering at the Pennsylvania State University. His research interests include system infor- matics and control for complex systems, modeling, and optimization based on spatial data and engineering statisti- cal learning.

Thor K. Hanson is a masters student in the Harold and Inge Marcus Department of Industrial and Manufac- turing Engineering at the Pennsylvania State University. He is a Major in the Logistics Corps of the U.S. Army, where he serves as an operations research systems analyst. His research interests include resource allocation optimiza- tion, trade space exploration, predictive analytics, process modeling, and decision-making strategies with applications in military operations and logistics.

Sharan Srinivas is a doctoral candidate in industrial engineering and operations research at the Pennsylvania State University, University Park. He received his mas- ter’s degree in industrial and systems engineering from Binghamton University, State University of New York, in 2013. His research interests include healthcare delivery sys- tems, transportation problem, scheduling, and inventory optimization.