Fluent Essays

Discussion: Optimization in Staffing Oscar is a healthcare administration leader who oversees the management of an ambulatory care clinic. Over the past 2 months, patient inflows have dramatically increased due to a recent shutdown of a neighboring care clinic. The patient inflows require that 90\% of all nursing staff work overtime to ensure effective healthcare delivery. However, having 90\% of the nursing workforce work overtime could be problematic in terms of patient quality and safety. Oscar would like to determine how he can best optimize his current staff holdings to ensure a balance between quality patient care and safety. For this Discussion, review and be sure to focus on the Bastian et al. (2015) article (attached). Reflect on the optimization problem mentioned in the article. By Day 3 Post a brief summary of the optimization problem presented in the Bastian et al. (2015) article (attached) in the resources for this week. Be sure to include an explanation of the objective function, as well as what the constraints actually mean. Then, explain what was done well in the article, and identify where you found shortcomings in the article. Be specific, and provide examples Minimum 500 words Articles in Advance, pp. 1–20 ISSN 0092-2102 (print) � ISSN 1526-551X (online) http://dx.doi.org/10.1287/inte.2014.0779 © 2015 INFORMS 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, [email protected] Pat McMurry AMEDD Personnel Proponency Directorate, U.S. Army Medical Department Center and School, Fort Sam Houston, Texas 78234, [email protected] Lawrence V. Fulton Center for Healthcare Innovation, Education and Research, Rawls College of Business Administration, Texas Tech University, Lubbock, Texas 79410, [email protected] Paul M. Griffin H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, [email protected] Shisheng Cui, Thor Hanson, Sharan Srinivas Department of Industrial and Manufacturing Engineering, Pennsylvania State University, University Park, Pennsylvania 16802 {[email protected], [email protected], [email protected]} 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 1 Bastian et al.: Optimizing AMEDD Workforce Planning Decisions 2 Interfaces, Articles in Advance, pp. 1–20, © 2015 INFORMS 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 Bastian et al.: Optimizing AMEDD Workforce Planning Decisions Interfaces, Articles in Advance, pp. 1–20, © 2015 INFORMS 3 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 Bastian et al.: Optimizing AMEDD Workforce Planning Decisions 4 Interfaces, Articles in Advance, pp. 1–20, © 2015 INFORMS 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}. Bastian et al.: Optimizing AMEDD Workforce Planning Decisions Interfaces, Articles in Advance, pp. 1–20, © 2015 INFORMS 5 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), Bastian et al.: Optimizing AMEDD Workforce Planning Decisions 6 Interfaces, Articles in Advance, pp. 1–20, © 2015 INFORMS 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) Bastian et al.: Optimizing AMEDD Workforce Planning Decisions Interfaces, Articles in Advance, pp. 1–20, © 2015 INFORMS 7 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, …