A Model for Efficiency-Based Resource Integration in Services

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1 University of St. Tomas, Minnesota UST Researc Online Operations and Supply Cain Management Faculty Publications Operations and Supply Cain Management 2012 A Model for Efficiency-Based Resource Integration in Services Seneeta W. Wite University of St. Tomas, Minnesota, wit6237@sttomas.edu Ralp D. Badinelli Virginia Tec, ralpb@vt.edu Follow tis and additional works at: ttp://ir.sttomas.edu/ocbopmtpub Part of te Business Administration, Management, and Operations Commons, and te Management Sciences and Quantitative Metods Commons Recommended Citation Wite, Seneeta W. and Badinelli, Ralp D., "A Model for Efficiency-Based Resource Integration in Services" (2012). Operations and Supply Cain Management Faculty Publications. 17. ttp://ir.sttomas.edu/ocbopmtpub/17 Tis Article is brougt to you for free and open access by te Operations and Supply Cain Management at UST Researc Online. It as been accepted for inclusion in Operations and Supply Cain Management Faculty Publications by an autorized administrator of UST Researc Online. For more information, please contact libroadmin@sttomas.edu.

2 European Journal of Operational Researc 217 (2012) Contents lists available at SciVerse ScienceDirect European Journal of Operational Researc journal omepage: Innovative Applications of O.R. A model for efficiency-based resource integration in services Seneeta W. Wite a,, Ralp D. Badinelli b,1 a University of St. Tomas, Operations and Supply Cain Management Department, 2115 Summit Ave, St. Paul, MN 55105, United States b Virginia Tec, Department of Business Information Tecnology (0235), 1007 Pamplin Hall, Blacksburg, VA 24061, United States article info abstract Article istory: Received 25 October 2010 Accepted 6 September 2011 Available online 16 September 2011 Keywords: OR in manpower planning Services Coproduction Dynamic programming Service processes, suc as consulting, require coordinated efforts from te service recipient (client) and te service provider in order to deliver te desired output a process known as resource integration. Client involvement directly affects te efficiency of service processes, tereby affecting capacity decisions. We present a matematical model of te resource-integration decision for a service process troug wic te client and te service provider co-produce resource outputs. Tis workforce planning model is unique because we include te extent of client involvement as a policy variable and introduce to te resource-planning model efficiency and quality performance measures, wic are functions of client involvement. Te optimization of resource planning for services produces interesting policy prescriptions due to te presence of a client-modulated efficiency function in te capacity constraint and subjective client value placed on participation in te service process. Te primary results of tis researc are optimal decision rules tat provide insigts into te optimal levels of client involvement and provider commitment in resource integration. Ó 2011 Elsevier B.V. All rigts reserved. 1. Introduction Inerent in complex, ig-contact client services, suc as consulting and business services, is a close client-provider relationsip. Tis relationsip is a determinant of te successful delivery of te service. Te service system for a consulting job, for example, typically starts wit a contract or service-level agreement between te client and te service provider, wic describes te acceptable lead time of te job, te structure of te job and payment, and te responsibilities of all parties involved (Dietric, 2006). Client resources are needed to provide information about te tecnical and business needs of te client s organization and to address any problems tat may arise during te service process. Trougout tis service system te client is an active participant. Our model captures te involvement of te client wen te client is seen as a co-producer of te service-system outputs. We assume tat, for eac client, tere is an establised agreement regarding deliverables and a delivery date for te work, and tat te service processes tat are needed to complete te job ave been identified. However, tere as been no agreement regarding te level of effort of te client. In tis paper we develop a resource-integration model for complex, ig-contact client services, and derive general forms of Corresponding autor. Tel.: +1 (651) ; fax: addresses: wit6237@sttomas.edu (S.W. Wite), ralpb@vt.edu (R.D. Badinelli). 1 Tel.: +1 (540) ; fax: +1 (540) policies for client involvement and provider involvement in a service process. We sall refer to tis model as te co-production resource integration model (CORIM). CORIM is a deterministic, resource-planning model. We explore CORIM bot teoretically and experimentally to sow te effects on policy of varying te labor constraints, process efficiency and process quality. We use dynamic programming to solve te resource integration problem. Te contributions of tis paper include: A resource-integration model tat describes te co-production of outputs by integration of resource inputs from te service provider and te service recipient. Te inclusion in te model of client involvement as a policy variable and client value as a performance measure, wic makes te resource-integration plan te result of a collaborative decision tat pursues te interests of bot te service provider and te service recipient. Te introduction of client-modulated efficiency and quality performance measures in te resource-integration model. Te exposition of tradeoffs and policy forms wic are unique to service processes due to te need to coordinate resource inputs from te service provider and te service recipient. Derivation of an optimal resource-integration policy and proof of a stationary policy form for te multi-period, resourceintegration problem. Business-to-business services and oter client-intensive services are managed across multiple deliverables for multiple clients /$ - see front matter Ó 2011 Elsevier B.V. All rigts reserved. doi: /j.ejor

3 440 S.W. Wite, R.D. Badinelli / European Journal of Operational Researc 217 (2012) over multiple time periods. Te ultimate goal of tis researc stream is te creation of model support for resource-integration planning in suc industries. Te current paper lays a foundation stone of tis model development by creating and analyzing a model for a single process stage of a service system. A sequel will extend te results presented erein to te multi-stage case. We review relevant literature in Section 2. In Section 3, we present our model for resource integration and derive basic caracteristics of te feasible region. In Section 4, we derive te form of te optimal policy troug a decomposition metod. In Section 5, we present managerial interpretations of te optimal resourceintegration policies. Section 6 discusses te contribution of te researc and future researc directions. 2. Related literature Te current study extends literature on resource planning of service systems, resource flexibility, and efficiency-based resource planning. Resource planning is a sequential decision process tat strives to apply an organization s capacity most efficiently to meet demand. Te seminal work of Holt, Modigliani, Mut, and Simon (HMMS) (Holt et al., 1955, 1956) laid te foundation for planning and control models. Gaimon and Tompson (1984), Anderson (2001), Anderson et al. (2006), and Bordoloi and Matsuo (2001) extend te HMMS model to include multi-sourced employee capacity in resource-planning models for services. Altoug te aforementioned models capture varying resource capabilities, we treat te variations in client capabilities more robustly by including in our model efficiency and quality measures wic are functions of client involvement in te service system. Troug tese functions we can explicitly capture te nature of co-production of output resources and co-creation of value. Martin et al. (2001) argues tat a measurement of productivity tat does not capture te client side of te service encounter is inadequate wen talking about a business service suc as consulting. A natural extension to planning models wit resources from multiple sources is allowance of resource flexibility. Wit flexible resources, processes can become more efficient. A few well-known papers on workforce agility in a service process are Abernaty et al. (1973), Campbell (1999), Brusco and Jons (1998), Van Oyen et al. (2001), Hopp and Van Oyen (2004), and Hopp et al. (2004). Altoug tese papers sow tat flexible resources can be used to accomplis differing tasks, tey fail to capture te sared value component of coproduced services. Roels et al. (2010) and Xue and Field (2008) capture service coproduction in wic te service recipients make coices about te engagement of various service providers as te recipient s self interests dictate. By contrast, te context for te model in te current paper is tat of a contracted engagement between te service provider and te client suc as tose in te service industries of consulting, IT development and oter knowledge-based, co-creation services. Generally, once te client commits to te service, te service provider is responsible for managing te service project, wic includes planning te extent and scope of client involvement. In making tis project plan, te provider specifies te resource integration tat will lead to successful service delivery. Data Envelopment Analysis (DEA), a metodology tat was designed for measuring te relative efficiency of decision making units witout assumptions are made about te underlying transformation process as been used for efficiency-based resource allocation. Atanassopoulos (1995) and Beasley (2003) model a form of a transformation function by using targets on inputs and outputs wile Golany and Tamir (1995) and Koronen and Syrjänen (2004) use proportional bounds and scaling. In Atanassopoulos (1998) and Golany et al. (2006), owever, an explicit linear transformation function is defined. Altoug DEA can be used successfully for capturing te effects multiple inputs and multiple outputs on planning and control decisions, it falls sort in capturing te nonlinearities tat exist in service firms and te need for information regarding te transformation function. Te model presented in tis paper captures te nonlinearity of service processes troug te clientmodulated efficiency and quality functions. To our knowledge, our work is one of te first initiatives to create a matematical model of client involvement in a service system using explicit efficiency and quality functions. Te efficiency and quality functions in tis paper are constructed along te lines of te production-function attributes used in Gaimon (1997), Carrillo and Gaimon (2004), and Napoleon and Gaimon (2004). Unlike tese workforce planning models for services, our efficiency function modulates a capacity constraint in a planning model, wic also includes demand, inventory and backlog of a service. Tese features produce a model tat is more realistic and relevant to te management of business services. 3. Co-production resource integration model Key components of te model are measures of efficiency and quality of te service process. Tese measures are functions of te level of client skill and of client intensity. We define client intensity as te ratio of te client s labor input to te service system to te provider s labor input to te service system. Literature reveals additional insigts into te effects of client intensity on efficiency and quality. Case (1978) determined tat at some level, furter client involvement is eiter ineffective or detrimental. Terefore, tere exists a level of client intensity at wic efficiency reaces, peraps asymptotically, its maximum or saturation level. Consequently, te efficiency and quality functions of our model eac strictly increase to a maximum value of 1.0 as a function of client intensity. Gaimon (1997) and Napoleon and Gaimon (2004) posit a set of general, reasonable assumptions about te functional dependencies of efficiency and quality. We augment tis structure troug te introduction of leverage tat is provided by client participation in te service process. Below, we summarize te assumed caracteristics of tese functions. Efficiency and quality are strictly increasing and concave in client intensity. Efficiency and quality are bounded above by a saturation level. Efficiency and quality are strictly increasing and concave in provider skill and in client skill. Efficiency and quality are strictly increasing in te quality of te oter resource inputs. Tese assumptions determine caracteristics of te sape of te efficiency and quality functions. See Fig. 1 for an example. In te current paper we are concerned wit te dependency of te efficiency and quality functions on client intensity. We define te deliverables of a service as service components. In te case of a software consulting firm, for example, service components could be database designs, web-page construction, code writing and testing, etc. We assume tat te service provider delivers a service troug a service system consisting of a network of linked processes eac process delivering one type of service component. Eac process of te network requires a certain number of cycles per unit of te service component tat is delivered to te client. Te cycle of eac process as a standard labor requirement. A key element of te provider s resource integration decision is te provider s assessment of te client s costs of involvement and

4 S.W. Wite, R.D. Badinelli / European Journal of Operational Researc 217 (2012) Efficiency te value to te client of involvement. Certainly, te time commitment of te client must be recognized as a marginally increasing penalty, as tis commitment increasingly interferes wit oter client resource needs. However, it is well known tat a client s involvement in service processes yields benefits in te form of stronger acceptance and ownersip of te service deliverables, willingness to embrace canges in beavior tat te service deliverables require and perceived value of te service. Te modeling of te client s value function is complex and te subject of muc current researc, placing tis modeling beyond te scope of te current paper (Heitz et al. (2009), Tang (2009) and Strong et al. (2009)). We only impose te modest assumption tat te client s value of involvement in te production of service is marginally decreasing. Tese penalties and benefits taken togeter can be viewed as a cost function on te client s involvement in te service system. Te provider must assess tis cost function in order to plan resource integration a callenging task tat is familiar to every consulting firm and software development team. Te development of assessment models is beyond te scope of te current paper. However, te relevance of tis assessment to resource planning and te impact of assessment accuracy are demonstrated by te model tat is presented erein. In consideration of te common practice of assigning service workers to multiple projects, implying part-time assignment to any given stage of a given service, we measure te provider s workforce in terms of te continuous measure of full-time equivalents (FTE) instead of te discrete measure of number of personnel. Similarly, labor-assignment variables are measured in terms of FTE. Te model development presented below is tat of te resource-integration plan for a single service process. We establis a stationary form for te optimal policy of te multi-period, resource-integration problem troug te modeling of a given time period as a stage of a dynamic program. All of te decision variables, performance measures and parameters are defined for te current time period for tis process. In order to simplify notation we suppress te subscripts tat would explicitly identify te time period and process Definitions and notation g Decision Variables w f y Efficiency as a function of client intensity Client Intensity Fig. 1. Example of an efficiency function. generation of process cycles of te service process (# of completed cycles) number of provider FTEs assigned to te service process number of provider FTEs ired for te service process number of provider FTEs fired from te service process client intensity = te ratio of client time to provider time spent on te service process We use te terms ire and fire to represent te assignment of personnel to a service process and te removal of personnel from a process, respectively. Tis assignment and removal can take many forms across different contexts of service provision. In some cases, tese actions can literally be in te form of iring and firing or laying-off. In oter contexts, te iring is in te form of assigning a portion of a service-worker s time to a service process and firing is in te form of re-assigning tis worker. Many different words can be used to indicate tese labor decisions, eac one relevant to a certain context. We settle te matter by coosing te terms ire and fire in a generic sense. i b State Variables inventory of te service component at te end of te current period = te number of units of te service component tat are completed earlier tan needed backlog of te service component at te end of te current period = te number of units of te service component tat are overdue Note tat inventory and backlog are real-valued to reflect te possibility of partial completion of a service component. Performance Measures e(y) efficiency of te service process as a function of client intensity q(y) quality of te service process as a function client intensity c c (yg) cost of client involvement in te service process = net effect of penalties and benefits of te time spent by te client, yg, in te service process c b (b) te loss of value of te service process due to delayed deliverables V(n, w) present value of te optimal plan after te current period if te current period ends wit net inventory, n = i b, and workforce,w z total profit of te resource integration plan for te service process over te current and future periods As we stated earlier, we assume tat te efficiency and quality measures are strictly increasing, concave functions of y and tat c c is convex. Te backlog variable measures te number of units of te service component tat are not completed by teir due dates. We impose a cost on tis backlog, c b, in order to capture te marginally increasing losses of client perceived value as well as postponed revenue to te service provider. Hence, we assume tat c b is convex and increasing. Backlog also increases future labor costs, wic are captured by te function, V. Tere is no explicit inventory olding cost. If service deliverables are completed prior to or later tan te times tat tey are needed, ten te cost of generating tese deliverables is recognized in te time periods of generation. Hence, early or tardy generation will affect net present value. We include in te objective function te optimal discounted future profit to capture te effects of te decision in te current period on future optimal performance. Once te costs of generation in te current period are accounted for, iger inventory or lower backlog at te end of te period as a positive marginal net present value, as it can reduce future labor cost and, potentially, future backlog costs. Tere is a limit to te

5 442 S.W. Wite, R.D. Badinelli / European Journal of Operational Researc 217 (2012) benefit of inventory wen te inventory level reaces tat wic covers all future demand. Define, n max ¼ te inventory level tat covers all future demand; g max ¼ rðn max þ d i 0 þ b 0 Þ ¼ te generation tat yields a net inventory level of: From tese considerations, we assume tat V is differentiable and non-decreasing in n < n max. We assume te V is at least quasi-concave in w, imposing a reasonable and weak condition on te tradeoffs for te existence of an optimal work-force. Parameters c c f c w v d r r a w a c w b y y q cost of iring an FTE of te service provider for te service process cost of firing an FTE of te service provider from te service process cost of wages per FTE-period of te service provider revenue per unit of te service component demand for te service component (units of te service component) required number of cycles of te service process per unit of te service component number of standard labor ours required per cycle of te service process available provider capacity (labor ours/fte-period) available client capacity for te service (labor ours/ period) maximum provider workforce level (FTEs) maximum allowed backlog of te service process (# units of te service component) minimum required client intensity maximum allowable client intensity minimum required quality level Note: r gy = number of ours allocated to te service by te client. Te resource integration problem can be stated as follows: Problem P. Maximize w;;f ;i;b;y;g zðw; ; f ; i; b; y; gþ ¼dv c c f f c w w c c ðygþ c b ðbþþvði b; wþ; Subject to : w w 0 þ f ¼ 0; ð1þ w w P 0; ð2þ b b P 0; ð3þ a w weðyþ r g P 0; ð4þ a c r yg P 0; ð5þ y y P 0; ð6þ y y P 0; ð7þ qðyþ q P 0; ð8þ i 0 b 0 þ g=r d i þ b ¼ 0; ð9þ g max g P 0; ð10þ w; ; f ; g; y; i; b P 0: Te objective function maximizes te net present value of all current and future resource commitments, inventory, backlog and revenue. In te spirit of co-production we view tis optimization as a joint venture between te provider and te client. Consequently, te client s costs and te provider s costs are combined in te objective function, and te functions, c c and c b, reflect client values. Constraint (1) is a workforce balance constraint. Constraint (2) ensures tat te maximum workforce level tat te provider can accommodate is not exceeded. Constraint (3) ensures tat te number of service jobs tardy (backordered) does not exceed te maximum level tat is allowed by te service level agreement. Constraints (4) and (5) are te provider and client capacity constraints, respectively. Te efficiency term in te capacity constraint (4) is needed to represent te effect of client involvement on te effective capacity of te workforce. Note tat te boundaries of eac of Constraints (4) and (5) define generation as a function of client intensity. We denote tese two functions as g 4 (y) and g 5 (y), respectively, and te inverses of tese functions as y 4 (g) and y 5 (g), respectively. In Constraints (6) and (7) we ave set minimum and maximum amounts of client involvement, respectively. We assume tat in business services suc as consulting and IT development te client will always be part te service process to a certain extent, and tat te client cannot be te sole labor source in any process. Typically, organizations bencmark temselves against competitors in terms of quality and establis internal quality standards. Constraint (8) imposes a minimum process quality level tat must be acieved. Constraint (9) is te conventional balance equation tat establises te functional dependencies among inventory, backlog, demand and generation. Constraint (10) sets an upper bound on generation tat is based on n max. Problem P can be solved efficiently wit standard non-linear programming metods. However, our purpose in tis paper is to reveal te form of te optimal policy. Consequently, we proceed wit a detailed analysis of Problem P and its optimization Conditions In order to ensure non-trivial solutions and to establis tigter bounds on feasible solutions, we derive several conditions on te parameters of te problem. Condition 1. Constraints (3) and (9) set a lower limit on generation. Define, g min ¼ rðd i 0 þ b 0 bþ: We assume tat demand is ig enoug to require g min >0. Constraint (3) can be replaced wit, g g min P 0: ð11þ Condition 2. Since q(y) is strictly increasing and bounded above by 1, Constraint (8) can be re-written as a lower bound on te client intensity. Tere is a unique value of y for wic q(y)=q. We sall denote tis value, y q.ifq(0) > q, ten we set y q = 0. Te igest level of client intensity tat is allowed, y, must be able to provide te minimal level of quality. Terefore, y q < y. Define, y min ¼ maxðy; y q Þ: Constraints (6) and (8) can be replaced wit te single constraint, y y min P 0: ð12þ Condition 3. Constraint (5) must provide enoug capacity to acieve te minimal required generation. Tis requirement imposes an upper bound on te client intensity. Define,

6 S.W. Wite, R.D. Badinelli / European Journal of Operational Researc 217 (2012) y 5 ðgþ ¼ ac r g ; y max ¼ minðy; y 5 ðg min ÞÞ: y 5 (g) = unique value of client intensity tat is specified by te boundary of Constraint (5) for any given generation level, g > 0. Terefore, y min 6 y 6 y 5 (g min ). Constraint (7) can be replaced wit te tigter constraint, y max y P 0: ð13þ Condition 4. Constraints (4), (11) and (13) place a lower bound on te provider s workforce level, wic we denote, w. Tis condition follows from te facts tat, e(y max ) is te igest level of efficiency obtainable and g min is te smallest number of cycles tat can be generated. Define, r g min w ¼ a w eðy max Þ ; w w P 0: ð14þ Condition 5. Constraints (4), (11) and (12) specify a workforce level below wic Constraint (4) cannot provide enoug capacity for te minimum required generation witout requiring more tan te minimum level of client intensity. We denote tis workforce level, w 0. r g min w 0 ¼ a w eðy min Þ : Condition 6. Constraints (4), (5), (10) and (12) specify a workforce level, wic we denote, w 00, above wic Constraint (4) is not binding. Constraints (4), (5) and (10) place upper bounds on te generation. Since te bound on generation tat is imposed by Constraint (4) is increasing in y and te bound on generation tat is imposed by Constraint (5) is decreasing in y,ifg 5 (y min )<g 4 (y min )or g max < g 4 (y min ) ten Constraint (4) is redundant for all y, y min 6 y 6 y max. Define, w 00 ¼ r minðg max ; g 5 ðy min ÞÞ : a w eðy min Þ Condition 7. Te marginal value of te workforce to te optimal future resource-integration plan, V, is bounded by te costs of iring and firing because, for any workforce level, w, tis state could be acieved from a workforce level of w 1 and iring one FTE or from a workforce level of w + 1 and firing one FTE. Terefore, c 6 c : Condition 8. We can eliminate te variables, and i b, and te equality Constraints (1) and (9) by substituting for tese variables. ¼ w w 0 þ f : Te constraint, P 0 becomes, w w 0 þ f P 0; i b ¼ i 0 b 0 þ g=r d: We define c b ( i 0 + b 0 g/r + d) = 0 for i 0 + b 0 g/r + d <0 ð15þ 4. Dynamic program Problem P is expressed as a multi-period dynamic program. We now decompose tis problem furter by separating te iring/firing/workforce decision variables from te rest of te decision variables. Te current period s problem is ten a two-stage dynamic program. Stage 1 is te optimization of te generation and client plan for a given level of te provider s workforce Problem P1, below. Stage 2 is te optimization of te combined workforce and generation-client plans Problem P2 below. Te state variable tat connects te Stage 2 to Stage 1 is te workforce level, w. Applying Condition 8 to tis two-stage formulation we define, z 1 ðy; g; w; i 0 b 0 Þ¼ c c ðygþ c b ð i 0 þ b 0 g=r þ dþ þ Vði 0 b 0 þ g=r d; wþ; z 2 ðw; f ; w 0 ; i 0 b 0 Þ¼z 1 ðw; i 0 b 0 Þ ðc f þ c Þf ðc w þ c Þw þ c w 0 þ dv: Problem P1 max y;g Subject to : Problem P2 max w;f z 1 ðy; g; w; i 0 b 0 Þ ð4þ; ð5þ; ð10þ; ð11þ; ð12þ; ð13þ; z 2 ðw; f ; i 0 b 0 ; w 0 Þ Subject to : ð2þ; ð14þ; ð15þ and f P 0: 4.1. Optimality conditions for Problem P1 ð16þ First, we derive te solution of Problem P1 for te optimal production plan, given a workforce, w, and initial net inventory, i 0 b 0. Once tis solution is obtained, we can solve Problem P2 to find te combination of te optimal provider workforce level and its associated generation-client plan. We now derive some basic expressions tat are related to te optimality condition of Problem P1. Define, X(w,i 0 b 0 ) = feasible region of Problem P1 Te gradient of te objective function for Problem P1 is, rz T 1 ¼ c0 c ðygþg; c0 c ðygþy þ 1 r c0 b ð i 0 þ b 0 g=r þ dþþ 1 r Te Lagrangian for Problem P1 is, L 1 ¼ z 1 þ k 4 ða w weðyþ r gþþk 5 ða c r ygþþk 10 ðg max gþ þ k 11 ðg g min Þþk 12 ðy y min Þþk 13 ðy : ð17þ Te necessary Karus Kun Tucker (KKT) optimality conditions (see Winston, 1990) include te following two, first-order conditions, c 0 c ðygþg þ k 4a w w de dy k 5r g þ k 12 k 13 ¼ 0; c 0 c ðygþy þ 1 r c0 b ð i 0 þ b 0 g=r þ dþþ 1 r k 10 þ k 11 k 4r k 5 r y ð18þ ð19þ

7 444 S.W. Wite, R.D. Badinelli / European Journal of Operational Researc 217 (2012) Fig. 2 sows an example of te constraints of Problem P1, wic will aid te reader in understanding te conditions and propositions tat follow. Te client value function, c c (yg), plays a critical role in determining te optimal policy. From conditions (18) and (19) we can see tat te optimal solution to Problem P1 depends significantly on te sign of c 0 cðygþ. Accordingly, we recognize te importance of te value of te argument of c c at wic te client value function reaces a maximum, wic we define te maximum client value point (MCVP). We define te maximum client value frontier (MCVF) as te locus of points in te y g plane tat satisfy te condition, c 0 c ¼ 0. Te MCVF is a yperbola tat is defined by, yg ¼ MCVP: ð20þ We note tat condition (20) is independent of all parameters oter tan te parameters of te client s cost and benefit functions and g max. For g > g max, te client would not experience any benefit from generation. Terefore, tere is an upper bound on MCVP, specifically, MCVP 6 y max g max : ð21þ We also note tat te yperbola defined by (20) is eiter coincident wit or non-intersecting wit te client capacity constraint (5). Te example of Fig. 2 sows tree fundamentally different cases of te MCVF. Tese cases lead to different forms of te optimal solution, wic are establised by te propositions below. We note tat te position of te MCVF relative to te feasible region is determined only by te client s utility function and is independent of all of te parameters of te problem. Terefore, a robust optimization of co-creative resource integration necessitates solutions for all possible client utility functions. Te propositions below determine te form of te solution for eac possible case. We can now prove several propositions wic lead to a specification of te optimal solution to Problem P1. Proposition 1 establises an important monotonicity property of te objective function, z 1, along te MCVF and te capacity constraint, (5). Proposition 1. Along any arc, yg = constant, z 1 is non-decreasing in g and non-increasing in y Proof. Te proof is in te Supplemental document. In case 1 of te example of Fig. 2, Proposition 1 implies tat te optimal solution lies on te left-most feasible point of te Constraint (5) if tis constraint is binding. Corollary 1. If Constraint (5) is binding at optimality, ten an optimal solution is located at te point on tis boundary of te minimal feasible value of y and te maximum feasible value of g. If te maximum client value is attained at iger intensities and generations tan are feasible, tat is, te MCVF lies above te feasible region, ten te optimal solution to Problem P1 is a point on te boundary of te feasible region tat bounds client intensity and generation from above. Refer to case 2 of Fig. 2. Te only constraints tat place an upper bound on te client intensity are Constraints (5) and (13). Along Constraint (5), by Proposition 1, te gradient favors iger generation. Along Constraint (13), te gradient also favors iger generation. Terefore, te optimal solution is found on eiter of tese constraints at te igest feasible generation and client capacity is maximally utilized. We formally establis tis fact in Proposition 2. Proposition 2. If, for all feasible solutions, yg 6 MCVP ten g P {gj(y,g) 2 X(w,i 0 b 0 )} and Constraints (5) or (13) binding. g max g Constraint #12 g min Constraint #4 y min Constraint #10 Constraint #5 MCVF Case 3 Proof. Te proof is in te Supplemental document. If te maximum client value is attained at lower intensities and generations tan are feasible, tat is, te MCVF lies below te feasible region ten te optimal solution to Problem P1 is a point on te boundary of te feasible region tat provides te minimum feasible client intensity. Refer to case 3 in Fig. 2. In tis case, te only constraints tat place a lower bound on te client intensity are Constraints (4) and (12) and te provider s capacity is maximally utilized. Proposition 3 establises te fact tat tese constraints tat are binding in tese two cases. Proposition 3. If, for all feasible solutions, yg P MCVP ten y 6 {yj(y,g) 2 X (w,i 0 b 0 } and Constraints (4) or (12) are binding. Proof. Te proof is in te Supplemental document. If te MCVF intersects te feasible region, ten, by Proposition 1, te optimal solution to Problem P1 is a point on or above tis line tat provides minimal client intensity and maximum generation. Refer to case 1 of Fig. 2. Proposition 4. If yg = MCVP for some (y,g) 2 X(w,i 0 b 0 ) ten y g P MCVP and Constraints (4), (10) or (12) are binding. Proof. Te proof is in te Supplemental document optimality conditions for Problem P2 Analysis of te optimality conditions for Problem P2 leads to te proof tat te optimal value of te objective function, z 2, is nondecreasing in te state variable, i 0 b 0, and quasi-concave in te state variable, w 0. Tis fact is instrumental in establising a stationary policy for te multi-period problem. We begin te analysis wit te beavior of te optimal value of te objective function to Problem P1, z 1 ðw; i 0 b 0 Þ, wit respect to i 0 b 0. Proposition 5. z 1 ðw; i 0 b 0 Þ is non-decreasing in i 0 b 0 < n max Proof. Te proof is in te Supplemental document. Proposition 5 allows us to determine te beavior of te optimal solution to Problem P2, z 2, wit respect to i 0 b 0. y max Fig. 2. Te feasible region for P1. Proposition 6. z 2 is non-decreasing in i 0 b 0 < n max MCVF Case 2 Constraint #13 MCVF Case 1 Constraint #11 y

8 S.W. Wite, R.D. Badinelli / European Journal of Operational Researc 217 (2012) Proof. Te proof is in te Supplemental document. From te necessary, first-order KKT conditions for Problem P2 we can prove tat te optimal value of te objective function of te single-period problem is quasi-concave in te initial work force level. Te lagrangian (Winston, 1990) for Problem P2 is, L 2 ¼ z 1 ðw; i 0 b 0 Þ ðc w þ c Þw ðc þ c f Þf þ c w 0 þ k 2 ð w wþ þ k 14 ðw wþþk 15 ðw þ f w 0 Þþk 16 f ¼ 0 ¼ c w þ c þ k 2 k 14 k 15 ; ð22þ 2 ¼ 0 ) k 15 þ k 16 ¼ c þ c f : Table 1 Experimental parameter data. Hiring cost (c ) 10,000 Provider capacity (a w ) 160 Firing cost (c f ) 5000 Client capacity (a c ) 160 Cost of wages (c w ) 8000 Max provider 30 workforce ð wþ Revenue (v) 20,000 Min client intensity 20% (y) Demand (d) 40 Max client intensity 90% ðyþ Required # of cycles (r) 1 Min quality (q) 70% # of standard labor ours/ cycle (r ) 80 Max backlog ðbþ 10% of demand Proposition 7. z 2 is quasi-concave in w 0 Proof. Te proof is in te Supplemental document Multi-period case A major result of tis paper is te establisment of a stationary policy for te multi-period resource-integration problem. Proposition 8 provides useful bounds on te sensitivity of te optimal value of te objective function of Problem P2 to te initial workforce level. Proposition 8. c f 0 6 c : Proof. Te proof is in te Supplemental document. Proposition 9. For every time period, V(n,w) is increasing in n and quasi-concave in w; c 6 c for every time period. Proof. Te proof is in te Supplemental document. Proposition 9 implies tat Propositions 1 8 old for every period, te form of te optimal policy as specified by tese Propositions is stationary and te policy depends only on te state variables, w 0, i 0 b Managerial interpretations For any given workforce, a service firm finds itself in one of tree cases, identified in eac of Propositions 2 4. In setting a policy for client intensity and service generation level, a trade-off between client costs and net inventory costs is made. Wat distinguises te tree cases is te nature of tis trade-off, wic we now explain Tradeoffs Tradeoffs of te resource-integration problem are non-linear and more complex tan tose of a manufacturing resource plan. We illustrate tis penomenon wit an example, using te data sown in Table 1. Fig. 3 illustrates te non-linear tradeoff between client slack capacity and provider slack capacity as functions of client intensity for tis example. Te MCVP specifies levels of client intensity and generation at wic te trade-off for te client between te cost of involvement and te benefits of involvement are optimal. Te tradeoff for te Slack Hours Client Intensity Fig. 3. Tradeoff Illustration. Provider Slack Capacity Client Slack Capacity service provider augments te client s tradeoff wit te value of inventory and te cost of te provider workforce. If te optimal tradeoff prescribes a level of client involvement tat is above te MCVP, ten, in order to support a more overall efficient delivery of te service, te service provider must ask te client to commit more time and resources to te service process tan is in te immediate interests of te client. As educators, consultants and medical-care specialists know, tis argument is usually callenging to make. However, te alternatives are sub-optimal policies tat consume excessive provider resources. Te service provider is well-advised to be aware of te position of te intensity-generation plan relative to te MCVP in order to prepare for te possibility of client dis-satisfaction. Te gradient of z 1 can favor lower client intensity, wic implies tat quality is sacrificed for cost. Tis penomenon is not surprising for a model tat places a bound on quality wile carging a variable cost for client involvement in te service process. Te optimal policy reduces client involvement until eiter Constraint (4) or Constraint (12) is binding, as tese are te only constraints tat impose a lower bound on client intensity. Terefore, te resource integrators coose to reduce client involvement in te service process until te minimum allowable quality level is reaced or until provider capacity is sufficient to maintain desired generation levels. An interesting speculation is tat te optimal resource integration policy would cange in favor of iger client intensity if resource integrators coose to evaluate quality on a ratio scale as opposed to a categorical scale, as a constraint on te quality level (8) implies. In oter words, a Taguci approac to service quality would motivate iger levels of client involvement in te service process. Genici Taguci s metodology, widely adopted in manufacturing quality control, views any deviation from quality targets as a financial loss function tat is increasing and continuous in te magnitude of te deviation (Evans and Lindsay, 2011).

9 446 S.W. Wite, R.D. Badinelli / European Journal of Operational Researc 217 (2012) Load leveling If c w c f w¼w0 < c w þ c ten, by condition (22), te optimal solution is, w = w 0, and te provider sould maintain te current workforce. Tis condition is intuitively consistent wit all workforce planning policies. Tat is, te stability of te optimal workforce plan increases wit te iring and firing cost parameters. However, in addition to te conventional motive for not canging te workforce level, diminising marginal value of client intensity due to te concavity of te efficiency function motivates keeping client intensity down, even at te expense of backlog or lower inventory. It pays to produce less and defer generation to a period wen te client s involvement is leveraged better because te generation can take place at a lower level of client intensity. Hence, a load-leveling resource plan is motivated not only by iring/firing cost, but also by te diminising effectiveness of client intensity Sensitivity analysis Sensitivity analysis is essential in te interpretation of te model s results because te model is based on a deterministic representation of te efficiency, quality and client-value functions. Altoug it is reasonable to assume tat oter parameters of te model may be estimated wit an accuracy tat approaces determinism, te same cannot be said of te parameters of tese tree functions. For most services, te natures of service efficiency and quality are poorly understood and client value is even more of a mystery. Terefore, we sould examine te potential effects of errors in estimating tese functions. Errors in estimating te quality function will affect te position of Constraint (12). Very simply, if te quality function is over-estimated, ten Constraint (12) will allow levels of client intensity tat will not support minimum required levels of quality. If te quality function is under-estimated, ten Constraint (12) will require a minimum level of client intensity tat exceeds te level tat can support te minimum required quality. Considering tat most service clients would prefer to err on te side of iger quality, we can advise all users of te model to under-estimate te quality function wen tere is any doubt about its true form. Te estimation of te efficiency function presents more callenging problems. Te effect of te efficiency function on a prescription for a resource-integration policy is seen troug te position and sape of Constraint (4). Te efficiency function influences te determination of te point on Constraint (4) were an optimal solution is found. We can deduce certain properties of te sensitivities to misestimations in te efficiency function in terms of te comparison of te policy tat te mis-estimated model would recommend, (y r,g r ), to te true optimal policy, (y,g ). If te efficiency function is over-estimated (under-estimated), ten te solution will be set at y r < y, g r > g (y r > y,g r < g ). In te case of an over-estimated efficiency, te policy prescription attempts to generate more service and engages less client time tan is optimal. Of course, te actual service tat is generated will be less tan expected, and, given te less-tanoptimal client intensity, te actual generation will even be less tan te optimal generation. Te effects of tese errors are a lower inventory or a iger backlog level and a iger client expense tan are warranted or expected by te resource planner. Higer client costs will be incurred in a future period wen te generation sortage is recognized. Tere could also be a surprising backlog cost or even a violation of te backlog constraint in tis case. Percent Loss True MCVP= In te case of an under-estimated efficiency, te policy prescription generates less service and engages more client time tan is optimal. Te effects of tese errors are a lower inventory or iger backlog level and a iger client expense tan are warranted. However, tere are no surprises in terms of differences between actual output and planned output. We conclude tat resource integrators sould generally err on te side of under-estimating efficiency. Finally, we recognize te crucial role of estimating te client cost function, c c, in setting a resource-integration policy. Recall tat tis function incorporates te client s perceived value of te service, an individualized and context-sensitive function. Te many issues involved in tis estimation and te myriad potential metodologies for performing te estimation are beyond te scope of tis researc. However, te model presented erein serves to identify tis function and to establis clearly its role in resource-integration planning. Fig. 4 sows an example of te comparison of actual performance of te solution to te resource-integration problem to te optimal performance as a result of mis-estimating te MCVP. Te grap sows te percent loss in te objective function due to mis-estimation. 6. Conclusions and future researc In tis paper, we ave developed a resource-integration model tat will serve as a foundation for future service-science researc. Our model is unique troug its incorporation of te client as a resource in te generation plan. We also capture te effects of client intensity on efficiency and quality in our model. Te policy recommendations of te model give service enterprises valuable information regarding teir iring and firing policies, level of client intensity, and service generation. We preserve te notion of inventory in resource planning troug te use of discounted cas flows and deferred revenues. Te generality of te results are due to te generally applicable assumptions about te efficiency and quality functions. Terefore, te model is robust wit respect to its breadt of applications as well as wit respect to te teoretical insigts we ave derived. A fruitful and exciting direction for future researc is te callenge of estimating te client value function, te efficiency function and te quality function. A furter extension of tis model is te incorporation of client learning and te potential of te service provider to influence canges in te client s value function over time. Appendix A. Supplementary data MCVP Estimate Fig. 4. Sensitivity of optimal objective function to MCVP. Supplementary data associated wit tis article can be found, in te online version, at doi: /j.ejor

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