Title: Stochastic models of resource allocation for services

Transcription

1 Title: Stochastic models of resource allocation for services Author: Ralh Badinelli,Professor, Virginia Tech, Deartment of BIT (235), Virginia Tech, Blacksburg VA 2461, USA, Phone : (54) , Fax : (54) , Web age: htt:// Submitted to the EIASM 29 NAPLES FORUM ON SERVICES i

2 Author bio sketch: Ralh Badinelli is a rofessor in the Deartment of Business Information Technology of the Pamlin College of Business of Virginia Tech. He received Ph.D. and M.S. degrees in Management from the Krannert Graduate School of Business of Purdue University, an M.S. degree in Physics from Purdue University, and a B.S. degree in Mathematics and Physics from Hofstra University. Prior to ursuing an academic career, Dr. Badinelli worked as Production Planning Suervisor for the otics division of a west-coast laser manufacturer. He has consulted with a number of comanies on lanning, forecasting, and software develoment for decision suort. Dr. Badinelli s teaching resonsibilities are in the areas of oerations management and quantitative methods for business. These resonsibilities extend over undergraduate, MBA and doctoral levels. He has taught many executive-develoment seminars on the subjects of roject management, quality imrovement, roduction and inventory control, and business strategy. He has received the R. B. Pamlin College of Business Certificate of Teaching Excellence. Dr. Badinelli has ublished refereed articles across a wide range of toics including inventory control olicies, revenue management, robability estimation, decision analysis and otimization. His ublications have aeared in Oerations Research, Management Science, Decision Sciences, Naval Research Logistics, Comuters and Oerations Research, International Journal of Production Research, Euroean Journal of Oerational Research, and the Journal of the International Academy of Hositality Research. He is a member of the Institute for Oerations Research and Management Science (INFORMS), Project Management Institute (PMI) and a certified member of the American Production and Inventory Control Society (APICS). He is the Vice- President/President-Elect of the INFORMS Service Science Section and a member of the SRII Leadershi Team. ii

3 Abstract: Purose: In this aer we develo a resource allocation model with general forms of service roduction functions, which describe the relationshi between inuts and oututs of a rocess of co-creation of value by a service rovider and a service reciient. The model develoment is directed at roviding useful olicy rescrition for service roviders and a foundation for research into the nature of resource allocation olicies in service industries. Design/methodology/aroach: The model develoment makes use of concets of robability theory, otimization theory and extant DEA models. Findings: A ractical otimization for allocating resources to service rocesses as well as insights into the comlexity of service resource management are obtained. Research limitations/imlications: The model resented in this aer is based on constant returns to scale of the service rocess. Originality/value: To date, service science lacks models for resource management that aroach the usefulness of resource-management models for manufacturing enterrises even though the service economy in the industrialized world is larger than the manufacturing economy. This aer initiates a stream of model-building research. Research Paer Keywords: resource allocation, disatching, service rocess iii

4 Stochastic models of resource allocation for services 1. Introduction This research aer resents the develoment of a mathematical model for allocating and disatching resources in a service enterrise. The starting oint for this model is the definition of a service rocess. We define a service rocess as a coordinated set of activities which transforms a set of tangible and intangible resources (inuts), which include the contributions from the service reciient, into another set of tangible and intangible resources (oututs). Fitzsimmons (1985), Bettencourt et al. (22), Lance et al. (22), Samson (27) and others have firmly established the defining characteristic of service rocesses as the co-roduction of service oututs through the joint effort by the rovider and reciient of the service. Tang & Zhou (29) have refined this concet by emhasizing the coordination of these joint efforts, introducing the word, taktchronicity, to indicate the requirement of choreograhed effort among multile articiants in the service rocess. Examles of the kinds of service enterrises that motivate this research are software develoment, consulting, education and roject management. Resource allocation is the managerial function of making available levels of caacity that can suort lanned oerations. Resource disatching is the managerial function of assigning resources to articular rocesses. In the case of service rocesses, the resources are rovided by both the service rovider and the service reciient. Furthermore, a tyical service rocess requires several different tyes of inuts, such as labor, material, information, equiment, and roduces several tyes of outut, such as money, information and software. The relationshi of the oututs of a rocess to the inuts is secified by a function that we call the service technology function. Unlike manufacturing rocesses, service rocesses are generally not well understood by either the service rovider or service reciient. In articular, high value-adding services, such as those that motivate this research, are rarely understood in enough detail to allow the ublication of a rocess sheet that describes detailed stes in the rocess along with accurate resource requirements and cycle times for each ste as one would find in a manufacturing environment. There are several sources of uncertainty for a tyical service rocess. Among these are, Uncertainty of client commitment Uncertainty of client quality Uncertainty of the knowledge of the service rocess estimation and secification of usage and yield rates Uncertainty of recognized uncontrollable factors which may cause changes to usage and yield rates The thrust of this research is the develoment of mathematical decision models for resource management in service enterrises. The long-term goal of this stream of research, which is consistent with the exhortations in Chase and. Garvin (1989), Fitzsimmons and Fitzsimmons (24), Sohrer et al. (27) and Machuca et al. (27), 1

5 is the accomlishment of model-based decision suort for service suly chains that is at least as sohisticated as that which is available to manufacturing enterrise. The contribution of the research resented herein is three-fold: 1. A modeling framework for service rocesses that can serve as a foundation for further model develoment 2. A useful otimization model for resource allocation and disatch 3. Some basic guidelines for otimal resource allocation/disatching, for client involvement and adatation of resource management to rocess learning Resource-management modeling in service industries is in its infancy. Most of the effort in this field has been devoted to data enveloment analysis (DEA). DEA rovides a tyically macroscoic view of service rocesses and focuses on estimating and comaring the economic efficiency of services. See Charnes et. al. (1994), Fare and Grosskof (2) and Golany et. al.(26) for ersective and overview of DEA. For the resource manager, a microscoic view is needed with the aim of determining the otimal assignment of resources to rocesses given their existing efficiencies. Few references are available for sho-floor resource management models. Korhonen and Syrjanen (24) utilize a DEA model of service efficiency in order to determine directions for changing the resource allocations to the service in ursuit of multile objectives. Their model is oriented towards a macro re-distribution of resources to an ongoing service enterrise. Gaimon (1997) takes a more rocess-level aroach to setting workforce levels overtime for IT and knowledge workers. White and Badinelli (29) extend this work to a workforce lanning model in which client involvement at its effect on quality and efficiency are exlicitly reresented. All of these resource-lanning models are deterministic. In the current aer we model a single-stage service rocess that requires an arbitrary number of tyes of inuts and roduces an arbitrary set of oututs. We achieve a higher level of realism than the contributions mentioned above because of the incororation of uncertainty into the rocess model. Our aroach is more secific than that rovided by Bordoloi and Matsuo (21), Carillo and Gaimon (24), Naoleon and Gaimon (24) and Dietrich (26), and focuses on the scheduling roblem of assigning resources to articular service rocesses as oosed to aggregate resource allocation. We begin with a model of a technology function of a single-stage service rocess roosed by Athanossooulus (1998), which was offered for a DEA study. A technology function for a service encounter is a function that effectively mas inuts to oututs according to the caabilities of the service articiants to transform inuts into oututs. We construct this functional relationshi by considering the inuts and oututs of a rocess to be functions of the volume, or number of service cycles, of the rocess which are simultaneously executed. 2. Technology functions Different technology functions tyes can roduce a given set of inuts and oututs. Efficiency is determined by a technology function. In service rocesses the combination 2

6 of inuts which roduces a secified set of oututs can vary. Hence, there may be more than one technology function tye available to a service rocess. We osit the following rinciles for any technology function that can reresent realistically a service rocess. 1. The service rocess has inuts from two sets one controlled by the rovider and one controlled by the client. 2. There are technology constraints in the form of standard usage and yield limits exressed in terms of quality units. The qualities of inuts and oututs are indeendent of the erceived utility of the inuts and outut. 3. The technology constraints may not be fully known to the rovider or the client, which makes the model stochastic. 4. The technology function is defined as the otimal solution to a game between the service rovider and the service reciient. Different assumtions about knowledge sharing and ower can roduce different game equilibria. 5. There are constraints in terms of resource allocations that determine the resource units that are actually rovided and the resource units that are actually yielded from a rocess. 6. Awareness the client may not have full knowledge of the rovider s resource commitments and the quality of the rovider s resource commitments. Similarly, the roviders may not have full knowledge of the client s resource commitments and the quality of the client s resource commitments. 7. The objective function that determines the otimal rocess inuts and oututs is the maximization of utility through a of the service articiants. Define, x = x 1,x 2,..., x m rocess y = y 1,y 2,..., y n rocess ( ) = ( ) = a vector of quantities of the m inut tyes required by a vector of quantities of the n outut tyes required by In general, a technology function for any rocess,, is a vector function that exresses the vector of oututs as a function of the inuts. T t, { x } = { y } i i S I j S O A linear, variable-returns-to-scale (VRS) technology function can be written, 3

7 Where T x + b = y T is a matrix of constants. A linear, constant-returns-to-scale (CRS) technology function can be written, T x = y The form of the technology function that we use for our model is a secial case of linear, CRS function that can be found in Athanassooulis (1998). We will refer to this tye of function as a recie function because it describes the relationshi of inuts to oututs in terms of usage rates and yield rates of a rocess cycle. For a recie-tye technology function, the x i ' s must be rocured according to usage rates of a rocess cycle and y j ' s are generated according to yield rates of a rocess cycle. The recie for inuts and oututs re rocess cycle forces all inuts to be in fixed roortions with resect to one another. The recie technology function form Note: The word benchmark secifies an ideal rocess that is 1% efficient 1 µ i = = benchmark usage of resource i er cycle of rocess βi 1 γ = = benchmark generation (yield) of resource j er cycle of rocess α β i = benchmark technological coefficient of inut i of rocess (number of rocess cycles er unit of resource) α = benchmark technological coefficient of outut j of rocess ( number of rocess cycles er unit of resource) v = volume of the rocess execution. Can be thought of as the number of cycles of rocess that are executed v µ i = x i,i = 1,...,m (1) v γ = y, j = 1,...,n (2) y x i v = = γ µ i, y β i γ x i = x i = x i = γ β i x i = α µ i α µ i, (3) Fixed roortions of inuts and oututs are inherent in these formulas for the elements of the technology function. For examle, 4

8 x i x k y y k y x i µ i v µ i β k = = = µ k v µ k β i γ v γ α k = = = γ k v γ k α γ v γ β i = = = µ i v µ i α A matrix form of the technology function The technology function, which exresses oututs as functions of inuts, can be written as a matrix multilication as follows: y = τ x where, T γ β τ = m γ β i 1 τ i = γ β i = = = µ i α α µ i The columns of the technology matrix are linearly deendent, each column being a multile of a single, base column of technology coefficients. Similarly, the rows are linearly deendent. Note that this version of the matrix, is more constrained than that of the general linear CRS case. In the case of the rocess model the matrix is a dyad. The dyad can be built from the usage and yield arameters or from the technology coefficients. From DEA we have the familiar general form of the relationshi between inuts and T T oututs of an efficient CRS technology function, u y = v x. One may wonder about the corresondence of the recie technology function to this relationshi. The following theorem establishes the generality of the recie model. Theorem 1: A technology function imlies the relationshi, the technology function is a rocess-model function. T T u y = v x, if and only if Proof: Suose y = τ x Multilying both sides by the vector, T T α y = nβ x T α, we obtain T T Now suose, u y = v x, where v = nβ, u = α Using the same definitions given in the develoment above, we can derive the result, 5

9 y T = α γ x Efficiency within the recie model We define the actual technology arameters ( b,u,a g ) for a given instance of a rocess from the ideal arameters ( β, µ, α, γ ) as follows: u µ, g γ b β,a α, For the reader who has a background in DEA, we make one final connection of the current model to this large body of knowledge. DEA studies are directed at measuring the overall efficiency of a rocess. For the recie model we can evaluate the difference between volume suorted by inuts to volume required by oututs to measure inefficiency. Suose, for a articular execution of a rocess, m β i = 1 i x i n α j = 1 y m i = 1 β i x i s = n α j = 1 y The surlus variable s reresents the overall level of inefficiency of the rocess. (See Athanassooulos, 1998). m s = (1 e ) β i x i, which effectively measures the amount of wasted inut i = 1 resources.the surlus of weighted inuts over weighted oututs reresents the overall inefficiency of the rocess. If the efficiency is 1%, then s =. e s = 1 β i x i i For resource allocation and disatching, we must consider efficiency as a multidimensional quantity. That is, the effectiveness of each inut resource on a rocess can be different and the roductivity of the rocess in terms of each outut resource can be different. We view inefficiency in terms of deviations from the benchmark recie. Hence, efficiency has a multi-dimensional foundation in the technology function. For resource lanning uroses, we need to measure the comonents of the efficiency in order to know which inut resources should be adjusted. Randomness and inefficiency in the recie model Inefficiency can be due to systemic shortcomings in the DMU as well as random variations in erformance. There is a arallel to the notions of common vs. secific causes of variation in modeling of manufacturing rocesses. We can reresent the inherent inefficiency of a articular service encounter as well as the inherent uncertainty in the erformance of the service rocess by defining non- 6

10 negative random variables to reresent the deviation of the rocess arameters from their ideal, deterministic values. u = µ + δ u g b = γ = β δ δ g b a = α + δ a where each delta random variable has non-negative suort. These exressions define the rocess arameters, ( b,u,a g ), as random variables. We assume that the inuts to the service rocess are controllable and inuts are disatched rior to the realization of random variations of rocess arameters. Oututs, on the other hand, are roduced through the realization of rocess arameters. Hence, the oututs, y, j = 1,..., n, are also random variables. Since, ( i b ) x i v b i x i = β δ = for i = 1,2,..., m. the random variables, b 1,b2,..., bm, must be erfectly correlated. That is, after the rocess inuts are disatched, the rocess usage rates mutually adjust to values that suort a certain volume and which are consistent with the inefficiencies and random variations of the usages. v Define, ν = β i x i and ε = ν 1 Then, i x i = ε (4) Each random variable, b i x i, can be relaced by the random variable ν Hence, the m decision variables, x 1,..., x m can be relaced with a single decision variable, ν. 3. Otimization The criteria for the resource disatching decision can be stated as follows: Problem P1 min { x } ŷ w + j subject to: r x T ( ŷ y ) f y ( y )dy c x 7

11 x for all ŷ = a vector of target oututs for rocess f = the distribution of y, which is a function of the resource allocations, x y r = vector of caacities of available resources The Loss Function The ortion of the objective function that catures the effects of failing to roduce enough oututs to reach desired targets is called the loss function. ŷ w ( ŷ y ) fy ( y ) dy (5) j The loss function in the objective function makes the otimization a rather comlex NLP. Lemma 1: The loss function increases with inefficiency Proof: The roof is established most easily from the form of the loss function, (5). Inefficiency is reflected in the robability distribution for y. If we consider any two service rocesses such that the second rocess has less efficiency than the first, then the robability distribution for y 2 j stochastically dominates the robability distribution for y 1 j. ŷ1 j ŷ 2 j ( ŷ1 j y ) f y ( y )dy ( ŷ y ) f ( y ) dy 1 j 2 j y 2 j Lemma 2: Loss is increasing in the targets, ŷ Proof: Obvious from original form of the loss function, (5). The Loss Function in terms of technology arameters By exressing y = g ε ν x, y in terms of their deendence on the volume through x =,, we can reduce the set of decision variables from { x,x,..., x } the single variable, ν. Furthermore, we introduce the robability distribution of the random arameters, g ε, which allows us to re-state the roblem more directly in terms of the sources of randomness. Define, 1 2 µ m ν to 8

12 z = g ε y Then, 1 Fy ( y ) = Fz and f y ( y ) = f z ν ν ŷ ŷ ŷ y y ŷ y f y y dy = f g ε ν ν ŷ Where ẑ = ν Problem P2 The resource disatching roblem becomes, y ν ( ) ( ) dy = ν ( ẑ z ) f ( z )dz ẑ T min w ν ( ẑ z ) f z ( z ) dz + c µ ν { ν } j subject to: r µ ν ν for all The loss function now can be maniulated into a form that is more convenient for comutation and further analysis. ẑ z ẑ ( ẑ z ) f ( z ) dz = ẑ F ( ẑ ) ( zf ( z )) Define, Gz Az Bz ( z ) ( z ) ( z ) z ẑ = Fz z = 1 Fz ( z ) = f z z = Gz ( s ) ds z = Az ( s ) ds z Az ( ) = E z Lemma 3: [ ] ( z )dz ( s ) ds Proof: See Hadley-Whitin (1963) z ẑ ẑ + Fz ( z )dz Denote E [ z ] = z. The loss function can be written, 9

13 ( ẑ z + A ( ẑ )) w ν z (6) j Lemma 4: Loss is decreasing and convex in volume Proof: ŷ Recall that ẑ = ν d dν = w Since Az ( w ν ( ẑ Az ( ) + A ( ẑ ))) z ( A ( ) + A ( ẑ )) + w ẑ G ( ẑ ) Az z ( ) + Az z is decreasing and convex, and ( ẑ ) < ẑ G ( ẑ ) z da z Therefore the loss function is decreasing in volume. 2 d 2 dν ( w ν ( ẑ A ( ) + A ( ẑ ))) z z dz = G, z ( w ( A ( ) + A ( ẑ )) + w ẑ G ( ẑ )) d = z z z dν ẑ ẑ 2 ẑ w Gz ( ẑ ) w Gz ( ẑ ) + w f z ( ẑ ) > ν ν ν Therefore the Loss function is convex in volume. KKT conditions The lagrangian for Problem P2 is, T T ( ) ẑ z + Az ( ẑ ) + c µ ν + λ µ ν r L( ν, λ ) = w ν j The convexity of the loss function ensures a unique solution. The first-order KKT conditions for the otimization roblem include, T T ( z + Az ( ẑ )) + w ẑ Gz ( ẑ ) + ( c + λ ) µ L = = w (7) ν j The derivative of the loss function in (7) reresents the marginal value of one unit of volume to the achievement of target levels of outut. We denote this marginal value, M ( ν ). ( z + A ( ẑ ) + ẑ G ( ẑ ) M ( ν ) = w z z ) j 1

14 Then (3) states, T T ( c ) M ( ν ) = + λ µ (8) The right-hand side of (8) is the marginal resource cost of a unit of volume. Note that, by Lemma 4, M ( ν ) is ositive and decreasing ensuring a unique solution to (8). Standard NLP methods such as KNITRO (see Byrd, et al., 26) can be used to find the otimal volumes of all available rocesses and, hence, the otimal disatch of inut resources to the rocesses. Theorem 2: Processes that have lower usage rates will be allocated higher roortions of available inut resources and achieve higher volumes under an otimal olicy. Proof: From the definition of ν x β, x = x i iν i β i = β i i Allocation of resources across rocesses The otimality conditions indicate that the otimal allocation of inut resources across rocesses that are different in terms of their efficiencies, uncertainties and/or outut targets is quite comlex and, in some cases counter-intuitive. The behavior of M ( ν ) as these characteristics of a rocess change is not uni-directional. Furthermore, the fact that M ( ν ) and the otimality condition (8) are based on vector inner roducts, there is the ossibility that different rocess can differ in one direction on some dimensions and in another direction on other dimensions. Therefore, simlistic guidelines for disatching resources across different rocesses are not to be exected. Conflict between service roviders and service reciients The resources that are disatched as inuts to a rocess are rovided by both the service reciient and the service rovider. Similarly, the benefits of the oututs of a service encounter are enjoyed by both arties, however, not necessarily in the same way. Secifically, the weights, w, and the targets, ŷ may be different for the two arties. Clearly, such differences will lead each arty to arrive at a different otimal olicy. Another oortunity for a conflict in the suort of a rocess rovided by the service rovider and the service reciient stems from their differences in the robability distribution of the rocess arameters, z. Clearly, any difference in the secification and estimation of these distributions will manifest themselves in differences in z + A ( ẑ ) + ẑ G ( ẑ ), which would imly different solutions for (8). z z 11

15 Solutions to Problem P2 for different arameter values and robability distributions reveal that the magnitude of differences in service-rocess suort and exectations of outcomes are bound to create mis-understandings and conflicts between roviders and reciients that are well-known in service enterrises such as consulting and education. A clear recommendation can be derived from this research. Service roviders and service reciients should make every attemt to educate themselves jointly about the nature of a service rocess before they engage in disatching resources to it. This joint understanding of a rocess has the benefits of minimizing the uncertainty about the rocess and, erhas more imortantly, create a consensus about secification and estimation of the rocess technology. Furthermore, we can recommend that, having achieved this consensus, the solution to Problem P2 be comuted by both arties so that the otimal lan for the service encounter is consistently executed. 4. Conclusion This aer rovides a model for resource disatching to secific service encounters within a service enterrise. The otimization is straightforward and achievable by standard NLP software. Therefore, the model has ractical utility. The model resented herein is a stochastic model of a service rocess which has the otential to cature randomness of service rocesses as well as uncertainty in the secification and estimation of rocess arameters by the rocess articiants. Consequently, the data collection and arameter estimation that is necessary to aly this model necessarily involves assessing the extent of the knowledge of the rocess by the service rovider and the service reciient. The model allows the investigation of the sensitivity of the olicy for resource disatch to rocess knowledge and the asymmetry of these olicies across articiants who differ in their understanding of the service rocess. This tye of investigation holds the romise of bringing service roviders and service reciients towards common ground in lanning service rocesses. Future research will exand this model into a multi-stage framework and the investigation of two-arty games as a means to describe the interlay of a service rovider and a service reciient in their decisions to disatch resources to a service rocess. 12

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