Towards a formalization of budgets

T o wards a F ormalizat ion of Budgets Jan A. Ber gst ra 1 Sanne Nolst Tr enit ´ e 2 Mark B. van der Zw aag 1 1 Section Software Enginee ring, Informatics Institute, Uni versity of Amsterdam 2 Faculty of Science, Uni versity of Amsterdam Email: { janb,sanne,mbz } @science.uva.n l Abstract W e go into the need for, and the requirements on, a formal theory of budg ets. W e present a simple algebraic theory of rational budg ets, i.e., budgets in which amounts of money are specified by fun ctions on the rational numbe rs. This theory is based on the tup lix c alculus. W e go into the importance of using totalized models for the rational numbers. W e present a case study on the educational budg et of a uni versity department offering master programs . 1 Introd uction The pro cess of budget design an d financial accounting is becom ing increasingly spe- cialized a nd exclusive. Unfortu nately , the n eed for an underly ing theo ry seems to be unreco gnized. Eco nomic theories of finance do not provide the simple insights need ed for managing small-scale operations. W e are currently witnessing the follo wing dev el- opments. 1. Financial work takes p lace in a con text of complex IT support systems, which are often poorly documented from a user p erspective. Documen tation is typ i- cally lim ited to u ser ma nuals, and d oes n ot g iv e co nceptua l description s of the underly ing b udg et theory and financial theories. 2. Financial competence is easil y confu sed with the ability to operate certain finan- cial systems. Because these sy stems are incre asingly co mplex, the competen ce to u se them is becom ing scarce and req uires train ing and exper ience. Neverthe- less th at co mpetence need not imply any deep er awareness of the variation of business logics that may or may not be s er ved with a gi ven system. 3. Financial plan ning is at the basis of many com plex o rganizational tr ansforma- tions. Its logic is intimately connec ted with novel structur al changes such as outsourcin g, insourc ing, b acksourcin g and o ffshoring. Or gan izational ch anges are often correlate d with chang es in b udget logic. 1 In this situatio n, we find it worthwhile to explore th e applicability o f m odeling tech- niques de veloped in the fields of inform ation scien ce and softw are engineering . Un like software arch itecture, finan cial arc hitecture seems to be a su bject to which relatively little attention is paid. It is worth a n ef for t to apply system description techniques from computer science to fi n ancial systems and to f acilitate systematic and co rrect reasoning about th em. W e believe that financial a rchitecture can profit from the same develop- ment strategy as sof tware architecture by m aking use o f a b asis of desig n patterns and by developing very clear modularizatio n techniques. In the other dir ection, we imagine a formaliza tion of b ud gets to be a helpful ingre - dient for the d ev elop ment o f sourcin g theory (see for instance [5] and [ 6]). Sou rcing theory requir es the pre sence o f so-called business cases for insour cer and outsource r to be available and scru tinized befor e any deal is made. After ou tsourcing has b een executed, services are expected to be delivered in acco rdance with an SLA (Service Lev el Agree ment). Budget info rmation is an essential p art of any SLA. No comp lete theory of sour cing is p ossible without some the ory o f SLAs an d their underly ing b ud- gets. SLAs con stitute a relativ ely new topic in compu ting and their meta-theory is still in an initial stage. W e expect that SLAs, or similar entities existing within a fur- ther ev olved terminology , will become a central corner stone of the emerging theory of service-orie nted computing . Overview . In th is ar ticle we define a simple algeb raic theo ry of rational budgets , that is, budgets in which am ounts o f m oney are specified by f unction s on the ration al number s. For this th eory we bo rrow from our experien ce with process alg ebras (mod- eling the behavior of computer systems) and ab stract d ata types. In Section 2 we go into the use of rational functions, and in particular on the importanc e of totalized mod- els f or the rational n umber s when de fining them as an abstract d ata ty pe. In Section 3 we reflect on the form alization of budgets, and this is followed by a simple theor y of budgets in Section 4. W e use the so-called tuplix calculu s [2] to model budgets. In Section 5 we present a case study on the educationa l b u dget of a uni versity departme nt offering master progr ams. 2 Rational Numbers It is a common misund erstanding th at because budget figures are to be un derstood as measuring q uantities of money expressed in terms of known currencies all semantic problem s will disappear . Th e issue is compara ble to a progr am no tation designed for progr amming co mputation s on natural numbers. In spite of the seemingly clear m athe- matical basis of the program notation, in the absence of a formally s pe cified semantics of the program notation at hand, nothing much can be said about what transform ations on natural number s a specific program denotes. For the definition o f a b udget the data type o f rational numbe rs is considered of cen- tral importance. All financial quantities will me measured in exact ra tional numbers. I f the question arises what exactly are the rational numb ers we refer to [3] which provid es a n ovel an d con cise initial algebra specification of this classical mathema tical structur e. W e hold that division plays an impor tant role in budgeting, because of the need to dis- tribute exp ected co sts over a number of expected u sers. If ten users will make mutually 2 compara ble use of a single shared s erv ice each of them will be expected t o pay 10% of its costs unless mo re s pec ific information is a vailable. In terestingly , di vision seems not to feature in acco unting and b ookkeepin g. In [ 3], division is given a first class status with an operator symbo l reserved for it, very muc h like additio n, multiplication and subtraction. Moreover we will insist, again st conventional pr actice, that d ivision is a total o perator . This lea ds to ‘meadows’ o f ration al num bers: a meadow is t h e well- know algebr aic structure ‘field’ with a total oper ator fo r division, so that division by zero pr oduces so me value in the do main of the field. In a z ero-totalized field d ivision is made total by ch oosing zero as th e result of d ivision by zero ( and, for example, in a 47-totalized field one has chosen 47 to represent the result of all divisions by zero). The relev ance for ou r theory of budgets is this: budgets will co ntain expressions for rational function s rath er th an ‘closed’ figures in full (fixed p oint) precision . From the c onv entio nal perspective on rational numbers, these fun ctions may be undefined for certain inpu t values, n amely , f or values lead ing to devision b y zero. In gener al, it will be far fro m tr ivial to decide wheth er function s are alw ays defined, and , if not, to establish the values for which it is. On the oth er side, the meadow of rational numbers constitutes a total algebra with tri vial type-checkin g properties, that provides us with a clear m eaning of expressions. This will b e of the highest im portanc e to ou r en deav or, because just like in the case o f specify ing compu ter programs, there will be no way to av oid explicit syntax and type-checking of e xp ressions. A down-side of working w ith zero-totalized field s is that some calculations will produ ce useless results. In m ost cases the oc currenc e of division by zero in the cou rse of a calculation still indic ates the pre sence of an error somewhere and err or detection technique s will be needed. Nevertheless the meta-theory of this form of er ror detection is considered far simpler than the meta-theory of partial algebras, th us creating a trade- off to the advantage of the use of calculation in zero-totalize d fields. A small d igression: on e m ight wonder why the issue to define division b y zero is so easily a voided in school mathematics and its academic sequel. T he answer is that in mathematics mo st specialists make no distinctio n between syntax and semantics. No syntactic expression is entitled to any a ttention o n th e sole groun ds of its f ormal exis- tence. If syntax is u sed its u se follows the development of semantics and th ere always is an intended meaning. Realistically , the q uestion ‘what is the intend ed mean ing of that piece of syn tax’ cannot be ev en posed. A reluctance to sepa rate syntax and seman - tics m ay become a weakness whe n co ncepts need to be defined which are viewed as constructs of a syntactic nature consisting of parts rooted in classical mathematics. 3 Theory and Practice of Budgets A cru cial po int in the d esign of a f ormal the ory of budgets is th e fo llowing separation of concern s. Ou r task is to giv e a conceptu al, mathematical definition of budgets. This definition (of what a budget is ) should be as much as possible independ ent of how and why budgets are u sed . A budge t will not b e assigned a behavior of its own a priori. One wants to a void defin itions like ‘n atural n umbers are a very prac tical co ncept th at has been in use since the need ar ose to c ount sheep, for which natural numbers turn out to work very well fo rtunately . ’ Clearly budgets are artifacts of a h uman origin , and, as 3 in th e case of natu ral n umbers o r d ata b ases, the ir u se is indepen dent of the artifact at hand. W e believ e that a formal theory of b udg ets is essential to the analysis and improve- ment of their use. Some e xam ples: (1) The p ractice of budget design can b e compa red to the pra ctice of computer progr amming: in general pro grammer s have no mean s av ailable to k now in ad vance what comp uters will do with the ir writings. This un certainty is m ainly due to a lack- ing theo retical b asis but equ ally to a com mon eth os wh ich acts aga inst the use of that theoretical basis even if it happ ens to exist and if it mig ht be re adily available at rea- sonable costs. In compu ter pr ogram ming, testing is the ma in though u nconvincing tool to fight this form of uncertainty . I n b ud get design the concept of testing is significantly harder to imagine, howe ver . Bud gets seem to be submitted to a number of static checks only . Th en after their use some fo rm of evaluation and a ssessment may p roduce new guidelines (design rules) fo r budgeting an d ne w budgets wil l be match ed with these new design r ules as well. Budget testin g comparab le to d ynamic testing of contro l code will r equire a simulation en viro nment. That en viron ment is quite specific to an organization and most organization s hav e no suc h tool av ailable to th em at the time o f this writing. Clearly , b udg et simulation is a hopeless endeav or without a formal theory of budgets. (2) Irrespective o f the objectiv es a b ud get d esigner has in m ind when writing a bud- get, it will evolve th rough a life-cycle. An organization may prescribe this life- cycle to its budget design ers in very m uch the same way as a software life-cycle may pla y a normative role in a computer software productio n factory . Like a m achine control code can be active (r unning , e xecuting) a budget can be used to c ontrol e vents within an organization. W e w ou ld hope that when we start with a clear f ormal definition, we may be able to explain (at least in princip le) h ow that for m o f control might work. Also, as budgets are often very co ntext-specific, they may be compared to dedicated com- puter prog rams. Renewing a budget on an ann ual basis may be com pared to comp uter software m aintenanc e ( althoug h that comparison may well underestimate the degree of innovation tha t a new budget requires). (3) When hard -pressed to qualify the writing o f a budget, th e f ollowing viewpoint might be rea sonable: budgets are prop osed in a co ntext in which their p roposal is b est viewed as a move fro m th e side of its author in a game which is imp licitly p resent in the mentioned context. Let us assume that a budget is designed for an acti vity called A . After h aving been designed an d worked out in detail, the distrib ution of a b udge t can be considered a move in a game. By introducin g a budget prop osal th e decision- making process is somehow in fluenced. One may assume that this process will ev entu ally lead to a validated budget for acti vity A . The budget p roposal m ay imp act the style o f budget design which will be ad opted fo r A and for similar acti vities. The intrig uing observation i s th at when time has come to write budgets for A , many dif fer ent and competing budget pro posals may be simultaneously p ut fo rward. T hus at b ud get design time there is n o such thing as ‘ the budget’ in very mu ch the same way as a computer progr am u nder construction leaves o pen many degre es of freedom. It may be fruitfu l to expe riment with writing q uite d ifferent budgets fo r th e fin ancial control of a single activity A . Ag ain, such analysis of the practice of budgeting should start with a formal theory of budgets. 4 4 A Simple Budget Algebra W e p resent a simp le alge bra for ra tional budgets. This algeb ra is an applicatio n of th e so-called tu plix calcu lus [2]. A tup lix ( plural: tuplices) is a datastru cture th at co llects attribute-value pairs. The tuplix c alculus provides signatur e an d axioms f or v ario us operation s on budgets, includ ing a m eans to express con straints o n budgets, the com- position of budgets, and en capsulation. So, budgets will be given by means of tu plix expressions. The design o f u ser-friendly syntax fo r budget expression s is ou tside the scope of this paper . Howe ver , the definition of b ud gets as tuplices provides a ru dimen- tary syntax which will suffice for explanatory purposes. 4.1 Entries and T ests The b asic building block s of b ud gets are entries and tests . An entry is an attribute-value pair of the form a ( p ) where a is an attribute from a given set A of attribute sym bols, and p is a d ata term. For the values we use th e da ta type o f rationa l numbers, which w e assume to be given by a zero-totalized field as explained in Section 2 (so p / q is alw ay s defined). An entry represents a paym ent: the attribute is used in the c ommun ication between pay er and payee, and describes or ide ntifies a transaction; we refer to the attribute as the channel of the tr ansaction, a nd shall also say that th e paym ent o ccurs along th e chann el. The term p r epresents the amo unt of m oney inv olved. An entry a ( p ) with p > 0 stands for an obligatio n to pay amo unt p along channel a . If p < 0, the entry stands f or the expected receipt of amount p alo ng a . A zer o test is a term of the form γ ( p ) for amo unt p . It acts as a cond itional: if the argument p eq uals zer o, then the test is void and disappears f rom c omposition s; if the test is no t eq ual to zero, it nullifies any composition containing it. Ob serve that an equality test p = q can be expressed as γ ( p − q ) . 4.2 Budget Composition W e define a budget as a (conju nctive) compo sition of entries and zero tests. This com- position is commutative an d associativ e: x  y = y  x , (1) ( x  y )  z = x  ( y  z ) . (2) There are tw o constan ts for budgets: th e empty budget , no tation ε , and the null budget , notation δ . T he empty budget stands for the absen ce of entries or tests, and the null budget is used to mo del an er roneo us situation wh ich nu llifies th e en tire co mposition 5 containing it. Axioms: x  ε = x , (3) x  δ = δ . (4) Entries with the same attribute can be combined: a ( u )  a ( v ) = a ( u + v ) . (5) Note in particular that the composition of a payment a ( p ) and the receipt a ( − p ) can be reduced to a ( 0 ) . W e shall see below th at encap sulation b oth enfor ces an d hides such synchro nizations. For the ax iomatization o f th e zero te sts, we use th e pro perty that in the ze ro- totalized field for the rational numbe rs, the division p / p y ields ze ro on ly if p is equal to zero; otherwise it yields 1. Axio ms: γ ( u ) = γ ( u / u ) , (6) γ ( 0 ) = ε, (7) γ ( 1 ) = δ . (8) For reasoning about b udgets with open data terms, we add the following tw o ax ioms: γ ( u )  γ ( v ) = γ ( u / u + v / v ) , (9) γ ( u − v )  a ( u ) = γ ( u − v )  a ( v ) . (10) 4.3 Encapsulation For set of attributes H ⊆ A , the operato r ∂ H ( x ) encapsulates all entr ies with attribute a ∈ H occurring in x . That is, if the accu mulation of quantities in entries with attribute a equals zero, the encap sulation on a is co nsidered successful and the a - entries disappear ; if the accumulation is not equal to zero, it yields the null budget δ . Axioms: ∂ H ( ε ) = ε, (11) ∂ H ( δ ) = δ , (12) ∂ H ( γ ( u )) = γ ( u ) , (13) ∂ H ( a ( u )) = ( γ ( u ) if a ∈ H , a ( u ) if a 6∈ H , (14) ∂ H ( x  ∂ H ( y )) = ∂ H ( x )  ∂ H ( y ) . (15) W e further adopt the identities ∂ H ∪ H ′ ( x ) = ∂ H ◦ ∂ H ′ ( x ) an d ∂ / 0 ( x ) = x . Example. Consider b ud get P def = a ( − 30 )  b ( 10 )  b ( 20 ) . 6 This b udg et specifies the expected receipt of amount 30 along channel a and payments of amoun t 10 and of amount 20 along b . W e compose it with b ud get Q def = b ( − 30 )  c ( 30 ) , which specifies that am ount 30 is received alon g b and sent along chan nel c . W e see that the pa yments of P w ill match the rec eipt of Q on ch annel b , so that e ncapsulation of b will hide these entries: ∂ { b } ( P  Q ) = a ( − 30 )  c ( 30 ) . T o derive this, first deri ve that ∂ { b } ( a ( − 30 )  c ( 30 )) = a ( − 30 )  c ( 30 ) using axioms (14) and (15). Then: ∂ { b } ( P  Q ) = ∂ { b } ( a ( − 30 )  b ( 10 )  b ( 20 )  b ( − 30 )  c ( 30 )) = ∂ { b } ( b ( 0 )  a ( − 3 0 )  c ( 30 )) = ∂ { b } ( b ( 0 )  ∂ { b } ( a ( − 30 )  c ( 30 ))) = ∂ { b } ( b ( 0 ))  ∂ { b } ( a ( − 30 )  c ( 30 )) = γ ( 0 )  a ( − 30 )  c ( 30 ) = a ( − 30 )  c ( 30 ) . Another example. When comp uting the encap sulation of more than one chann el, we split the encapsulation up, and compute them one by one. Recall th at we defined ∂ H ∪ H ′ ( x ) = ∂ H ◦ ∂ H ′ ( x ) . For example, we derive ∂ { a , b } ( a ( 0 )  b ( 0 )) = ∂ { a } ◦ ∂ { b } ( a ( 0 )  b ( 0 )) = ∂ { a } ( a ( 0 )) = ε. 4.4 Constraints In the case stud y in this ar ticle, we assume that an absolute ope rator | | (d efined b y | p | = p if p ≥ 0, and | p | = − p otherwise) is part of the sign ature fo r rationa ls. W ith this operator we can express inequalities: γ ( | q − p | − ( q − p )) expresses the test p ≤ q . F or ine quality tests we sh all then simp ly write γ ( p ≤ q ) . W e sometimes write γ ( p = q ) for γ ( p − q ) . For example, we may design a budget under the constraint φ def = p ≤ q , and we then com pose th e b udg et with the test γ ( φ ) . For th e c omposition under multiple constraints, say φ and ψ , we may use the notation γ ( φ ∧ ψ ) def = γ ( φ/φ + ψ/ψ ) (9) = γ ( φ )  γ ( ψ ) . 7 5 Case Study: MSc Program Bud gets W e co nsider a university department that m aintains the three MSc pro grams A, B and C. Each program of fers a 1-year, 60 EC 1 curriculu m. These programs need a ne w bud- get because of changes concern ing budget guidelines, financial reporting, r isk manage- ment and business accou nting. Below w e develop budge ts f or the pr ogram s. Having fixed these b ud gets, the three program managers should ne go tiate the setting of certain variables. Having d one that the prog ram man agers are free to develop their prog rams within the c onstraints of the budget. This p rocess m ay be viewed as a game aim ed at the d esign of a single budget wher e coa litions try to get things the ir way by imposing preferr ed variable settin gs o n other participants. In ou r case, the program mana gers will not hesitate to get money their way at the expense of the other programs o r to prove other programs financially unsound , should they fi nd possibilities to do so in the new s ystem. 2 5.1 Generic Structu re o f an MSc Pr ogram Each of the three program s has the following s tru cture: 1. An introductio n week providing g eneral info rmation. 2. 4 courses of 10 EC each. A course consists of 300 working hou rs c omposed from these ingred ients: • Between 40 and 160 hou rs of teaching by senior s taff. • W orking grou p meetings supervised by junior staf f. • Unsupe rvised student team meetings. • Unsupe rvised indi vidu al e xp erimental work. • Unsupe rvised indi vidu al homew ork . • Participation in an examination. 3. 2 proje cts of 10 EC each. A pr oject is supervised in one of the following ways: • Between 5 and 10 hou rs of internal s en ior staff supervision . • At most 20 hou rs of junior staf f supervision . • At most 5 hou rs of external staff super vision (p erform ed outside th e insti- tution). A project en ds with a 30-minute pre sentation (with at least two senio r staff me m- bers present). 1 An EC is a unit of student acti vity/learni ng outco me in the European Credit Transfer System. One E C stands for 28 hours of work. 2 Mark Burgess (see for instan ce [4]) advocates the m echani sm of auton omous agents making promises which const rain their actions on a voluntary basis only . A bud get proposal might be vie wed as a promise conditi onal under the counter promise by other parties that they will go along with it. Finding protocol s for distributed budge t design in a contex t of volunt ary cooperat ion is an interesti ng challenge for further research . 8 4. A formal final degree ceremony . Each pr ogram offers at least two man datory courses f or its own stude nts, an d may offer a num ber of optional courses that can also be fo llowed b y s tud ents from the other progr ams. An expe nsiv e way to imp lement this is to offer six dedicated co urses in th e progr am and to make two of these compulsory while lea vin g the students the option to choose two fr om the other f our co urses. A rea son to u se this k ind of p lanning may be to let research staff lectu re abou t their advanced to pics in order to recr uit futu re Ph D students. Another r eason might b e to ma ke su re th at th e en tire stud ent popula tion ac- quires a wide body of kno wled ge representative of the field as a whole while accepting that each in dividual student has acqu ired knowledge in a m ore limited scope . A much cheaper op tion is to offer only the 2 m andator y courses and to ask the studen ts to take electiv es fr om co urses o ffered by other prog rams. Reason for d oing so may b e a lac k of staff o r financial reso urces. Another re ason mig ht b e th e inten tion to edu cate a ho - mogene ous gro up of exper ts who will be able to coop erate effectiv ely in forthcomin g projects. 5.2 Joint B udget In this case study we specify four b ud gets: budgets A , B , and C , f or the respecti ve progr ams A, B, and C, and one joint budget J . W e start with the joint b udg et. All inco me o f the prog rams fr om external sour ces is specified in the jo int budget, and the se incoming am ounts ar e received via chann el in . The task of the jo int budge t is to spe cify the distribution of the incom e between the three program s and shared costs. Payments fr om the joint budget to th e individual pr ogram budgets run via the respective channe ls a , b , and c . The only sh ared costs are the payments to the so-called education al service center (student consulting, time tabling, lecture hall reserv ation, facility management, administration); these payments are done via channel e . Picture: in   J e > > | | | | | | | | a          b   c   ? ? ? ? ? ? ? ? A B C When we co nsider the com position o f the fou r budgets, payments along the c hannels a , b and c are co nsidered to be internal. Th e channels in and e ar e external; they ‘link’ to parties for which we do not have the b udg ets. Notation. T he letter X ran ges over A , B , C (den oting the progra ms). W e use lo wer- case italics for variable names, and uppercase roman for abbreviations. V ar iables. The v ariab les used in the spe cification of the join t budget J are listed in Figure 1. The values f or th e variables X : nec and X : ndg are determined b y me asurement and monito ring (in pr actice one may take last years number s instead) . The values fo r the v ariab les bbp p an d k are d etermined by n egotiation between the p rogr am manage rs. W e shall return to the consequen ces of the setting of these variables. 9 Set by external authority: cpec The compensation per EC . Amount obtained w hen a warding one EC. cpdg The compensatio n per de gree . Amount ob tained when awarding one degree. escf The educationa l service center fraction (value between 0 and 1) of the overall incom e to be tran sferred to the ed ucational service center . Set by measurement: X : nec T otal n umber of EC awarded in cour ses o ffered by p rogram X. X : ndg Number of degrees a warded in progr am X. Set by bu dget designer/neg otiatio n: bbpp A fixed amou nt that serves as th e basic budget per pr ogram . This amount is equal for each program and i s u sed to pay for the program manager, various co mmittee tasks, market- ing and commu nication. k Frac tion (value between 0 an d 1) of the b bpp which is taken from the p art of the overall income stemming fro m de gr ee compen sation. The rem aining fraction 1 − k of the bb pp is taken from the overall EC compensation. Figure 1: V ariab les used in b udg et J 10 Income. The income, received v ia channel in , consists of two parts. First, there is th e overall EC com pensation (ECC), d efined as th e tota l numb er o f EC credits awarded in the three progr ams, multiplied by the compen sation per credit: NEC def = ∑ X X : nec , ECC def = NEC · cpec . Similarly , there is the overall d egree compensatio n (DGC), defined as the total number of degrees aw arded in the three prog rams, multiplied by the compensation per de g ree: NDG def = ∑ X X : ndg , DGC def = NDG · cp dg . Expenses. Th e educ ational service c enter (ESC) takes c are of all data base h an- dling, tim e tabling, logistics, co mmun ication and marketing, help desk s of various kinds, in ternational relatio ns an d form al cer emony managemen t. There is a join t pay - ment to the ESC, consisting of the fraction escf of the overall income: ESC def = escf · ( ECC + DGC ) . The r emainder ( 1 − escf ) · ( ECC + DGC ) of the overall income is distributed a mong the three progr ams. First, each pro gram r eceiv es the am ount bbpp . Of course, this amou nt canno t b e more than one third of the a vailable money , so we adopt the constraint φ 1 def = bbpp ≤ ( 1 / 3 ) · ( 1 − e scf ) · ( ECC + DGC ) . Furthermo re, we require that fr action k of the bbpp is taken fr om th e overall degree compen sation, and the remaining part ( 1 − k ) of the bbpp is taken fro m the overall EC compen sation, leading to the following constraints: φ 2 def = k ≤ ( DGC · ( 1 − escf )) / ( 3 · bbpp ) , φ 3 def = ( 1 − k ) ≤ ( ECC · ( 1 − escf )) / ( 3 · b bpp ) . Apart fro m the fixed amoun t bbpp , that provides each pr ogram with a financial basis indepen dent of its own student number s (assuming th at th e oth er p rogr ams h av e suf- ficient num bers of students), each program gets a share o f the r emaining p art of the overall EC and degree compensation. These shares are proportio nal to the contribution that the program has in the overall compensation . The remain ing part of the degree compen sation, after subtraction of the expenses on ESC and bbpp , is DGC · ( 1 − escf ) − 3 · k · b bpp , and the share of this amount awarded to program X is X : nd g / NDG. So we define X :DGC def = ( DGC · ( 1 − escf ) − 3 · k · b bpp ) · X : ndg / NDG . 11 Similarly , we define X :ECC def = ( ECC · ( 1 − escf ) − 3 · ( 1 − k ) · bbpp ) · X : nec / NEC . Each program X receiv es from the joint budget the amount X :ST AFF def = bbpp + X :DGC + X :ECC . Budget. Putting e verythin g t o gether, the join t b ud get J is defined by J def = γ ( φ )  in ( − ECC )  in ( − DGC )  e ( ESC )  a ( A :ST AFF )  b ( B :ST AFF )  c ( C :ST AFF ) , where φ def = φ 1 ∧ φ 2 ∧ φ 3 . Notes. The joint b ud get has been designed with the follo wing prop erties in mind. • By takin g b bpp low (or simply zero ) ea ch budget ge ts as much as po ssible re- sources prop ortional to its production in EC and in degrees. By taking bbpp higher each program budget is provided with a minimum funding with the effect that each progra m gets less return on in vestment for a single EC or degree. For example, a p rogram with relatively few students may strive for a significant fixed budget b asis bbp p (maybe even b bpp = ( 1 / 3 ) · ( 1 − escf ) · ( DGC + ECC ) in an extreme case). • By taking k lo w ( or simp ly zero) a ma ximal reward is provided for programs with a h igh yield in terms of degre es. By taking k high (or simp ly 1 ) a maximal reward is given to program s that get as many as possible ECs to studen ts irrespectiv e of their program and irrespective of whether or not the y will complete their degree. For e xamp le, again for a program with relati vely few students: choo se fraction k as close as possible to 1 (for all values of bbpp , k = ( 1 / 3 ) · ( DGC · ( 1 − escf )) / b bpp seems to be a reasona ble choice). By tak ing k as close as p ossible to 1 , the progr am will profit the most from students from the other programs following its courses. • The f ollowing constraints need no t be imp osed as they follow from the d efining equations, that is b y add ing these constra ints the meaning of the budget will n ot change: ∑ X X :DGC = D GC · ( 1 − escf ) − 3 · k · b bpp , ∑ X X :ECC = E CC · ( 1 − escf ) − 3 · ( 1 − k ) · bbpp . 12 Set by external authority: sscph Senior staff marginal in tegral cost per hour . jscph Junior staff marginal integral cost per hour . Set by program manager: X : lpf Lecture prep aration factor (the numb er of ho urs used to pr e- pare one hour of lecturing) . X : sset Senior staff examin ation time: time needed to set an d mark an exam. X : sspst Senior staff pro ject supervision time (n umber of ho urs spent by senior staff supervising a single student projec t). X : jspst Junior staff project supervision time (nu mber of hours sp ent by junior staff super vising a sing le stud ent project) in addi- tion to senior staff supervision . X : ssot Senior staff overhead in total (ho urs per year). X : pmt Progra m management time (hours per year spent by progr am manager ). X : C : sslt Sen ior staff lecturing time (numb er of hours) for cour se C . X : C : jsst Ju nior staf f s up ervision ti me (num ber of hours) for c ourse C. Figure 2: V ariables used in program b udg ets 5.3 Budgets per Program In the individual p rogram budgets, th e am ount r eceiv ed fr om the joint budget is spent on the following costs: • Senior Educational Staff (SES). Compensation fo r educ ational working ho urs by senior staff. • Junior Educatio nal Staf f (JES). Com pensation for educational w ork ing hours by junior staff. • Progr am Manag ement (PM ). Compen sation fo r all for ms o f progr am m anage- ment perfor med by education al staf f. V ar iables. T he v ariab les used in the program b udg ets are lis ted in Figur e 2 . Wi thin the constrain ts set by the ed ucational budget, a program manage r ca n vary th e values for th e seco nd set of variables. Needless to say th is leads to a combinator ial explo- sion of optio ns. Setting th ese variables lo w imp lies a sound budget but introduc es risk with student success ra tes, with student satisfaction m onitoring an d per iodically with external quality contro l author ities. Most importan tly , ho wever , setting th e o ther vari- ables very low will cause senio r staff to complain ab out u nrealistic requ irements and workloads. Budgets. Define the senior staff working hours (SSH) and jun ior staff w ork ing hours (JSH) as f ollows, with C rang ing over the set C X of cou rses offered by pro gram 13 X . X :SSH def = ∑ C ( X : C : sslt · ( 1 + X : lpf ) + X : sset ) + X : ndg · 2 · X : sspst X :JSH def = ∑ C ( X : C : jsst ) + X : ndg · 2 · X : jspst The factor 2 in the summands f or project supervision stem s from the fact that each student do es two projects. Note th at p roject sup ervision g enerates co mpensation o nly when students ob tain their degree . Failed projects or p rojects for stud ents failing else- where in the progra m will not generate financial resources. The expen ses for sen ior and ju nior staff are f ound b y multip lying their hours b y their respective marginal integral costs per hour : X :SES def = X :SSH · sscph , X :JES def = X :JSH · jscph . Finally , t he staff costs for progra m managem ent are giv en by X :PM def = ( X : ssot + X : pmt ) · sscph . Budgets. A def = a ( − A :SES − A :JES − A :PM ) B def = b ( − B :SES − B :JES − B :PM ) C def = c ( − C :SES − C :JES − C :PM ) Notes. 1. Both senior staff mem bers and junior staf f member s may spen d two kinds of hours: regu lar of fice hours and spare time hours. The second kind of work is un- paid. No one will be ev er forced to work witho ut com pensation but a culture may exist where this is done on a regular basis. It is quite common to perform unpaid research work outside regular ho urs. This is po ssible for teaching ju st a s well. Nev erth eless ther e ar e som e constraints. All formal teachin g, all introductory activity , all examin ations and all senio r an d jun ior staff supervision must take place within of fice hours and the cost of this staf f time is gi ven by fixed rates per hour . Office hours are either classified as edu cational office hours or as research office ho urs o r as unclassified office h ours. If ed ucationa l activity is per formed in r esearch o ffice h ours it need not be p aid fr om the educational budget but it will be paid f rom the research budge t instead. This mechanism a llows a research group to sub sidize its teach ing activities fr om its research budget. That su bsidy may be justified in the case ed ucating a small g roup of students b rings abo ut a few PhD students who might be very hard to find otherwise. 2. In these budgets per program a formidable amount of freedom e xists because all variables deter mining the amount of jun ior and senio r staf f time spen t o n courses 14 and projects can me defined specifically for each of the progr ams. I f projects are very close to staff r esearch they m ay b e sup ervised in (seeming ly) less tim e be - cause additional unspecified research time is used for the superv ision as well while th ose hou rs are not paid f rom this ed ucationa l b udget. If for instance pr o- gram A has rather few stu dents in c omparison with the o ther two the following variable settings (or rather suggestion s for setting v ariables) can be helpful: (a) Set low s enio r staff hours for project supervision, and compensate that set- ting with higher junior staff hours and wit h time paid for from the research budget which is not vie wed as a part of these budgets. (b) Reduce the n umber of contact ho urs in course lecturing while req uiring a sign ificant amou nt of auto nomou s w or k fro m the stud ents an d setting difficult exam pape rs. This stra tegy may backfire when students from the other program s are s up posed to attend the same lectures, howe ver . 5.4 Synchr onization of the Budgets W e have pr esented the four budgets A , B , C , and J . The combined budget is the syn- chroniza tion ∂ { a , b , c } ( A  B  C  J ) . W e derive that it equals γ ( φ ∧ ψ A ∧ ψ B ∧ ψ C )  in ( − ( DGC + ECC ))  e ( ESC ) where ψ X def = ( X :ST AFF = X :SES + X :JES + X :PM ) . Conclusion. This example shows h ow a mo dular decomp osition o f a budget can be design ed. The deco mposition is valid u nder a nu mber of conditions o nly . If these condition s are not met, further r efinement of the budgets is need ed. That can b e d one by means of the same notation of course. These con ditions describe an ab straction level in the sense that they rule o ut cir- cumstances which may b e of practical interest but which are considered an u ndesirable overhead at a certain stage of design. 5.5 Further Reflections Methods of Cost Measurement. Ev en if o ne ob serves th e course of action wh en a particular program is ru n in all detail it is still difficult to ma ke a precise statement concern ing its costs. A first d ifficulty is how to count or incorpo rate free time hour s made by staff me mbers. A second difficulty is how to decide if any official research time is used for educational purpo ses. A third issue is to determine the border between research and preparation for lecture s and pr oject supervision . A furth er co mplication for co st m easuremen t based on o bservations on how the work is actu ally d one is th is: if staf f members are mad e aware of ho w their time inv estment is cou nted they may change their b ehavior . Suppose a staff memb er often works addition al unpaid ho urs 15 at home to get research do ne and th en find s out th at a cou nting system h as detected many h ours spent on teaching within the in stitution. This may lead to the conclu sion that th e position will be r eclassified into a teaching position with a lim ited r esearch task o nly . Then o f course th is staff m ember may be inclin ed to in terchang e a numbe r of educational support acti vities (marking exams, preparing lectur es, etc) with r esearch activities that were don e ou tside th e office hou rs. Thu s a co unting system should be stable in the sense th at its introd uction sho uld not by itself influe nce staff behavior in such a way that the results of co unting are mod ified. In ord er to obtain this f orm of stability staff memb ers shou ld be gi ven a v ariety o f optio ns for formally accou nting their time. Budgets versus Co sts and Planning. Giv en the co mbinato rial explosion of options for plann ing an MSc prog ram in the f ormats given above it is an unre asonable request to ‘offer the progra m in a cheap er fo rm’ unless very clear goals ar e stated in adv an ce. T he way in which a curriculum (that is, the listing of course titles and of project propo sals) can b e o ffered and planned implies that it is useless to sug gest a ch ange on financ ial groun ds unless a quite clear model of costs and re venues is a vailable and unless a clear target in that mod el has been set in adv ance. 6 Conclusion W e h av e motiv ated our interest in a fo rmal theo ry of budgets, and we have prop osed a simple algebraic theory of budgets b ased on the tuplix calculu s [2 ]. Quantities are expressed as fun ctions on the ratio nal number s, which we have modeled as a totalize d field [ 3]. As a c ase study , we modeled budgets and their compo sition f or a u niversity departmen t offering master programs. W e ha ve kep t the theory simple, but thin k of extensions su ch as op erators for c hoice (as present in the tuplix calculus), binding of rational variables, a theory of interfaces and hiding, etc. As a preliminary conclusion we state that budgets are amenable to formalization in the the d ata typ e tradition o f theor etical comp uter science. The examp le demonstrates that form alization can b e helpfu l to specify d etails which ar e likely to be missed in a less f ormal trea tment and which are help ful fo r a prop er und erstanding . At the same time, while work ing on the examp le, we hav e d rawn the conclu sion that designing budgets in a modula r fashion is not a n obvious matter and that many mo re cases studies will be needed to obtain a stable and formalized structure theory of b ud gets. Refer ences [1] J.A. Bergstra a nd A. Po nse. Interface group s for financial transfer ar chitectures. arXiv .org, arXiv:0707.163 9 v 1, 2007. [2] J.A. Bergstra, A. Ponse, and M.B. van de r Zwaag. T uplix Calculus. arXiv .org, arXiv:0712.342 3v1 [cs.LO], 20 07. 16 [3] J.A. Bergstra and J.V . T ucker . The rational num bers as an abstract data type. Journal of the A CM 54 (2), 2007. [4] M. Burgess. An approach to understan ding po licy based on autonom y an d v olun- tary coo peration . 1 6th IF IP/IEEE Distributed S ystems Ope rations and Manage- ment (DSOM 2005) , LNCS 3775, Springer-V erlag , 2005. [5] G.P .A.J. Delen. Decision- en contr olfactor en voo r sour cing van IT (in Dutch). PhD The sis, University of Amsterdam (van Haren Publishing Zaltb ommel), 2 005. [6] D.B.B. Rijsenbr ij and G.P .A.J. Delen. Enterp rise-architectu ur is een no odzakeli- jke voorwaarde v oo r verantwoor de outsou rcing (in Dutch). In IT service manage- ment best practices (r ed. J. van Bon), van Haren Pub lishing Zaltbo mmel, 35– 58, 2004. 17