Algorithms for radiotherapy treatment booking

Transcription

1 Algorithms for radiotherapy treatmet bookig Saja Petrovic *, William Leug *, Xueya Sog * ad Sathaam Sudar # * Automated Schedulig, Optimisatio ad Plaig Research Group, School of Computer Sciece ad IT, Uiversity of Nottigham, Nottigham, UK. # Dept. of Ocology, City Hospital Campus, Nottigham Uiversity Hospitals NHS Trust, Nottigham, UK * {sxp,wwl,xxs}@cs.ott.ac.uk, # sathaam.sudar@uh.hs.uk Abstract Two algorithms for bookig courses of radiotherapy treatmet sessios for the dyamic arrival of patiets i a parallel machie eviromet are developed. Patiets vary by due date, by which they should start their treatmets, cliical category ad treatmet machie requiremet. The first algorithm, mimickig curret practice, books patiets forward from the release date (i.e. the date whe the patiet ca start radiotherapy treatmet). The secod algorithm books patiets backwards from the due date. Feasible schedules of treatmet sessios are geerated for each patiet with the aim to miimise the total umber of patiets who do ot meet waitig time targets, the total legth of waitig time breaches, ad the total umber of iterruptios to treatmet. The algorithms icorporate practical costraits that arise i real-life problems faced by the Nottigham Uiversity Hospitals NHS Trust, UK. 1 Itroductio Cacer is a group of diseases characterised by the ucotrolled growth ad spread of abormal cells. With a wide rage of cacer types, there is a variety of approaches to treat cacer. The most commo forms of treatmet are surgery, chemotherapy, ad radiotherapy. Typically, several forms of treatmet are combied to icrease the chace of success. For example, surgery is ofte followed by chemotherapy ad/ or radiotherapy to esure the elimiatio of cacerous cells. Radiotherapy ofte forms part of the treatmet for the patiet due to its miimally ivasive ature. It has bee estimated that about 50% of all cacer patiets have radiotherapy as part of their treatmet, Delaey et al. (2003). A patiet beig prescribed radiotherapy treatmet will geerally have to first udergo several processes before the treatmet: simulatio (localisatio of treatmet fields) usig a CT scaer or simulator, plaig (cliicias decidig o the best way of deliverig the amout of radiotherapy eeded), ad fially verificatio (of plaig) usig a simulator. Treatmet fields are the precisely defied areas where the beams of radiatio, varyig by agle ad itesity, will be targeted durig treatmet of the patiet to destroy the cacerous cells. Oly after these processes are completed ca treatmet sessios give o a daily basis, called fractios, begi. Durig treatmet, ioisig radiatio is delivered by a liear accelerator (liac) to those precisely defied treatmet fields o the patiet to shrik the cacerous cells. Each fractio lasts a set amout of time. The key performace metric is the time elapsed from decisio to treat (the date of the cosultatio i which the patiet ad cliicia agree the treatmet pla for the first time) ad first treatmet sessio, see RCR (2006) for a detailed explaatio. Ay delay to the start of treatmet or uscheduled iterruptio durig treatmet is likely to adversely affect the mortality rate, Dodwell ad Crelli (2006), RCR (2002). Various compesatory strategies for missed treatmets leads to icreased toxicity, Hedry et al. (1996). Hece appropriate schedulig would improve radiatio cure rates ad miimise complicatios from radiatio. The first recorded study of schedulig withi the related field of radiology ecoutered by the authors was by Lev ad Caltagiroe (1974). It describes a discrete evet simulatio model of patiet flow i Temple Uiversity s Radiology Departmet (USA) writte usig GPSS simulatio laguage with FORTRAN subrouties icorporated to hadle complex decisio processes ad data maipulatio. The first recorded study of schedulig of patiets for radiotherapy treatmets was by Larsso (1993). His schedulig system records waitig lists ad patiet details. However, it uses simple formulae rather tha ay schedulig heuristics. There is a scarcity of papers o schedulig i radiotherapy, though the major cotributio that it ca make has bee recogised by the NHS (2006). Sice the, there has bee a growig eed to automate the maagemet of the resources ivolved. Various iformatio maagemet systems have come ito commo use, such as Ocetra Cliic ( Such systems provide tools to hadle the icreasigly complex equipmet ad treatmet techiques ad attempt to provide some schedulig ad maagemet support. However, oe of them exploits the full stregth of state-of-the-art schedulig ad optimisatio methods. Oe ca draw a aalogy betwee radiotherapy schedulig ad a job shop schedulig problem where patiets, treatmet sessios, ad liacs are the jobs, operatios, ad machies respectively. Patiets follow a route partially dictated by the locatio ad severity of the cacer. Liacs ca be of high ad low eergy ad they are

2 treated as parallel machies. The best approximatios to the measures of performace for radiotherapy treatmets are: weighted umber of tardy jobs ad total weighted tardiess, which measure the umber of patiets that do ot start their treatmet o time, ad the delay with the treatmet, respectively. Traditioally, departmets operate by bookig the patiet as soo as the referral arrives. Therefore, of particular iterest i this research are various dispatchig rules developed for job shop problems. Vepsalaie ad Morto (1987) discussed the effects of various dispatchig rules i a job shop eviromet with the objective to miimise weighted tardiess. They ivestigated the performace of dispatchig rules uder differet loads of the shop floor. Rajedra ad Holthaus (1999) coducted a study of dispatchig rules aimig to miimise percetage of tardy jobs ad variace of tardiess. Ovacik ad Uzsoy (1995) discuss the relative merits of local ad global dispatchig rules i a parallel machie eviromet. This paper cosiders a real-world schedulig of treatmet sessios that is faced by the Nottigham Uiversity Hospitals NHS Trust (UK). It itroduces two differet algorithms that ca be used to schedule treatmets for patiets of differet categories o a daily basis ad details differet criteria that ca be used to assess the performace of those algorithms. The paper is orgaised as follows. I Sectio 2, the radiotherapy problem is described. I Sectio 3, the two algorithms ad criteria are preseted. I Sectio 4, the results from our experimets are preseted. The coclusio ad a discussio of future work are give i Sectio 5. 2 Problem statemet The otatio used i the problem statemet ad throughout the paper is as follows: N P C r d F I I m - umber of patiets eterig the treatmet bookig system o the curret day - patiets, = 1,,N - category of patiet P : A, B,C - release date of the patiet P, i.e. the date whe the patiet ca start his/ her treatmets oce all prior processes are completed - the due date by which the patiet P has to start the treatmet determied by the imum acceptable waitig time for that category of patiets - umber of prescribed treatmet sessios for patiet P - umber of iterruptios to treatmet - imum allowed umber of iterruptios to treatmet for patiet P determied by the category C - eergy of liacs o which patiet P has to be treated o; high, low M H L mh h ml l V Hd V Ld - set of liacs M = {mh h h = 1,,H} {ml l l = 1,,L} - umber of high eergy liacs - umber of low eergy liacs - h = 1,,H high eergy liacs - l = 1,,L low eergy liacs - the capacity of high eergy liacs o day d (d = 1,,7) give as the imum umber of sessios - the capacity of low eergy liacs o day d (d = 1,,7) give as the imum umber of sessios [s, f, m ] F - a vector assiged for each patiet P, = 1,,N, which shows the days for F treatmet sessios, f = 1,,F, for each treatmet sessio of patiet P o liacs of eergy m T U - umber of days betwee the release date ad the due date of the treatmet of patiet P - determies whether patiet P violates the due date of the start of his/ her treatmet We cosider a daily schedulig problem i which N patiets eters the treatmet bookig system which is partially booked with previously scheduled patiets. Each patiet has to be allocated F treatmet sessios, o liacs, of high or low eergy, that is deoted by m. If this m is high (low) the liacs to be cosidered for treatmet are mh h, h = 1,,H (ml l, l = 1,,L ). A patiet caot have two treatmet sessios o the same day. Patiets are of differet categories: (A) emergecy patiets; (B) palliative treatmets for pai alleviatio; ad (C) radical treatmets for curative itet. For each category, the Joit Coucil for Cliical Ocology (JCCO) set the waitig time targets that is measured from the time that the decisio to give radiotherapy was made to the first treatmet sessio. Table 1 shows the waitig time targets for each category of patiets. Table 1: Waitig time targets JCCO category Descriptio Maximum acceptable waitig times A Emergecy 2 days B Palliative 14 days C Radical 28 days Liacs of high or low eergy are treated as parallel machies. Each treatmet o a liac is of fixed duratio: 15 miutes o high eergy liacs ad 12 miutes o low eergy liacs, apart from the first fractio which takes 5 miutes loger (due to patiet iductio). The capacity of each liac measured as the imum umber of sessios per day is calculated based o the workig shifts that are differet o weekdays ad weekeds, ad the duratio of

3 sessios. A slot is reserved at the ed of each weekday o each liac for emergecy cases. I this paper we will assume that each patiet P fiishes all processes prior to treatmet sessios by the release date deoted by r. The radiographer has to book F treatmet sessios described by the vector [s, f, m ] F, f = 1,,F, for each patiet P. The bookig process has to satisfy the followig costraits: 1. The first treatmet sessio s, 1, m of patiet P, = 1,,N has to be set after its release date r ; 2. Palliative (B) ad radical (C) patiets are ot treated o weekeds; 3. Emergecy (A) patiets ca be treated o ay day of the week; 4. Radical (C) patiets do ot start treatmets o Fridays, so that at least two treatmets are give before the first (weeked) iterruptio to treatmet; 5. If the umber of treatmet sessios F is less tha or equal to 5, the treatmet must ot have a iterruptio, i.e. the treatmet must take place i a sigle week i F cotiguous days; 6. If the umber of treatmet sessios F is greater tha 5, patiets ca have a imum of I iterruptios weekdays without treatmet; 7. No two treatmets ca be booked o the same liac at the same time; 8. The capacity of each liac must ot be exceeded; 9. Two sessios of oe patiet s treatmet caot be booked i the same day. The followig criteria are used to evaluate the quality of the booked treatmets. 1. The umber of patiets who do ot meet the waitig time targets: C 1 = U where 1 s,1, m d U 0 otherwise 2. The total legth of waitig time breaches of the patiets: C 2 = T T s 0,1, m where d s,1, m d otherwise 3. The umber of iterruptios of the scheduled treatmets: C 3 = I where I couts for patiet P the umber of adjacet days i the treatmet which are defied as breaks of treatmet process. These deped o the category ad umber of treatmet sessios of the patiet. 3 Methodology The bookig process cosists of two phases: i the first phase patiets are prioritised for bookig, ad i the secod phase all of their required sessios are booked (scheduled). Schedulig is doe separately for liacs of low ad high eergy. Therefore, a separate prioritised list of patiets is maitaied for high eergy ad low eergy liacs. 1. The prioritisatio of patiets: i the first phase patiets P, = 1,,N, are prioritised for bookig. The rule for prioritisatio assigs high priority for all emergecy patiets (category A), while the remaiig patiets are prioritised by their due dates, i.e. Earliest-due-date (EDD) dispatchig rule is used. Without loss of geerality, we ca assume that the patiets are reidexed so that vector [P ] N presets a prioritised list of patiets. 2. Bookig: i the secod phase, the required umber of treatmet sessios, F, are booked for each patiet, startig from the patiets with highest priority. Two algorithms are developed for treatmet bookig: As soo as possible (ASAP) ad Just-i-time (JIT). They are described i more detail below. 3.1 ASAP algorithm ASAP algorithm assigs to each patiet from the prioritised list [P ] N the earliest feasible start date s, 1, m of the first treatmet (feasible i the sese that it does ot violate costraits listed i Sectio 2). The pseudo-code of the ASAP algorithm is give i Figure 1. The first treatmet s, 1, m caot start before prior processes are completed (step 1). Emergecy patiets (category A) are booked o the first available day (step 2). Other patiets have to be treated o weekdays (step 3). Patiets give a course of treatmet of five or less sessios are treated withi oe week without a weeked break; if this is ot possible treatmet start is delayed util the followig week (step 4). Radical patiets (category C) are ot permitted to start o Fridays, so starts are delayed util the Moday (step 5).

4 Oce a feasible startig date is foud, step 6 esures that there exists a feasible schedule for that start date, i.e. that there are liac slots available for all the treatmet sessios with the allowable umber of iterruptios for that category of patiet. If the origial start date caot provide a feasible schedule, the start date is moved forward oe day, ad the availability of liac slots is checked agai. For = 1,,N (1) s, 1, m = r (2) If C = A (emergecy) go to (6) (3) If s,1,m is a weeked day s, 1, m = ext Moday (4) If F 5 If all treatmet sessios caot fit ito oe week s,1,m = ext Moday (5) If C = C (radical) ad s,1,m is Friday s, 1, m = ext Moday (6) Test start date s, 1, m ad book all treatmet sessios I = 0 f = 1 do If capacity of liacs V Hs,1,m ( V Ls,1,m ) of eergy m = high (low) is ot exceeded for day s, f, m s, f+1, m is ext_day ( s, f, m ) f = f + 1 I = I + 1 If I I // the umber of iterruptios is acceptable s, f, m is ext_day ( s, f, m ) // too may iterruptios // ew start date of the treatmet is determied s, 1, m is ext_day ( s, 1, m ) go to (2) util f F retur [s, f, m ] F = 1,,N ext_day ( s, f, m ): If C = A (emergecy) retur ext day of s, f, m retur ext weekday of s, f, m Figure 1: Pseudo-code of the ASAP algorithm 3.2 JIT algorithm JIT algorithm assigs to each patiet from the prioritised list [P ] N the latest feasible start date s, 1, m of the first treatmet. The idea is to cosider firstly the day before the due date d (the due date itself is ot used because of the risk of breakdows o that date) ad to check backwards for a possible start date of treatmets. The pseudo-code of the JIT algorithm is give i Figure 2. For = 1,,N (1) s, 1, m = d - 1 (2) If C = A (emergecy) go to (6) (3) If s, 1, m is a weeked day s, 1, m = Friday before (4) If F 5 If all treatmet sessios caot fit ito oe week s, 1, m = week before o such a day so that all treatmet sessios ca fit i oe week (5) If C = C (radical) ad s, 1, m is Friday If capacity of liacs V Hs,1,m ( V Ls,1,m ) of eergy m = high (low) is exceeded s, 1, m = Wedesday before // treat o Wedesday ad Thursday, see costrait 4 s, 1, m = Thursday before // treat o Thursday ad Friday, see costrait 4 (6) Test start date s, 1, m ad book all treatmet sessios I = 0 f = 1 do If capacity of liacs V Hs,1,m ( V Ls,1,m ) of eergy m = high (low) is ot exceeded for day s, f, m s,f+1,m is ext_day ( s, f, m ) (ext_day is give i Figure 1) f = f + 1 I = I + 1 If I I // the umber of iterruptios is acceptable s, f+1, m is ext_day ( s, f, m ) // too may iterruptios // ew start date of the treatmet is determied s, 1, m is oe_day_before ( s, 1, m ) If s, 1, m < r apply ASAP algorithm to patiet P go to (2) util f F retur [s, f, m ] F = 1,,N oe_day_before ( s, f, m ): If C = A (emergecy) retur previous day of s, f, m retur previous weekday of s, f, m Figure 2: Pseudo-code of the JIT algorithm

5 The first treatmet s, 1, m starts the day before the due date (step 1). Emergecy (A) patiets are booked o the first available day (step 2). Other patiets have to be treated o weekdays (step 3). Patiets give a course of treatmet of five or less treatmet sessios are treated withi oe week without a weeked break; if this is ot possible treatmet start is moved back to the week before (step 4). Radical patiets must have two treatmets before their first weeked break; therefore they start o later tha Thursday (step 5). Oce a feasible startig date is foud, step 6 makes sure that there exists a feasible schedule for that start date, i.e. that there are liac slots available for all the treatmet sessios with the allowable umber of iterruptios for that category of patiet. If the origial start date caot provide a feasible schedule, the start date is moved back oe day, ad the availability of liac slots is checked agai. If o start date ca provide a feasible schedule, the the ASAP algorithm is applied to the patiet. The mai differece betwee the JIT ad the ASAP algorithms is that JIT starts from the latest feasible start date ad books backwards whereas ASAP starts from the earliest feasible start date ad books forwards. 3.3 Objective fuctio Satisfactio grades are itroduced to reflect the satisfactio of the radiographer with the value achieved for each criterio. Satisfactio grades take values from [0, 1] iterval, where 0 ad 1 represet full dissatisfactio ad satisfactio with the achieved criterio value, respectively. The values of the criteria used to evaluate the quality of the booked treatmets are of differet ature. For example, criterio C 1 couts the umber of patiets (the value betwee 1 ad P ) while C 2 is expressed i days. Satisfactio grades eable aggregatio of criteria values ito a sigle measure. I practice it is very difficult to book the treatmets i such a way as to satisfy waitig targets for all the patiets. I order to defie the satisfactio grade of criterio C 1 (measures the umber of patiets who do ot meet the waitig time targets) we itroduce a parameter p which expresses the threshold of the acceptable percetage of the patiets who do ot meet the targets. The satisfactio grades liearly decrease from 1 to 0 o the [0, p] iterval. Parameter p has value 0%, 33%, 53% for emergecy, palliative ad radical patiets, respectively (we used figures from the RCR s (2006) atioal audit to set the values for parameter p). The satisfactio grade of C 1 is calculated as the average of the satisfactio grades for all categories of patiets. The satisfactio grade S 2 of criterio C 2 (the total legth of waitig time breaches of the patiets) is calculated as the average of the patiet satisfactio grades S 2,, =1,,N. The patiet satisfactio grade is defied for each category of patiet; it has value 1 if the patiet meets the waitig time target ad liearly decreases to 0 whe the waitig time is 2, 28 ad 56 days for emergecy, palliative ad radical patiets, respectively. As a illustratio Figure 3 shows the satisfactio grade for radical patiets. The satisfactio grade S 3 of criterio C 3 (the umber of iterruptios) is calculated as the average of the patiet satisfactio grades S 3,, =1,,N that is defied i the followig way: S 3, S 2, radical I Waitig time (days) Figure 3: Satisfactio grade for waitig time of radical patiets I 1 I I I 0 I I 4 Results This sectio reports o the experimets desiged to evaluate the performace of the proposed algorithms. The algorithms were implemeted i the C++ laguage ad executed o a 2.4GHz Petium 4 with 1024MB of RAM. Real-world data about patiet referrals provided by the City Hospital for the period Jauary to March 2006 have bee used i the experimets. The umbers of patiets of differet categories are preseted i Table 2. Table 2: Categories of patiets, i the period Jauary March 2006 Patiets Ja 06 Feb 06 Mar 06 Emergecy Palliative Radical Total Each algorithm was tested uder three differet load coditios: Normal load of the treatmet bookig system; Light load ad Heavy load, which are 5% less ad 5% more (i absolute terms o a week by week basis) tha the Normal load, respectively. The percetages of already booked sessios i the treatmet bookig system whe N

6 patiets arrive for bookig, uder differet loads are give i Table 3. Table 3: Percetage loads of the treatmet bookig system Week Light Normal Heavy > Figure 4 shows the percetage of patiets late uder the Normal load. For all load coditios, o emergecy patiets were late. Comparig the two algorithms, the mai differece is i palliative patiets, amely the JIT algorithm produces betwee 28% to less late palliative patiets tha ASAP. 60% 50% 30% 20% 10% 0% 48% 46% 12% 50% 34% 33% 10% Ja Feb Mar 37% 29% 25% 6% ASAP palliative ASAP radical JIT palliative JIT radical Figure 4: Percetage of late patiets uder the Normal load of the treatmet bookig system Table 4 shows the satisfactio grades for the umber of late patiets (criterio C 1 ) uder the Normal load, achieved by the ASAP ad JIT algorithms. Oly the umber of late palliative patiets is less tha fully satisfactory. Satisfactio grades for criterio C 1 for all patiet categories for all moths (umber of late patiets) were equal to 1 (full satisfactio) for the JIT algorithm. Table 4: Satisfactio grades for C 1 uder the Normal load of the treatmet bookig system Patiets Ja 06 Feb 06 Mar 06 Emergecy Palliative Radical Average Table 5 shows the satisfactio grades for breach legth (criterio C 2 ) uder the Normal load of the treatmet bookig system. Comparig the two algorithms, JIT s satisfactio grades are 0.28 to 0.4 higher for palliative patiets tha ASAP. However, ASAP produces margially higher satisfactio grades for radical patiets. Fially, average satisfactio grades are higher for JIT. Table 5: Satisfactio grades for C 2 uder the Normal load of the treatmet bookig system Patiets Ja 06 Feb 06 Mar 06 Emergecy Palliative Radical Average Figure 5 shows the percetage of late patiets uder the Light load of the treatmet bookig system. Similarly to the Normal load, JIT produces betwee 29% to 30% less late palliative patiets tha ASAP. However ASAP produces up to 11% less late radical patiets tha JIT. 50% 30% 20% 10% 0% 44% 39% 38% 9% 36% 28% 28% 7% Ja Feb Mar 35% 24% 13% 6% ASAP palliative ASAP radical JIT palliative JIT radical Figure 5: Percetage of late patiets uder the Light load of the treatmet bookig system Table 6 shows the satisfactio grades for the umber of late patiets uder the Light load, achieved by both algorithms. Agai, oly the umber of late palliative patiets, uder ASAP, is less tha fully satisfactory. A 5% reductio i load has resulted i a 3% to 27% icrease i satisfactio grade for palliative patiets.

7 Table 6: Satisfactio grades for C 1 uder the Light load of the treatmet bookig system Patiets Ja 06 Feb 06 Mar 06 Emergecy Palliative Radical Average Table 7 shows the satisfactio grades for breach legth (criterio C 2 ) uder the Light load. JIT produces betwee 43% to 48% higher satisfactio grades for palliative patiets tha ASAP, leadig to less variability tha uder the Normal load. ASAP produces margially higher satisfactio grades for radical patiets. Agai, average satisfactio grades are higher for JIT. Table 7: Satisfactio grades for C 2 uder the Light load of the treatmet bookig system Patiets Ja 06 Feb 06 Mar 06 Emergecy Palliative Radical Average Figure 6 shows the percetage of late patiets uder the Heavy load. JIT produces betwee 27% to 41% less late palliative patiets tha ASAP. 60% 50% 30% 20% 10% 0% 51% 49% 46% 19% 55% 39% 14% Ja Feb Mar 43% 35% 33% 14% ASAP palliative ASAP radical JIT palliative JIT radical Figure 6: Percetage of patiets late uder the Heavy load of the treatmet bookig system Table 8 shows the satisfactio grades for the umber of late patiets (criterio C 1 ) uder the Heavy load. Uder the ASAP algorithm, a 5% icrease i load has resulted i a 9% to 11% reductio i satisfactio grade for palliative patiets. Table 8: Satisfactio grades for C 1 uder the Heavy load of the treatmet bookig system Patiets Ja 06 Feb 06 Mar 06 Emergecy Palliative Radical Average Table 9 shows the satisfactio grades for breach legth (criterio C 2 ) uder the Heavy load. JIT produces betwee 48% to 93% higher satisfactio grades for palliative patiets tha ASAP, showig more variability tha uder the Normal load. ASAP produces margially higher satisfactio grades for radical patiets. Table 9: Satisfactio grades for C 2 uder the Heavy load of the treatmet bookig system Patiets Ja 06 Feb 06 Mar 06 Emergecy Palliative Radical Average The satisfactio grades for the umber of iterruptios, preseted i Table 10, show little variatio betwee either loads or algorithms. Table 10: Satisfactio grades for C 3 Load Ja 06 Feb 06 Mar 06 Light Normal Heavy We ca coclude, based o the performed experimets that overall JIT achieves higher satisfactio grades due to its superior performace with palliative patiets. 5 Coclusio ad future work This paper presets two algorithms that ca be used i radiotherapy treatmet bookig; the ASAP algorithm which books forward from the earliest feasible start date ad the JIT algorithm which books backwards from the latest feasible start date. Some practical costraits that arise i the bookig process are described.

8 Our experimets o real-world data provided by the Nottigham Uiversity Hospitals NHS Trust idicate that JIT achieves higher satisfactio grades due to its superior performace with palliative patiets. A iterestig combiatio worth tryig may be the JIT algorithm for palliative patiets ad ASAP algorithm for radical patiets, ad vice versa. I our future research work we will itroduce adjuvat patiets for whom radiotherapy is a additioal treatmet for cacer. The further differetiatio of patiets will take ito cosideratio certai patiet sites ad itroduce high priority cases amog patiets of the same category. The processes prior to treatmet sessios have to be scheduled also. I real-world treatmet bookig there is a eed for rebookig because of did ot atteds (caused by patiets forgettig, beig o holiday ad loss of the patiets) ad short otice cacellatios (due to chages i coditio, disease status ad treatmet pla). This ca lead to the loss of valuable liac slots. It also reduces flexibility to accommodate acute/ urget patiets. Thus future research work will ivestigate the reschedulig of liac slots. Patiet prefereces for liac slots will also be take ito cosideratio. Ackowledgmets The authors would like to thak the Egieerig ad Physics Sciece Research Coucil (EPSRC), UK, for supportig this research (Ref No. EP/C549511/1). We would also like to ackowledge the support of the Nottigham Uiversity Hospitals NHS Trust (UK). Lev, B., ad Caltagiroe, R.J Evaluatio Of Various Schedulig Disciplies By Computer Systems. I Proceedigs of the 7th coferece o Witer simulatio, , Washigto, DC.: Witer Simulatio Coferece. NHS The how to guide: Achievig Cacer Waitig Times. Lodo: The Cacer Services Collaborative Improvemet Partership, NHS. Ovacik, I.M., ad Uzsoy, R Rollig horizo procedures for dyamic parallel machie schedulig with sequece-depedet setup times. Iteratioal Joural of Productio Research 33(11): Rajedra, C., ad Holthaus, O A comparative study of dispatchig rules i dyamic flowshops ad jobshops. Europea Joural of Operatioal Research 116(1): RCR Guidelies for the Maagemet of the Uscheduled Iterruptio or Prologatio of a Radical Course of Radiotherapy. 2 d editio. Lodo: Royal College of Radiologists. RCR Re-audit of radiotherapy waitig times Lodo: Royal College of Radiologists. Vepsalaie, A.P.J., ad Morto, T Priority rules for job shops with weighted tardiess cost. Maagemet Sciece 33(8): Refereces Delaey, G.P.; Jacob, S.; Featherstoe, C.; ad Barto, M.B Radiotherapy i cacer care: estimatig optimal utilisatio from a review of evidece-based cliical guidelies. Liverpool Hospital, Sydey: Collaboratio for Cacer Outcomes Research ad Evaluatio (CCORE). Dodwell, D., ad Crelli, A Waitig for radiotherapy. BMJ 332(7533): Hedry, J.H.; Betze, S.M.; Dale, R.G.; Fowler, J.F.; Wheldo, T.E.; Joes, B.; Muro, A.J.; Slevi, N.J.; ad Robertso, A.G A modelled compariso of the effects of usig differet ways to compesate for missed treatmet days i radiotherapy. Cliical Ocology 8(5): Larsso, S.N Radiotherapy patiet schedulig usig a desktop persoal computer. Cliical Ocology 5(2):