crew scheduling prob

8/7/2019 Crew scheduling prob

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a b l c to answer a numbcr o f impor t an t "wha t ir' ques t ions , i nc lud ingln~cgcr 1.incar f'rogramrrting

m Wh:11 i f a s t r ike knocks out a kcy faci l i ty '?

What i f a cer ta in regional produci goes nat ional?

W h a t i f increas ing fuel costs or changes i n fcdcral regulatory policicss ignif icant ly a lter the curren t ra te s t r uc tur es of t l lc t ruck or ra i l carr iers usedto t ranspor t the products?

Development of the mode l an d i t s a s soc i a t ed da t a r equ i red abou t s ix ca l enda rmon ths o f w ork by a p ro jec t t eam invo lv ing pe r sonne l f rom accoun t ing , da t a p ro -ces sing , marke t ing , managemen t s c i ence , p roduc t ion , an d o the r a r ea s o f t he com-

pany. Add i t iona l expenses r e su lt ed f rom t he r epea ted com pute r so lu tions o f t he ylarge-scale IL P u si ng a lt e r n a ti v e d a t a . F o r t u n a t e l y, ho w ev e r, t h e e h r t a n d ;I,expense were fully justified by an nu a l cos t s av ings a t t r i bu ted to t he model o f over

.;

S 1,000,000.

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9.11 A C R E W SCHEDULING PROBLEM

Th ere ex i s t man y app l ica tions o f i n t ege r l i nea r p rog ra mm ing t o deci s ion p rob lemsa r i s ing in t he a i r l ine indus try. Su ch app l i ca t ions inc lude th e sel ect ion o f t he m ix o fa i r c r a f t t y p e s k g . , 727, 747, a n d 767) th at wi l l comprise a n a ir l ine' s f lee t , the .rou t ing o f a i r c r a f t t o mee t t he a i r li ne ' s R igh t s chedu le , and the schedu l ing o f c r ews ,tto mee t an a i rl ine 's f l ight schedule . W e wil l i l lus t ra t e th e la t ter appl icat ion by con- ':/ s ider in g t he following scenar io:

Walt Dumbo is responsible for the routing of aircraft and the scheduling of flightcrews for Flying Elephant Airlines (FEA), an air cargo carrier serving three cities( A ,B , an d C). FEA operates 18 Rights daily: a morning, an afternoon, and an eveningflight in each direction between each p air of cities. Ta ble 9.13 sum mar izes FEA 'sflight schedule. (For simplicity. we a ssu me all flights are t\ceo hours long.) T o meetFEA's schedule. Walt must solve two problems:

1. the optimal routing of aircra ft, andt

2. the optimal scheduling of flight crews,

where e ach flight requires a crew of two pilots. As is typicali n the airline industry,Walt solves these problems separately because aircraft requirements (e.g., for mainte-nance) are quite different than crew req uirements (e.g.. for rest). In general, then, acrew does not spend its entire day with the same aircraft.

Table 9.14 displays Walt's solution to the aircraft routing problem. To meet the 18scheduled flights, FEA will use six aircraft routes, each consisting of three nights.(You may wish to verify that each route permits at least two hours between eachlanding and subsequent takeoff in order to unload cargo, refuel the aircraft, performnecessary maintenan ce, and load new cargo.) Altho ugh each of the six aircra ft routesoriginates in the morning at a city direrent from the one at which i t terminates atnight, two aircraft always termin ate the da y in each c ity. Conscquently, the same sixroutes may be repeated daily, although a particular aircraf~wil l not always fly the

same route.Unfortunetely. Walt is biiable to solve FEA's crcw scl~cduiingprcible~n1)). :~ s s ign~ngonc crcw to each of the si x aircraf t routes. Sucli a crcw \chcdulc would violi~tcalabor agrccnlcnl with the pilots' union stipulating 111i1t ;I crew 1 1 1 ~ ~ 1i~I\v:~ys\ ; t ; \ r [ ; ~ n dcnd its day nt thc same city ( A , H . or C ) , herea f~c rrcfcrrcd to a s t l l c crew's hrr.re.


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T A B L E 9.13 F E A ' s Flight S c h e d u l e

TABLE 9.14 FEA's Aircraft Routing

Flight

N u m b e rI A BI A C

M o r n i n g 1B Adepartures I B CI C AI C B

2 A B2 A C

Afte rnoon 2BAdepartures 2BC

2 C A2 C B

3 A B3 A C

Evening 3 B Adepartures 3 BC

3 C A

3 C B

IRoute Originates TerminatesRig h t SequenceN u m b e r a t C i t y a t C i t y

Thus, the labor agreement bars a crew fro m sleeping overnight in a city other than itsbase.

A s a preliminary step to solving FEA's crcw scheduling problem, Walt has generated30 potential daily schedules an individua l crew mi gh t rollow. I n airline parlance, eachor these is called a rorarion. Each rotation calls Tor a crew to fly two o r more Rightsduring the day, always starting and ending the d ay at the same base. Serving as acrew for a f l ight involves one-hall hour o l prefl ight duty, two hours of infl ight duty.and one-halr hour of postf l ight duty. O n d u ty t im e costs FE A S40 per hour per two-member crew. Upon completion or postflight dut y, a crew must have a rest period o lone hour berore its next preflight duty . Unde r the labor agreement with the pilots'union, this rest period is regarded as on-duty time and, therefore, costs FEA S40 pcrhour per two-member crew. Ti me i n excess or this one-hour rest period that precedesthe crew's next preflight du ty is referred l o as layover rin;e. Layover time costs FEAS20 per hour per IWO-member crew. Som e rotati ons involve the common ind ust rypractice or deadheading. whereby a crew is reposir~oned lor larcr usc by Ilying ahpassengers on one or FE A's flights.' Ti me spent deadheading is regarded as layovertime and, there lore, costs FE A %20 per hour per two-m enibcr crcw. Table 9.15 sum-

Departure

F r o m Ti m e

A 10:W a.m.A Il : 00 a.m.

B 7:00 a.m.B 6:00 a.m.C 8:00 a.m.C 9:00 a.m.

A 5:00 p.m.A 12 00 p .m.B 2:00 p.m.B 3 9 0 p. m.C 4:00 p.m.C 1:00 p.m.

A 6:00 p.m.A 11:OO p.m.B 9 9 0 p.m .B 7:00 p.m.C 8:00 p.m.

C 10:00 p.m.

Arrival

A t T i m e

B 12:00 p.m.C l:00 p.m.

A 9:00 a.m.C 8:00 a.m .A 10:OO a. m.B 11:OO a.m .

B 7:00 p.m.C 2 0 0 p .m .A 4:00 p.m.C 5:00 p.m.A 6:00 p.m.B 3:00 p.m.

B 8:00 p.m .C 1:00 a.m.A 11:OO p.m .C 9:00 p.m.A 10:OO p.m .

B 12:00 a.m.


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TA B L E 9.15 P o t e n t i a l R o t a t i o n s f o r FEA ' s C r c w s

Rotal ion On -du tyTi mc t LayovcrTimcS To ta lC o :Nu mb er Flight Sequence (hour s) (hours) (%I-

I IAB-2BC43CA I I 2 48 02 I BA-2AC-3CB I I 7 5803 ICB-2BA-3AC I I 6 5604 (I BA)-IAB-2BA-3AB I I 2.5 4905 I AB-2BA-3AB-(3BA) I I 2.5 4906 (1CA)- IAC-2CA-~AC 1 1 6.5 5707 I BC-ZCB-3BC-(3CB) 11 7.5 5908 ICA-ZAC-3CA-(3AC) I I 6.5 5709 (1 BC)-ICB-2BC-3CB I I 7.5 590

:O IAB-2BA 7 0 280* I I IAC-2CA 7 I 300

I2 IBA-2AB 7 6 40013 IBC-2CB 7 3 34014 ICA-2AC 7 0 280IS ICB-2BC 7 2 3 2016 2AB-3BA 7 0 28017 2AC"3CA 7 4 36018 2BA-3AB 7 0 28019 2BC-3CB 7 3 34 020 2CA-3AC 7 3 340

21 2CB-3BC 7 2 3 2022 (IAB)-ZBC-3CA 7 5.5 39 023 I A B-(2 BC )- 3C A . 7 6 4 0024 IAB-2B C-(3CA ) 7 5.5 39025 ( I BA)-2AC-3CB 7 10.5 49026 I BA-(ZAC)-3CB 7 I I 50027 I BA-2AC-(3CB) 7 10.5 49028 (ICB)-ZBA-3AC 7 9.5 47029 I C B - ( ~ B A ) - ~ A C 7 1 0 4 8 030 ICB-2BA-(3AC) 7 9.5 470

t Includes rest periods.$ Includes deadheading.

Corresponding decision variable equals I in the ILP's optimal solution.

m a r i z e s t h e 30 p o t e n t i a l c r e w r o t a t i o n s , w h e r e p a r e n t h e s e s a r o u n d a f l i g h t n u m b e ri n d i c a t e t h e c r e w i s d e a d h e a d i n g o n t h e f li gh t. (For t h e m o m e n t , i g n o r e t h e a s t e r is k st h a t p r e c e d e s o m e r o ta t io n s .) F o r e x a m p l e , r o t a t i o n 29 ca l l s fo r the fo l lowing c rews c h e d u l e a n d c os t:

Timc Period-8:30 a.m. - 9:00 a.m.9:00 a.m. - 1 1:00 a.m.

11:OO a.m. - 1 1.30 a.m.11.30 a.m. - 12:30 p.m.12:30 p.m. - 29 0 p .m.

2:00 p.m. - 4:00 p.m.4:00 p.m. - 10:30 p.m.

10:30 p.m. - Il:00 p.m.1 l:00 p.m. - I:00 a.m.

1:00 a.m. - I:3O a.m.

--

H ow S p e n ~

Preflight duty on Fl~ght1CBInflight duty on R ight IC BPostflight duty on Flight 1CBRcst periodLayovcr timeDeadheading on Flight 2BALayovcr timePreflight duty on Flight 3ACInflight duty on Flight 3ACPostflight duty on Flight 3AC

Cost ($17----

X 4 0 =2 x 4 0 =

f x 4 0 =1 X 4 0 =

I f x 2 0 =2 x 20 =

6 f x 2 0 =f x 4 0 =2 X 4 0 =f x 4 0 =

--Total Cost =' Although our example docs not allow for thc possibility. i n practicc an airline may permit dcadhc.~dingon thc schcdulcd flights of another airline.


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429Inlcgcr 1.1ncar Programming

(You may wish to verify the data forp6mc of the other rotations.) Rotations notappearing in Tablc 9.1 5 arc absent bccausc thcy arc infeasible, either becausc they arcphysicall v impo ssible or becausc thcy violate rcdcral regulations, company poli cy, orthe labor agrecmcnr \r,~rhthe p~lots 'unlon.

Walt must now decide which of the 30 rotations to assign crews to i n order to nlecrFEA's eighteen-flight schedulea t minimal cost.

T o fo rm ula te h is dec is ion p rob lem as an ILP, Walt first defines 30 binaryvariables XI , X Z , . . . , X M , where

I denotes that a crew is assigned to rotationJ ,0 denotes that a crew is not assigned to rotationJ

Wa lt th en form ulate s FEA's crew schedul ing problem as the fol lowing pure-binaryI LP:

Minimize 4801, +580xl +SMlx, +490x4 4-4Wx 5 +570x6 +590 ~, +570x1 + 5 9 h v +280xI0+30 0x ,, +4001,, +340x1, + 280x14 + 3ZOxl, +280x16 +360x11 +280x11 +340x19 + 340x1,+ 320x,, + 390x,, +400x1, + 390x,, +490x1, +5W x16 +490x1, +470 rn +480x, +470x,

subjcct to, X , + x 4 + X J + Xlp + 111 + XII = I (IAB)x, + X , I = I ( IAC)XI + X l i + XI, + I 1 7 = 1 ( 1 8 ~ )XI i XI ] = I ( IBC)XI + X ~ 4 = 1 (!CAIx + x +XI, +In + 1s = I (ICB)x i] + I 1 6 = I (2AB)x + x + ~ 1 4 + X l 7 +.XU + x17 = I (2AC), + x + x, + x,, + x + x,, + r, = I ( 2 B dx , + x, + x + x,, + 1 = I (2BC)x, +XI , + x m = I (2CA)x, + X I , + ~ 1 1 = I (2CB)x4 +x 5 +XI, = I (3AB)x, + x 6 + X, f x,, +I, = I (3AC)X1 6 = I (3BA)

1 7 + XZ I = I (3BC)x, + x, I + x,, + XI, = I ( ~ C A )XI + xe + 1 1 9 + XIJ + X l6 = I (3CB)

an d x , . r l . . . . . x, ,=OOrl .

T h e 1 8 c on s t r a i n ts e n s u r e t h a t e a c h o r t h e18 flights has exactly one crew on duty .T o see this ; observe th at there is one constraint corresponding to each of t he 18dai ly f l ights , and tha t the lef t -hand s ide of the constraint corresponding to a pa r t ic-ular f l ight con tains only t he binary var iables corresponding to the rotations whosecrews would be on-d u ty fo r tha t f ligh t. Requiring these binary variables to sum toexact ly 1 ensur es th at the f l ight will have exact ly one crew on-duty. For example,requir ing in the f i rs t constraint thatX I , x 4 , XS , XI ,-,. X23, and x24 sum to exac t ly 1ensures t ha t F l ight I A B will have exact ly on e crew on duty. the crew assigned toei ther rotat ion 1 , 4, 5, 10, 23, and 24.. Observe that ~ 2 2does not appear in thef i rst const raint because a crew assigned to rotat ion 22 would be deadhe ading onFl igh t IAB.

Wh en Wa lt solves the LP relaxat ion or his ILP. he receives a pleasantsurpr ise . All decis ion var iables assume integer values! Thus, Wa lt is for t unateenou gh t o have found his ILP's opt imal solut ion by simply solving i ts L Prelax ation -som ethin g tha t. while still rare , occurs with more rrequcncy Tor airli ncc rew schedu l ing p rob lem s than fo r the "average" I L P. T h e a s ~ e r i s k s a t t h ccx t remc l e f t o f Tab lc 9. IS indicatc thc cight rotations t l i ; l t corrcspond to the onlybinary var iables assuming the value of I i n t h e I L P ' s o p t i ~ ~ l a lsolution; tlic optiliialobject ive vnluc is % 31 90. As indicated by the aster isks . thc opt imal so lu ~i onc:ills


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for the assignment of eight crews, one to each of the rotations I,7 , 1 I, 14. 16, 18, '.76 2 n d 2 9 . Conseauently. FEA will employ eight crews: three crewswil l be basedi n city A and fly rotations I, I I , and 16; three crews will be based i n city B andwill Ry rotations7, 18, and 26 ; and two crews will be based in cityC and fl y rota-tions 14 an d 29. Observe that th e sa me crew need not fly the sa m e rotation each

day. For example, the two crews based in cityC

can alternate ona

daily basis,,

between rotations 14 and29 , there by equalizing the workloads between cr ews.FEA's crew scheduling problem is a simplification of an actual crew schedul-

ing problem that arose at the Flying Tiger Line( I T L ) , a cargo airl ine headquar-tered i n Los ~ n ~ e l e s . ~A typical crew scheduling problem at FTL was morecorn-plex than our example for several reasons, including

'*d

A weekly (rather than a daily) planning horizon. I-

A complex Right schedule. '1

The possibility of a crew overnighting at a city other than its base.= The possibility of deadheading on the scheduled flights of several passenger

airlines.

Complex federal regulations, company policies, and labor agreements with the

pilots' union governing such thing s as a crew's maximum h ours on- duty a ndminimum hours of rest.

As a result of these complexities in FTL's problem, the integer linear programming,formulation of a typical crew scheduling problem had78 structural constraints(corresponding to flights) and over 14,000 decision variables (corresponding to rota-3t ions). To solve such a la rge sca leILP, WL had to use a special-purpose algo-:.Irithm . Even so, obtaining th e ILP's optimal solution required over 3 0 minutes of?C PU time on a largeIBM com pute r. Fortunately, the large com pute r bill proved

'

to be a wise investment. FTL's imp lem enta tion of the optimal solution resu lted in'

ann ual savings of over $300,000. Furtherm ore, FTL enjoyed several indirectbenefits, such as the ability to quickly evaluate (during negotiations) the impact ofproposed changes in the pilots' labor contract.

T he crew scheduling problems of th e Flying Tiger Line are smalli n compar-

ison with those for a large passeng er airline. It would not be unusual for one of;those to result in an IL P having ov er 300 0 constraints and over 15 ,000 variables.,Consequently, a large passenger airline will usually not attempt to optimally solve':the resulting I LP but will instead s eek a n acceptable close-to-optimal solution usinga special-purpose heuristic. Alternativ ely, the airline may seek an acc epta bleclose-to-optimal solution by decomposing the crew scheduling problem into severalsubproblem s (e.g., one subproble m for each aircraft type) and then optimallysolving each subproblem.

T he crew scheduling problem is just one of many diverse applications tha tgive rise to three important classes of ILPs: thesel parli lioning p rob len ~, th e s e l -covering problem, and the se t -pack ing problem. Th e context in which the s et par -titioning problem arises can be broadly described as follows:

A set of m requirements (e.g., the flights in the crew scheduling problem) must be

satisfied. There existn alternative activities (e.g., [he rotationsin the crew schcdulingproblem). eachof which saiisfies at a known cost a direrent subset of the rcqulr e-ments. The objective is to choose the minim alcost combinationof activities t h a t willsatisfy each requirement exactly once.thcreby partitioning (dividing) these t ofrequirements among the various activities.

8 R . E. Marsten.M . R Muller.a n d C. L. Kil l ion. "Crcw P l ; ~ n n i n ga1 Flying T~ger- A Succe.;sTul 4pp l i .caclon or I n t e ge r Progran~ming , "Monogemen! S c i c n c e . Vol. 25. No . 12 (Dccembcr 1979). 1 175-1 183


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43 1Imcgcr Linear Programming

Upon letting x, denote a binary variable that equals I i f and only if activity jis chosen, the set-partitioning problem can be expressed as the following pure-billdry ItP:

Minimize ~ 1 x 1+ ~ 2 x 2 + . . - + C.X.subject to o i l x l + a i l x l + - - - + a,.x. = I for i = I , 2, ..., rn

and xj = Oor 1 for j = 1, 2 , ..., n ,

where cj denotes the cost of alternative j and where the coefficient a i j equals 1 i factivity j satisfies requirement i and equals 0 otherwise. As with Walt's formula-tion of FEA's crew scheduling problem, each of the rn structural constraints ensuresthat the corresponding requirement will be satisfied exactly once.

The set-covering problem is identical to the set-partitioning problem exceptthe relationships in all structural constraints are 2 rather than =. The presenceof the 2 constraints permits a requirement to be "oversatisfied"-that is, satisfiedby more than one activity. Exercise - involves a simplification of an actual set-covering problem.9 Th e problem involved the selection (from 112 potential sites) ofa combination OF sites for t he location of fire companies to serve 246 geographicalsubdivisions of the city of Denver. In contrast to FEA's crew scheduling problem,where each flighi requires exactly one crew, it is permissible (and even d'esirable)

for a geographical subdivision to be covered (served) by more tha n one fire com-pany..

set-packing problems differ from set-partitioning problems in that the rela-tionships in the structural constraints are all of the < type an d th e objective func-tion involves maximization.

As we have seen, th e formulation of a set-partitioning, a set-covering, or aset-packing problem results in an integer linear program. However, as illustratedby FEA's crew scheduling problem, computational experience by practitionerssuggests that the optimal solution of the LP relaxation results in an all-integer solu-tion with a frequency greater than that for th e "average" ILP. Even if the LP-relaxation's optimal kolution has a fractional value for some decision variables, arounding strategy can often produce an acceptable close-to-optimal solution.

9.12 CLUSTER ANALYSIS FOR MARKETING RESEARCH

A fundamental tool in marketing research (and, indeed, many other diversc discip-lines such as biology and linguistics) is known as cluster analysis. Cluster analysisis a generic label applied to a set of techniques that partit ion a set of objects intodistinct clusters (groups), where two objects i n the same cluster have a high degreeof similarity and two objects in different clusters have a low degree of similarity.Marketing applications of cluster analysis include the clustering of customers intodifferent market segments.

A wide variety of cluster-analysis techniques exist, each dimering from theothers in the way it measures the similarity between two objects and/or in the way

it uses the measure of similarity to form the clusters. Research by Mulvey andCrowder established that integer linear programming deserved to be on the list ofcluster-analysis techniques.I0 To illustrate, we will consider the following scenario:

'O J . M . Mulvey and H. P Crowder. "Cluster Analysis : An Appl icat ion of Langrangian Rel~ xat ion ."Managerncnr Science. Vol . 25 . N o . 4 (April 1979). 329-340.' D . K. Plane and T. E. Hendrick, "Mathematical Prograrnm~n g and the Location of Fire Cornpanics forthe Denver Fire Departrnenl." Opcra~ ions Research. Vol. 25. No. 4 (July-August 1977) . 563-578.

crew scheduling prob

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