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Fantasy OR: Sangraal (CP and MIP models) Description The Sangraal problem is an example of a mathematical problem
embedded in a computer fantasy game. The description of the problem and the mathematical model introduced below draw on a publication by M.Chlond: M.J. Chlond, Fantasy OR,
INFORMS Transactions on Education, Vol. 4, No. 3, 2004.
http://ite.pubs.informs.org/Vol4No3/Chlond/ When the Sangraal (Holy Grail) is almost won the hero arrives at a castle where he finds 8 imprisoned knights. He is facing the task to bring the largest possible number of knights for the arrival of the Sangraal in twenty minutes' time. The time required for freeing a knight depends on his state of binding. A freed knight then needs a given amount of time to wash and recover himself physically. For every knight, the durations of freeing and preparing are given. The problem of deciding in which order to free the knights is a standard scheduling problem, or to be more precise the problem of sequencing a set of disjunctive tasks. Typical objective functions in scheduling are to minimize the completion time of the last task (the so-called makespan) or the average completion time of all tasks. The objective to maximize the number of knights who are ready by a given time makes the problem slightly more challenging since we need to introduce additional variables for counting the knights who are ready on time. - The MIP model (sangraal.mos) defines binary decision variables x(k,j) indicating whether knight k is the j'th knight to be freed. A formulation alternative uses indicator constraints (sangraalind.mos).
- The CP model (sangraal_ka.mos or sangraal2_ka.mos) uses the notion of `tasks' with fixed durations and variable start times. These tasks need to be scheduled subject to precedence relations (freeing before preparing) and the disjunctive use of a resource (the hero's time).
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sangraal_ka.mos (!**************************************************************** CP example problems =================== file sangraal_ka.mos ```````````````````` Sangraal scheduling problem. When the Sangraal (Holy Grail) is almost won the hero arrives at a castle where he finds 8 imprisoned knights. He is facing the task to bring the largest possible number of knights for the arrival of the Sangraal in twenty minutes' time. The time required for freeing a knight depends on his state of binding. A freed knight then needs a given amount of time to wash and recover himself physically. *** This model cannot be run with a Community Licence for the provided data instance *** (c) 2008 Fair Isaac Corporation author: S. Heipcke, 2005, rev. Jan. 2018 *****************************************************************!) model "sangraal (CP)" uses "kalis" parameters K = 8 end-parameters forward public procedure print_solution setparam("kalis_default_lb", 0) declarations KNIGHTS = 1..K NAMES: array(KNIGHTS) of string ! Knights' names FREE, PREP: array(KNIGHTS) of integer ! Durations of freeing/preparing startF: array(KNIGHTS) of cpvar ! Start of freeing each knight startP: array(KNIGHTS) of cpvar ! Start of preparing each knight ontime: array(KNIGHTS) of cpvar ! ontime(i)=1 if knight i finished ! within 20 minutes, 0 otherwise Disj: array(range) of cpctr ! Disjunction betw. freeing op.s totalFreed,freedLate: cpvar ! Objective function variables Strategy: array(range) of cpbranching ! Branching strategy end-declarations NAMES:: ["Agravain", "Bors", "Caradoc", "Dagonet", "Ector", "Feirefiz", "Gareth", "Harry"] FREE :: [1, 1, 2,2, 3, 4, 5,6] PREP :: [15,5,15,5,10,15,10,5] MAXT:= sum(i in KNIGHTS) FREE(i) + max(i in KNIGHTS) PREP(i) ! Define binary variables forall(i in KNIGHTS) do ontime(i) <= 1 startF(i) <= MAXT startP(i) <= MAXT end-do ! Every knight must be freed before he can prepare himself forall(i in KNIGHTS) startF(i) + FREE(i) <= startP(i) ! Scheduling freeing operations (all disjunctive, i.e., one at a time) ct:=1 forall(i,j in KNIGHTS | i<j) do Disj(ct):= startF(i) + FREE(i) <= startF(j) or startF(j) + FREE(j) <= startF(i) Disj(ct) ! Post the constraint ct+=1 end-do ! ontime(i) = 1 if knight i is freed and prepared within 20 minutes forall(i in KNIGHTS) equiv( ontime(i)=1, startP(i)+PREP(i) <= 20 ) ! Maximize number of positions finished within 20 minutes totalFreed = sum(i in KNIGHTS) ontime(i) freedLate = sum(i in KNIGHTS) (1-ontime(i)) ! Define an enumeration strategy Strategy(1):= assign_and_forbid(KALIS_LARGEST_MAX, KALIS_MAX_TO_MIN, ontime) cp_set_branching(Strategy) ! Uncomment the followng line to use automatic linear relaxation ! setparam("kalis_auto_relax",true) ! Solve the problem cp_set_solution_callback("print_solution") ! if not cp_minimize(freedLate) then if not cp_maximize(totalFreed) then writeln("Problem is infeasible") exit(1) end-if cp_show_stats !******************************************************************** ! Solution printing public procedure print_solution writeln(" Freed Ready <=20 min") forall(i in KNIGHTS) writeln(strfmt(NAMES(i), -12), strfmt(getsol(startF(i)+FREE(i)),2), " ", strfmt(getsol(startP(i)+PREP(i)),2), " ", if(getsol(ontime(i))=1,"yes","no")) writeln("Number of knights freed on time: ", getsol(totalFreed), "\n") end-procedure end-model | |||||||||||||||||||

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