Filling a theatre in times of corona
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Filling a theatre in times of corona
Danny Blom, Rudi Pendavingh, Frits Spieksma
Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands,[email protected], [email protected], [email protected],
In this paper, we introduce an optimization problem posed by the Music Building Eindhoven (MBE) todeal with the economical consequences of the COVID-19 pandemic for theatre halls. We propose a modelfor maximizing the number of guests in a theatre hall that respects social distancing rules, and is based ontrapezoid packings. Computational results show that up to 40% of the normal capacity can be used for asingle show setting, and up to 70 % in case artists opt for two consecutive performances per evening.
Key words : Integer programming; COVID-19
1. Prologue
All around the world, the corona-crisis has hit the cultural sector hard. Festivals are can-celled, orchestra’s are at the brink of bankruptcy, choirs have stopped performing, andtheatres are struggling to survive. Different countries, or regions, have imposed differentrules in an attempt to stop the spread of the virus. We do not aim here to overview theprecise (dynamic!) contents of all these rules, and their impact on the cultural sector;a number of descriptions of such rules and their impact can be found on governmentalwebsites (e.g. [Australia (2020)], [Germany (2020)], [Sweden (2020)], [UK (2020)] and[USA (2020)]), and in other contributions (such as [Jacobs (2020)]).The situation in the Netherlands is not atypical from other countries or regions. StartingMarch 12, 2020 until June 1, 2020 all performances were cancelled or suspended. From June1 onwards, a relaxation of the rules has allowed performances with at most thirty guests,as long as non-family members were seated at least 1.5 meters apart. The upper bound onthe number of guests for indoor performances was eventually increased to one hundred onJuly 1, 2020; a description of the current rules can be found at [The Netherlands (2020)]. a r X i v : . [ c s . A I] S e p lom, Pendavingh, and Spieksma: Filling a theatre in times of corona Article submitted to
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Clearly, these rules have a dramatic impact on the operation of any theatre, and despitegovernmental efforts theatres are struggling to survive. As a consequence, many employeesin this sector risk losing their jobs.Indeed, for many theatres, the challenge is to find a way to welcome their guests whilesatisfying the distance rules, and still be commercially viable. Many creative efforts haveresulted in a number of ideas that are being experimented with (see, for example, the use ofa so-called nebulizer device, see [Berliner Ensemble (2020)]). Here, we focus on the questionto what extent large audiences can still be accommodated in a theatre when distance rulesmust be satisfied. We describe a mathematical model that, given the layout of the seats ina theatre and the distribution of the demand, computes a safe seating arrangement thatattains the maximum occupation of the theatre.The Music Building Eindhoven (MBE), located in the city of Eindhoven in the Nether-lands, features a “Grand Room” (1250 seats) and a “Small Room” (400 seats). This theatrehas served as a motivation for this study, and all our computational efforts are based onits two rooms. Our findings have been implemented by the MBE, allowing them to remainopen.In Section 2 we give a precise problem description, and in Section 3 we phrase theproblem in terms of packing of trapezoids. In Section 4 we give our model, and in Section 5we show solutions of the model on instances coming from the MBE. Upper bounds arediscussed in Section 6, and we conclude in Section 7.
2. Problem description
Here, we describe the crucial ingredients of our problem. Seats, distances and forbiddenzones are discussed in Section 2.1, and target profiles are explained in Section 2.2. Thisallows us to arrive at a problem statement given in Section 2.3.
When a theatre wants to offer a corona-proof experience to its customers, a few constraintsneed to be taken into account. Obviously, safety is of utmost importance and therefore,the subset of seats that could be used for reservations needs to be chosen according to theguidelines provided by the government. We realize that these guidelines vary for differentcountries. However, a common denominator between different countries is that members ofdistinct families (or bubbles ) should keep a prespecified distance from each other to prevent lom, Pendavingh, and Spieksma:
Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) the spread of the COVID-19 virus. In the Netherlands, two people should be seated atleast 1.5 meter apart (unless they are members from the same household), as establishedby the Dutch Government [The Netherlands (2020)].Figure 1 shows a sketch of four consecutive seats, viewed from the front and from above,and the corresponding interseat distances of seats in the MBE. Figure 1 Front and upper view of four seats in MBE, with corresponding measures. The height difference ofconsecutive rows is 0.31m, due to acoustics and visibility reasons. .
51 m . m . m Due to the exception of the distance rules for family members, another relevant factoris the size t ∈ T of a family, with T ⊂ Z + the set of allowed family sizes. In particular, wewill call a family of size 1 a singleton , of size 2 a pair , of size 3 a triple , and a family of size4 a quad . Guests from the same household are allowed to sit next to each other, within the1.5m bound; in fact, we assume that a family of size t occupies t consecutive seats. Basedon the distances in Figure 1, it follows that whenever a certain seat is occupied, there isa “ring” of seats around it that are forbidden for use by a member of another family. Theforbidden ring corresponding to a pair is depicted in red in Figure 2.Consider a theatre, and let S denote the set of seats of the theatre. Each seat is specifiedby its row r , and its position s in row r . Formally, a seat is a pair of integers ( r, s ) ∈ Z × Z and the set of seats is a collection S ⊆ Z × Z . Typically, the seats in each row arenumbered starting with s = 1 , , , . . . , so that the relative position of the seats ( r, s ) and( r (cid:48) , s ) will depend on where rows r and r (cid:48) start. For the description of our model it will beconvenient to assume that the seats in each row are numbered such that for each s ∈ Z ,the seats { ( r, s ) ∈ S : r ∈ Z } are in a straight line. Figure 2 also illustrates this conventionfor { ( r, ∈ S : r ∈ Z } . lom, Pendavingh, and Spieksma: Filling a theatre in times of corona Article submitted to
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Again, in a typical theatre (such as MBE), consecutive rows are shifted relative to eachother (for reasons of visibility), so that the four seats ( r + 1 , s − , ( r + 1 , s ) , ( r − , s ) , ( r − , s + 1) form the corners of a rectangle with center ( r, s ). This seat renumbering methodis illustrated in Figure 2 for ( r, s ) = (3 , a , andthe spacing of consecutive rows is denoted by b , then the distance between the centers ofseats ( r, s ) and ( r (cid:48) , s (cid:48) ) is d (( r, s ) , ( r (cid:48) , s (cid:48) )) = (cid:115)(cid:18) ( s − s (cid:48) + 12 ( r − r (cid:48) ) (cid:19) a + ( r − r (cid:48) ) b . Assuming that members of different families may not be seated within distance c , the‘forbidden zone’ for members of other families surrounding a person in seat (0 ,
0) is F := { ( r, s ) : d (( r, s ) , (0 , < c } . Taking distances a = 0 . m and b = 0 . m as in Figure 1, and a forbidden distance of c = 1 . m , a calculation reveals that F is a collection of 13 seats: the occupied seat itself,two seats on each side in in the same row, and four nearby seats in each adjacent row. Ingeneral, the forbidden zone for a family of size t consists of 13 + 2( t −
1) = 2 t + 11 seats.In Figure 2, we indeed see that the forbidden zone of a pair consists of 15 seats, as thereis an extra forbidden seat in both adjacent rows. Figure 2 The red seats cannot be occupied whenever the green seats are occupied by a pair. The seats with seatnumber 3 in consecutive rows are situated on a straight line. This example is based on the measuresof the MBE.
The forbidden zone surrounding each other seat ( r, s ) is just a shifted version of thisforbidden zone around (0 , F r,s := { ( r, s ) } + F = { ( r + r (cid:48) , s + s (cid:48) ) : ( r (cid:48) , s (cid:48) ) ∈ F } . lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) Members of the same family are allowed to be in each other’s forbidden zone, but theunion of their forbidden zones is forbidden for members of other families. A family of size t located at ( r, s ) will occupy the seats S r,s,t := { ( r, s + i ) : i = 0 , . . . , t − } and will have aforbidden zone F r,s,t := S r,s,t + F = { ( r + r (cid:48) , s + s (cid:48) + i ) : i = 0 , . . . , t − , ( r (cid:48) , s (cid:48) ) ∈ F } . (Notice that we use the Minkowski sum when adding two sets A and B , i.e. A + B = { a + b : a ∈ A, b ∈ B } .) A subset A ⊆ S × T is a seating arrangement if each ( r, s, t ) ∈ A indicates a possible location of a family of size t at ( r, s ), i.e. if S r,s,t ⊆ S for each ( r, s, t ) ∈ A and S r,s,t ∩ S r (cid:48) ,s (cid:48) ,t (cid:48) = ∅ for each distinct pair ( r, s, t ) , ( r (cid:48) , s (cid:48) , t (cid:48) ) ∈ A . Definition 1.
A seating arrangement A is safe if S r,s,t ∩ F r (cid:48) ,s (cid:48) ,t (cid:48) = ∅ for each distinct pair ( r, s, t ) , ( r (cid:48) , s (cid:48) , t (cid:48) ) ∈ A .Thus, a seating arrangement A is safe if no member of a family is in the forbidden zone ofanother family.For any seating arrangement A , let n t ( A ) := |{ ( r, s, t ) ∈ A : ( r, s ) ∈ S}| capture the num-ber of families of size t in A , t ∈ T . Definition 2.
The size of a seating arrangement A is (cid:80) t ∈ T t · n t ( A ).Thus, the size of a seating arrangement corresponds to the number of customers present. Apart from providing a safe environment for the audience while enjoying a performance,a theatre needs to consider its booking strategy. In general, multiple factors play a rolewhen deciding upon such a strategy (see Baldin and Bille [Baldin and Bille (2018)], andthe references contained therein). One option is to sell the individual seats (perhaps aftersegmentation into classes) chosen by customers in a first-come first-serve manner. The riskof such a strategy is that customers choose seats that do not lead to a maximum occupancy.Another option is to simply sell tickets, and only reveal very shortly before the start ofthe performance which particular seats are assigned to which individual customers. Thisallows the theatre flexibility to find a maximum occupancy, yet customers might find itunattractive not to be able to choose their specific seats. Without going into details of lom, Pendavingh, and Spieksma:
Filling a theatre in times of corona Article submitted to
INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) the various considerations, we have opted, in collaboration with the MBE for a policythat (i) allows customers to choose their seats, and (ii) uses a so-called target profile totake the size of families visiting the performance into account. Indeed, prior informationon the distribution of the customers over singletons, pairs, triples and quads is valuableinformation and can serve as a proxy for customer behaviour.
Definition 3. A target profile for a seating arrangement A is a vector (cid:126)p = ( p t ) t ∈ T ∈ [0 , T such that (cid:80) t ∈ T p t = 1.Each entry p t indicates a targeted proportion of the reservations corresponding to familiesof size t , i.e. we aim for a seating arrangement A for which n t ( A ) (cid:80) t (cid:48) ∈ T n t (cid:48) ( A ) ≈ p t , ∀ t ∈ T. (1)A target profile can be determined through statistical analysis or machine learning mod-els applied on historical data. We show in Section 2.3 how we formalize this aim. When we use the target profile to proxy customer behaviour as input, we can describe theproblem in the following way:
Problem:
MAXIMUM PROFILED SEATING ARRANGEMENT
Instance: a tuple ( S , T, (cid:126)p, (cid:15) ) consisting of a set of seats S ⊆ Z × Z , a set of allowed familysizes T ⊂ N + , a target profile (cid:126)p = ( p t ) t ∈ T and (cid:15) ∈ (0 , Goal:
Find a safe seating arrangement A of maximum size such that the followingconditions hold: ( p t − (cid:15) ) (cid:88) t (cid:48) ∈ T n t (cid:48) ( A ) ≤ n t ( A ) ≤ ( p t + (cid:15) ) (cid:88) t (cid:48) ∈ T n t (cid:48) ( A ) , ∀ t ∈ T. Notice that there is no unique way to model a condition as provided in Equation (1).One alternative is to consider strict lower bounds on the number n t ( A ) of families of size t .Furthermore, (cid:15) is used as a multiplicative threshold parameter, but one could also use it asan additional parameter instead. In the following section, we describe a nontrivial connec-tion between finding a safe seating arrangement and the problem of packing a maximumweight set of trapezoids. lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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3. Trapezoid packings
We describe in Section 3.1 a nontrivial connection between finding a safe seating arrange-ment and the problem of packing a maximum number of trapezoids in a polygonal shape,and in Section 3.2 we prove a bound on the number of families that fit in a theatre. InSection 3.3 we analyze, as an intermezzo, a theatre with an infinite number of rows havingeach an infinite number of seats, and in Section 3.4 we discuss the occupation density ofseating arrangements in large theatres.
Given the structure of the forbidden zone F , we are able to formulate a model based on trapezoids . The trapezoid based at (0 ,
0) is the collection of seats T := { (0 , − , (0 , , (0 , , (1 , − , (1 , } . The trapezoid based at ( r, s ) then is T r,s := { ( r, s ) } + T = { ( r + u, s + v ) : ( u, v ) ∈ T } , and the trapezoid of a family of size t located at ( r, s ) is T r,s,t := S r,s,t + T = { ( r + u, s + v + i ) : i = 0 , . . . , t − , ( u, v ) ∈ T } . Figure 3 The trapezoid T , , together with its associated forbidden zone F , , (given by red and green seats). The trapezoid T is chosen so that F = T + ( −T ) , (2)where T + ( −T ) := { ( u, v ) − ( u (cid:48) , v (cid:48) ) : ( u, v ) , ( u (cid:48) , v (cid:48) ) ∈ T } . This key property will allow us toshow: lom, Pendavingh, and Spieksma: Filling a theatre in times of corona Article submitted to
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Theorem 1.
Let
A ⊆ S × T be a seating arrangement. Then A is safe if and only if {T r,s,t : ( r, s, t ) ∈ A} is a collection of pairwise disjoint trapezoids. Theorem 1 forms the basis of the integer programming models in Section 4, which solvethe main problem of this paper for the MBE, and which in principle apply to theatres S ofany given shape or irregular form. Theorem 1 also enables us to analyse, in the remainderof this section, the limiting behaviour of optimal arrangements for large ’square’ theatres.The proof of Theorem 1 takes the form of 2 lemma’s. Lemma 1.
Let ( r, s ) , ( r (cid:48) , s (cid:48) ) ∈ S . Then ( r, s ) ∈ F r (cid:48) ,s (cid:48) ⇐⇒ T r,s ∩ T r (cid:48) ,s (cid:48) (cid:54) = ∅ . Proof.
Necessity: Suppose ( r, s ) ∈ F r (cid:48) ,s (cid:48) . Since F r (cid:48) ,s (cid:48) = ( r (cid:48) , s (cid:48) ) + F = ( r (cid:48) , s (cid:48) ) + T + ( −T )by (2), it follows that there are ( u, v ) , ( u (cid:48) , v (cid:48) ) ∈ T so that ( r, s ) = ( r (cid:48) , s (cid:48) ) + ( u (cid:48) , v (cid:48) ) − ( u, v ).Then T r,s (cid:51) ( r, s ) + ( u, v ) = ( r (cid:48) , s (cid:48) ) + ( u (cid:48) , v (cid:48) ) ∈ T r (cid:48) ,s (cid:48) , so that T r,s ∩ T r (cid:48) ,s (cid:48) (cid:54) = ∅ , as required.Sufficiency: Suppose T r,s ∩ T r (cid:48) ,s (cid:48) (cid:54) = ∅ . Then ( r, s ) + ( u, v ) = ( r (cid:48) , s (cid:48) ) + ( u (cid:48) , v (cid:48) ) for some( u, v ) , ( u (cid:48) , v (cid:48) ) ∈ T . Then ( r, s ) = ( r (cid:48) , s (cid:48) ) + ( u (cid:48) , v (cid:48) ) − ( u, v ) ∈ ( r, s ) + T + ( −T ) = F r (cid:48) ,s (cid:48) , asrequired. (cid:3) Lemma 2.
Let ( r, s, t ) , ( r (cid:48) , s (cid:48) , t (cid:48) ) ∈ S × T . Then S r,s,t ∩ F r (cid:48) ,s (cid:48) ,t (cid:48) (cid:54) = ∅ ⇐⇒ T r,s,t ∩ T r (cid:48) ,s (cid:48) ,t (cid:48) (cid:54) = ∅ . Proof.
We have S r,s,t ∩ F r (cid:48) ,s (cid:48) ,t (cid:48) (cid:54) = ∅ if and only if there are i ∈ { , . . . , t − } , i (cid:48) ∈ { , . . . , t (cid:48) − } so that ( r, s + i ) ∈ F r (cid:48) ,s (cid:48) + i (cid:48) . By the previous lemma, this is equivalent to T r,s + i ∩ T r (cid:48) ,s (cid:48) + i (cid:48) (cid:54) = ∅ for some i ∈ { , . . . , t − } , i (cid:48) ∈ { , . . . , t (cid:48) − } . In turn, this is equivalent to T r,s,t ∩ T r (cid:48) ,s (cid:48) ,t (cid:48) (cid:54) = ∅ . (cid:3) By this lemma, we may replace the asymmetrical condition S r,s,t ∩ F r (cid:48) ,s (cid:48) ,t (cid:48) = ∅ in thedefinition of a safe arrangement by the equivalent symmetrical condition T r,s,t ∩ T r (cid:48) ,s (cid:48) ,t (cid:48) = ∅ .This proves Theorem 1. lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) Under normal circumstances, it is clear how guests of a theatre use the available resources:each guest needs one seat. With social distancing rules, it is not immediately clear to whatextent guests claim the resources. There will be many empty seats in any safe seatingarrangement, and it is not obvious which guest to blame or charge. Theorem 1 is helpfulin this sense, as it makes clear that each family with t members at ( r, s ) blocks the seats T r,s,t ∩ S , and so is responsible at least for the emptiness of these seats. If ( r, s ) is sufficientlyfar away from the boundary of the theatre, so that T r,s,t ⊆ S , this amounts to blocking |T r,s,t | = 2 t + 3 seats. Ignoring the boundary effect, this gives a rough upper bound on thenumber of families of size t that can fit the theatre safely: |S| / (2 t + 3).The following consequence of Theorem 1 describes how we may take the boundary of acollection of seats S into account when estimating the capacity of a theatre. Theorem 2.
Let A be a safe seating arrangement in S . We have: ∞ (cid:88) t =1 (2 t + 3) n t ( A ) ≤ |S + T | . Proof.
For each ( r, s, t ) ∈ A , the family of size t which is located at ( r, s ) will occupythe seats S r,s,t ⊆ S and hence for the corresponding trapezoid we have T r,s,t = S r,s,t + T ⊆ S + T . By Theorem 1, each seat of S + T is in at most one of trapezoids {T r,s,t : ( r, s, t ) ∈ A} , andhence |S + T | ≥ (cid:88) ( r,s,t ) ∈A |T r,s,t | = (cid:88) t (2 t + 3) n t ( A ) , as required. (cid:3) Note that the set ( S + T ) \ S consists of a rim of seats adjacent to the boundary of S in Z × Z .For large theatres with a relatively simple boundary, the collection S + T of the theoremis only marginally larger than S . For example, let S k be a block of k rows of k seats each,then |S k | = k and |S k + T | ≤ ( k + 1)( k + 2). Then |S k + T ||S k | → lom, Pendavingh, and Spieksma: Filling a theatre in times of corona Article submitted to
INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) as k → ∞ .We analyse the limiting case of this sequence in Section 3.3, and then we return tosafe arrangements in large theatres S k in Section 3.4. Though the square theatre S k isperhaps artificial, very large venues such as sport stadiums are sufficiently similar to meritcomparison. Figure 4 Floor plan of the ground floor of the Grand Room of the MBE. The set of seats S is colored lightblue,with the virtual rim of seats ( S + T ) \ S colored in black. Inspired by the famous thought experiment of Hilbert on the concept of infinity, the
Hilberttheatre has seats S ∞ = Z × Z : it is an infinite sea of regularly spaced seats.For each family size t , the seating arrangement A t := { u (2 , −
1) + v (1 , t + 1) : u, v ∈ Z } is such that the corresponding collection of trapezoids {T r,s,t : ( r, s, t ) ∈ A t } covers each seatin S ∞ exactly once. lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) Figure 5 The safe seating arrangement A in the Hilbert theatre. Hence A t is a safe seating arrangement, and the average density of occupied seats in A t equals the proportion of occupied seats within each trapezoid |S r,s,t ||T r,s,t | = t t + 3 = 12 − t + 6 =: d t . We cannot hope to attain a better density than d t in any arrangement with families of size t . This shows that in the Hilbert theatre, the maximum density when packing families ofsize t is d t ; notice that this value increases with t and will never exceed . The followingtable shows the density d t and its reciprocal 1 /d t for small family sizes t . t · · · ∞ d t .
20 0 .
29 0 .
33 0 .
36 0 .
38 0 . · · · . /d t . .
75 2 . . · · · /d t ) decreases steeplyin this initial range. Finite theatres tend to approximate the Hilbert theatre as they become larger. For a seatingarrangement A in a finite theatre S , we formally define the occupation density as d ( A ) := (cid:80) ( r,s,t ) ∈A t |S| . lom, Pendavingh, and Spieksma: Filling a theatre in times of corona Article submitted to
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In our square theatre S k , there is a seating arrangement of families of size t which arisesby restricting A t to the locations available in S k : A t,k := { ( r, s, t ) ∈ A t : S r,s,t ⊆ S k } . Then each row of S k will see at least (cid:98) k t +3 (cid:99) families in A t,k , and hence A t,k attains anoverall density of occupied seats of at least d (cid:0) A t,k (cid:1) = tk (cid:98) k t + 3 (cid:99) /k ≥ t (cid:18) k t + 3 − (cid:19) /k = d t − tk . Evidently, this lower bound on the density tends to d t as k → ∞ .For an upper bound on the occupation density of any arrangement A in S k consistingfamilies of size t only, we may bound the number n t of families in A by applying Theorem 1.This gives (2 t + 3) n t ≤ |S k + T | ≤ ( k + 1)( k + 2) . Since each family has t members and the total number of seats in S k is k , we obtain anupper bound on the occupation density of d ( A ) = tn t k ≤ t ( k + 1)( k + 2)(2 t + 3) k ≤ d t (cid:18) k + 3 k (cid:19) . So the occupation density of such A may marginally exceed the density d t of A t for lowvalues of k , but the upper bound tends to d t as k → ∞ . In particular, we find that thedensity of A t,k tends to d t as k → ∞ , since both lower and the upper bound converge tothis value. So, the upper bound from Theorem 2 ultimately dictates the maximum densityof arrangements in sufficiently large theatres. This extends to arrangements where therelative proportion of families of size t is restricted, in the following precise sense. For anarrangement A of S , let p ( A ) : N → R record the relative proportion of families of size t : p ( A ) : t (cid:55)→ n t ( A ) / |A| . For any p : N → R + , put D ( p ) := (cid:80) t p t d t . Theorem 3.
For any safe arrangement A of S , we have d ( A ) ≤ D ( p ( A )) |S + T ||S| . Moreover, for any p : N → R + such that (cid:80) t p t = 1 , there exists a sequence of arrangements A k of S k such that p ( A k ) → p and d ( A k ) → D ( p ) as k → ∞ . We omit the proof, noting that it is a straightforward extension of the argumentationabove. lom, Pendavingh, and Spieksma:
Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) Remark on special seating arrangements
The highest possible occupation densities areattained by arrangements where there are families placed in each row of the theatre (seeSection 5). For comparison, it is interesting to consider safe seating arrangements in S k where each occupied row is sandwiched between two empty rows. If A is such an arrange-ment with families of size t , then A is safe if and only if there are two empty seats betweenany two families in the same row. Restricting to such simpler arrangements may havepractical advantages. It will be easier to guide guests to their seats, and in fact it becomespossible to rearrange the order in which families take place in their row on the fly. This maytranslate to a lesser need for personnel hosting guests. Also, since safety of an arrangementdepends only on a condition within each row, finding safe arrangements becomes so easythat it can be done manually or with software applying straightforward strategies. Withoutthe need to incorporate a ’black box’ advanced solver in the process of selling seats, theflexibility of this process may be greatly increased.Let us analyze the densities d (cid:48) t that can be obtained for such special seating arrangementsfor a fixed t . Since A will have (cid:100) k/ (cid:101) occupied rows, we have d (cid:48) t (1 − o (1)) = 1 k (cid:100) k (cid:101) t (cid:98) k + 2 t + 2 (cid:99) ≤ d ( A ) ≤ k (cid:100) k (cid:101) t (cid:100) k + 2 t + 2 (cid:101) = d (cid:48) t (1 + o (1))as k → ∞ , where d (cid:48) t := t/ (2 t + 4). This is worse than d t = t/ (2 t + 3), but not by much.To assess the loss of density when using these special seating arrangements, we includea table with the densities d (cid:48) t , the inverse densities 1 /d (cid:48) t , and the relative densities d (cid:48) t /d t . t · · · ∞ d (cid:48) t .
17 0 .
25 0 .
30 0 .
33 0 .
36 0 . · · · . /d (cid:48) t .
33 3 2 . . · · · d (cid:48) t /d t
83% 88% 90% 92% 93% 94% · · ·
Remark on a shorter safe distance
The above analysis can be straightforwardly adapted toa safe distance of 1 meter. Then the forbidden zone becomes F := { ( − , , ( − , , (0 , − , (0 , , (0 , , (1 , − , (1 , } . Taking T := { (0 , , (0 , , (1 , } we will again have F = T + ( −T ), i.e. (2). Then Theorem1 holds, and we obtain |T r,s,t | = 2 t + 1 for each t ∈ N and d t = |S r,s,t ||T r,s,t | = t t + 1 . lom, Pendavingh, and Spieksma: Filling a theatre in times of corona Article submitted to
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There are safe seating arrangements in the Hilbert theatre that attain this density, andTheorem 3 remains valid. For suboptimal seating arrangements with empty rows betweenoccupied rows, we obtain the densities d (cid:48) t = t t +2 .We have so far used our model to investigate a square and sufficiently large theatre S k . We will next explain how our model yields a computational strategy to find optimalpackings for any specific theatre.
4. An integer programming model to maximize the size of a seatingarrangement
We describe our model in Section 4.1, and in Section 4.2 we show how the concept ofmultiple shows can be embedded in the model. In Section 4.3 we indicate how we speedup the solution process of the model.
With any collection
A ⊆ S × T , where T ⊆ N + is a finite collection of allowed family sizes,we can associate a characteristic vector y ∈ { , } S× T with y r,s,t = 1 if and only if ( r, s, t ) ∈ A .Then A is a seating arrangement in S if and only if y r,s,t = 0 whenever S r,s,t (cid:54)⊆ S . (3)This seating arrangement A is safe if and only if (cid:88) ( r (cid:48) ,s (cid:48) ,t (cid:48) ): T r (cid:48) ,s (cid:48) ,t (cid:48) (cid:51) ( r,s ) y r (cid:48) ,s (cid:48) ,t (cid:48) ≤ , for all ( r, s ) ∈ S + T . (4)From a geometric point of view, constraints (4) ensure that each seat ( r, s ) ∈ S + T iscovered by at most one of the trapezoids it is contained in. Finally, A accommodates n t families of each size t ∈ T if (cid:88) ( r,s ): S r,s,t ⊆S y r,s,t = n t , for each t ∈ T. (5)Thus, the feasibility of a safe seating arrangement that simultaneously accommodates n t families of size t for t ∈ T translates to an integer linear feasibility problem in variables y r,s,t and n t . However, without a priori conditions on the number of families of each size t ,the optimal solutions of this problem will tend towards including many large families andfew small families. This is intuitively clear, since a family of t together ’wastes’ a trapezoid lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) of 2 t + 3 seats, so that 2 + 3 /t (= 1 /d t ) seats are taken per person in a family of size t . Inthe extreme case that T includes large enough sizes to fill entire rows of seats with a singlefamily, then a solution where the even rows are empty and each odd row is filled with asingle family is feasible, and similar for leaving the odd rows empty and filling the evenrows. One of these solutions then is optimal, and uses at least half of the seats in S . Indeed,now that we are letting our imagination roam free, we can fill the entire theatre with asingle large enough family if we also let go of our restriction that families must be seatedin the same row. To ensure that we find safe seating arrangements that approximatelycorrespond to the typical sizes of families that book seats for a performance, we use thetarget profile, as introduced in Section 2.2. Recall from the problem statement that thetarget profile imposes the condition( p t − (cid:15) ) (cid:88) t ∈ T n t ≤ n t ≤ ( p t + (cid:15) ) (cid:88) t ∈ T n t , for each t ∈ T. (6)In this way we obtain an integer linear program that maximizes the size of a seatingarrangement over all safe seating arrangements in S :max (cid:40)(cid:88) t ∈ T tn t : (3) , (4) , (5) , (6) , y ∈ { , } S× T , n ∈ Z T (cid:41) . (7)Notice that the LP relaxation of (7) gives an upper bound which, by the safety con-straints (4), is informed that each family of size t occupies at least 2 t + 3 seats from S + T .Evaluated with (cid:15) = 0, the LP relaxation will be at least as good as the bound of Theorem3. One of the ideas that the MBE has implemented to remain commercially viable is toperform the same show during the same evening twice, each time for a different audience.We refer to this phenomenon as consecutive shows . Clearly, this puts a burden on theperforming artist(s); in many cases however, this is a realistic option. The MBE, however, isnot able to clean the seats in between the shows. This creates an interdependence betweenthe two seating arrangements for each individual show as each seat can be used at mostonce in each of the two seating arrangements. lom, Pendavingh, and Spieksma:
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However, it is relatively straightforward to extend our model to find k consecutive seatingarrangements A v for v ∈ V = { , . . . , k } , k ∈ Z + , so that no seat is used in two differentarrangements, i.e. if v, v (cid:48) ∈ V are distinct, then S r,s,t ∩ S r (cid:48) ,s (cid:48) ,t (cid:48) = ∅ for all ( r, s, t ) ∈ A v and ( r (cid:48) , s (cid:48) , t (cid:48) ) ∈ A v (cid:48) .To model the problem of finding such consecutive seating arrangements, we use binaryvariables y ∈ { , } S× T × V , and integer variables n ∈ Z T . The condition that each A v is aseating arrangement of S becomes y r,s,t,v = 0 , whenever S r,s,t (cid:54)⊆ S . (8)Safety of each A v is modelled by (cid:88) ( r (cid:48) ,s (cid:48) ,t (cid:48) ): T r (cid:48) ,s (cid:48) ,t (cid:48) (cid:51) ( r,s ) y r (cid:48) ,s (cid:48) ,t (cid:48) ,v ≤ , for all ( r, s ) ∈ S + T , v ∈ V. (9)We also need to ensure that no seat is used more than once. (cid:88) v ∈ V (cid:88) ( r (cid:48) ,s (cid:48) ,t (cid:48) ): S r (cid:48) ,s (cid:48) ,t (cid:48) (cid:51) ( r,s ) y r (cid:48) ,s (cid:48) ,t (cid:48) ,v ≤ , for all ( r, s ) ∈ S . (10)Letting the n t count the overall number of families of size t is accomplished by writing (cid:88) v ∈ V (cid:88) ( r,s ) ∈S y r,s,t,v = n t , for each t ∈ T. (11)The profiling condition (6) need not change at all.Maximizing the number of guests in consecutive arrangements in S , whilst respecting aprofile p ∈ R T up to a fixed (cid:15) >
0, is then modelled as the following ILP:max (cid:40)(cid:88) t ∈ T tn t : (6) , (8) , (9) , (10) , (11) , y ∈ { , } S× T × V , n ∈ Z T (cid:41) . (12)This model is rather flexible: many additional wishes can be formulated. For instance,upper bounds on n t for some t ∈ T , or a balance between the distribution in different shows,or specific (monetary) weights to maximize the revenue that could be gained, seats can allbe arranged through standard modifications of the integer linear program. lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) For some instances of ILP formulation (12), the corresponding LP relaxation leads to longrunning times of the solver. Hereunder we propose two methods that ameliorate solverperformance, namely (i) adding a class of valid inequalities to strengthen the LP relaxationand (ii) using a symmetry breaking method for formulation (12).
Strengthening the linear relaxation
The ILP formulations (7) and (12) only take into account for each individual seat ( r, s ) ∈ S the trapezoids T r (cid:48) ,s (cid:48) ,t (cid:48) that contain ( r, s ). Using the adjacency of seats, Lemma 3 finds a setof new valid inequalities. Lemma 3.
Let ( r, s ) ∈ S such that X = { ( r, s ) , ( r + 1 , s − , ( r + 1 , s ) } ⊆ S . Then, foreach t ∈ T : (cid:88) ( r,s,t ): |T r,s,t ∩ X |≥ y r,s,t,v ≤ , v ∈ V. (13)Notice that X consists of three seats. Thus, from all trapezoids that contain at least twoseats from X , one can select at most one.To show that inequalities (13) are valid for nontrivial theatres, consider the examplein Figure 6, where T = { } , p = 1, and V = { } . The theatre consists of a set S of fiveseats, represented by the non-white squares, with the rim of virtual white seats aroundit. Each of these four dark grey seats is contained in at most three trapezoids. so the LPsolution with each of the four trapezoids chosen with 1/3 is feasible for (7), with value 4/3.However, a safe seating arrangement A can clearly contain at most one seat. Figure 6 A valid inequality for this example is y , , + y , , + y , , + y , , + y , , ≤ . It separates the previouslyfeasible LP solution ( , , , , ) . Symmetry breaking techniques
The presence of symmetry in a (mixed) integer programming formulation oftenposes a computational challenge, see e.g. Margot [Margot (2010)] and Hojny andPfetsch [Hojny and Pfetsch (2019)]. Indeed, naive implementations can be unsuccessful, as lom, Pendavingh, and Spieksma:
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) many equivalent problems need to be solved in the branch-and-bound procedure to ensureoptimality.Consider now a sequence of consecutive safe seating arrangements ( A v ) v ∈ V . Then, foran arbitrary permutation σ ∈ Sym ( | V | ), the sequence ( A σ ( v ) ) v ∈ V is again feasible. Thechoice of permutations can even be done independently for each segment of the theatre,i.e. the ground floor and its separate balconies. The following lemma provides a class ofinequalities that drastically reduces the feasible region by removing symmetries caused bythese permutation groups. Proposition 1.
Let a segment (cid:96) = 1 , . . . , n in a theatre have a set of seats S (cid:96) ⊂ S . Let ≺ be the standard lexicographic ordering relation on S . For each ( r (cid:48) , s (cid:48) ) ∈ S (cid:96) , v (cid:48) ∈ V , a classof symmetry breaking inequalities is (cid:88) t (cid:48) ∈ T y r (cid:48) ,s (cid:48) ,t (cid:48) ,v (cid:48) ≤ (cid:88) ( r,s ) ∈S (cid:96) ( r,s ) ≺ ( r (cid:48) ,s (cid:48) ) (cid:88) t (cid:48) ∈ T y r,s,t (cid:48) ,v , ∀ v ∈ V : v < v (cid:48) . (14)Suppose we are given a seat ( r (cid:48) , s (cid:48) ) ∈ S (cid:96) , v (cid:48) ∈ V and v < v (cid:48) . The left hand side considersfamilies of size t starting on seat ( r (cid:48) , s (cid:48) ) for show v (cid:48) . The inequality tells us that we can onlyplace at family starting at ( r (cid:48) , s (cid:48) ) for show v (cid:48) whenever in an earlier show v < v (cid:48) a familywas placed starting on some seat ( r, s ) ≺ ( r (cid:48) , s (cid:48) ). Most importantly, we claim without proofthat it is a correct symmetry breaking method. Lemma 4.
There exists an optimal solution to (12) that satisfies inequalities (14).
5. Computational results
We implemented the models (7) and (12) above in Julia 1.3.0, using the modelling languageJuMP to build the optimization model, with Gurobi as the lower level LP and MIP solver.Experiments were run on a computer equipped with an Intel Core i7-7700HQ CPU @ 2.8GHz with 32 GB of RAM. For our experiments we considered four different target profiles (cid:126)p = ( p t ) t ∈ T , with T = { , , , } , based on requests made by the MBE: • Historical data on reservations: mge1 : (cid:126)p = (0 . , . , . , . • Pairs only: mge2 : (cid:126)p = (0 , , , • Singletons and pairs: mge3 : (cid:126)p = (0 . , . , , • Pairs and quads: mge4 : (cid:126)p = (0 , . , , . lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) We solve the integer programming models in (7) and in (12) both for the Grand Roomand the Small Room, where we considered two scenarios for the set of consecutive shows: V = { } (single show) and V = { , } (double show). Both the basic versions of both mod-els ( vanilla ), and the versions with the speedup techniques ( speedup ) are considered andcompared. The speedup version is implemented by adding all inequalities described inSection 4.3 to the ILP formulations. We set ε = 0 .
02 for the constraints in (6), as we empir-ically observe this choice for ε to give a suitable tradeoff between solver performance andsolution structure with respect to target profiles. The same forbidden areas are consideredfor both theatre rooms, as the interseat distances coincide for both rooms. Additionally, weadapt the Gurobi parameters Symmetry , Cuts and
Presolve to their aggressive settings.Table 1 and Table 2 provide the densities d ( A ( (cid:126)p )) for the Grand Room and the SmallRoom respectively, where A ( (cid:126)p ) is an optimal safe seating arrangement with respect to anindicated target profile (cid:126)p , both for the single show and double show case. Table 1 Densities d ( A ( (cid:126)p )) (in %) of maximum safe seating arrangements A ( (cid:126)p ) in the Grand Room, according to the targetprofiles. The reported numbers in the columns vanilla and speedup represent time in seconds (rounded to two decimal places). Target profile Single show Double showDensity vanilla speedup
Density vanilla speedupmge1
32 3.39 1.50 63 532.69 48.28 mge2
29 0.28 0.10 56 6.67 2.49 mge3
30 1.39 0.97 58 2107.68 6.05 mge4
36 5.29 1.10 70 4485.33 726.11
Table 2 Densities d ( A ( (cid:126)p )) (in %) of maximum safe seating arrangements A ( (cid:126)p ) in the Small Room, according to the targetprofiles. The reported numbers in the columns vanilla and speedup represent time in seconds (rounded to two decimal places). Target profile Single show Double showDensity vanilla speedup
Density vanilla speedupmge1
34 1.19 0.41 64 8.18 13.82 mge2
31 0.02 0.02 58 0.30 0.35 mge3
31 0.22 0.07 59 2.19 0.89 mge4
37 0.08 0.11 70 5.46 9.17
Let us first comment on the densities found in Tables 1 (Grand Room) and 2 (SmallRoom). For each of the four target profiles, the differences in density between the GrandRoom and the Small Room are small, both for the single show and for the double showsituation. This is to be expected as the interseat distances from Figure 1 apply to both lom, Pendavingh, and Spieksma:
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Real life implementation: alternating empty rows
Recall that in Section 3.3, we analyzed a setting where in each show, seats in row r couldbe occupied by guests whenever row r − r + 1 (if existent) were empty, for any row r . lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) Tables 3 and 4 report the optimum densities of safe seating arrangements A (cid:48) ( (cid:126)p ) that havethe property of using only one of two consecutive rows at a time per show, for the GrandRoom and the Small Room respectively. For the double show case, this means in the firstshow, all odd-numbered rows are used, and all even-numbered rows in the second. We onlyreport computation times for the vanilla descriptions, as these speedup techniques nowonly yield redundant inequalities. The column called “Loss (%)” indicates the percentualloss of occupied seats, which can be seen as a proxy for the loss in revenue. Table 3 Densities d ( A (cid:48) ( (cid:126)p )) (in %) of maximum safe seating arrangements A (cid:48) ( (cid:126)p ) in the Grand Room, according to the fourtarget profiles when one of every two consecutive rows might be occupied by guests. Target profile Single show Double showDensity Loss (%) vanilla (s) Density Loss (%) vanilla (s) mge1
29 -8.5 0.15 57 -9.4 0.33 mge2
27 -7.8 0.01 52 -7.5 0.05 mge3
27 - 9.1 0.05 52 -9.8 0.07 mge4
34 -5.8 0.04 65 -6.0 0.12
Table 4 Densities d ( A (cid:48) ( (cid:126)p )) (in %) of maximum safe seating arrangements A (cid:48) ( (cid:126)p ) in the Small Room, according to the fourtarget profiles when one of every two consecutive rows might be occupied by guests. Target profile Single show Double showDensity Loss (%) vanilla (s) Density Loss (%) vanilla (s) mge1
30 -15.7 0.06 58 -9.8 0.11 mge2
26 -14.8 0.00 52 -9.6 0.01 mge3
26 -16.8 0.02 52 -11.8 0.08 mge4
34 -13.5 0.01 66 -5.7 0.03
Clearly, as the results in Tables 3 and 4 correspond to a more restricted setting of ourproblem, the realized densities are always smaller than those achieved for the setting whereall rows can be used for all shows. Indeed, we observe that for all instances, especially theones based on the Small Room, the percentual loss of occupied seats is rather significant,with no percentual loss smaller than 5.7% of occupied seats.Computation times for this setting are much smaller. This is caused by the much smallersize of the resulting instances and much fewer dependencies between the variables.The solutions that correspond to the occupancies for the target profile mge1 given inTables 1 to 4 are provided in the appendices. We used the color red to indicate that thoseseats are forbidden for use by guests and the color green (and blue for two consecutiveshows) to indicate seats that can be occupied by guests. lom, Pendavingh, and Spieksma:
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6. Upper bounds for the occupancy in MBE
In Table 5, we list the number of seats |S| and the number of virtual seats (the “rim”) | ( S + T ) \ S| , in each of the two rooms of the MBE. By applying Theorems 2 and 3 to therooms of the MBE, we can find upper bounds on the achievable occupancy. Table 5 Number of (virtual) seats in the Grand Room and the Small Room.
Grand Room Small Room |S| | ( S + T ) \ S|
458 133
From Theorem 2, we can deduce the following bound for a single safe seating arrangement A for the Small Room: (cid:88) t ∈ T (2 t + 3) n t ( A ) = 5 n ( A ) + 7 n ( A ) + 9 n ( A ) + 11 n ( A ) ≤
400 + 135 = 533 . Analogously, we have for the Grand Room5 n ( A ) + 7 n ( A ) + 9 n ( A ) + 11 n ( A ) ≤ . We consider these volume bounds on the realized safe seating arrangements A max cor-responding to the data in Table 1 and Table 2 on the single show setting, for the GrandRoom and the Small Room respectively. The following table provides the left hand sides onthese volume bounds to illustrate the strength of these bounds. Recall the right hand sidesfor the Grand Room (1708) and Small Room (533). The interpretation of the numbers Table 6 Left hand sides of the volume bounds of Theorem 2 for the differenttarget profiles on the MBE theatre rooms.
Target profile Theorem 2 LHSGrand Room Small Room mge1 mge2 mge3 mge4 depicted in Table 6 is the number of seats in S + T that are covered by the correspondingtrapezoid packing. We observe that approximately 75 to 85 % of (virtual) seats are covered lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) by a trapezoid. The gap on the volume bound can be explained by the fact that the trape-zoid forms do not allow for a perfect packing of seats, and this effect is amplified by theinclusion of target profiles, which further restrict the set of possible trapezoid packings.Secondly, we consider the bound given by Theorem 3. Now, we consider the upper boundfor occupation densities d ( A ( (cid:126)p )) of this bound corresponding to maximum safe seatingarrangements in the model of (12). For convenience, we include the results on the realizedoccupation densities for the single show setting using all rows again in Table 7. Table 7 Upper bounds on the occupation density d ( A ( (cid:126)p )) (in %), where A ( (cid:126)p ) is a maximum density safeseating arrangement satisfying target profile (cid:126)p , together with actually realized densities. Target profile Occupation densityUB d ( A ( (cid:126)p )) d ( A ( (cid:126)p )) (Grand Room) d ( A ( (cid:126)p )) (Small Room) mge1 mge2 mge3 mge4 We observe that the realized optimal densities are not very far away from the respectiveupper bounds in Table 7. Nevertheless, the bound of Theorem 3 is based on perfect tilingsof trapezoids of one family size in a suitable chosen theatre architecture, which is not thecase for typical inputs. This is a possible explanation for the gap between the bound andthe realized density
7. Conclusion
The 1.5 meter constraint has a huge impact on the occupancy when filling a theatre. Incase of the MBE, when performing a single show on an evening, occupancy will not exceed40% (both for the Grand Room and the Small Room). However, allowing two shows perevening, it is possible to reach an occupancy of 70% while satisfying the constraint that noseat is sold twice. A more logistically suitable solution is to use alternating empty rows,but this comes at the cost of losing at least 5% on the number of occupied seats. Thecorresponding solutions, together with other innovations, may offer some hope to theatresto remain competitive.
Acknowledgements:
Omitted for anonymization lom, Pendavingh, and Spieksma:
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Cinemas, theatres, concert halls, drive-in cinemas , [Baldin and Bille (2018)] Baldin, A. and T. Bille (2018), Modelling preference heterogeneity for theatre tickets: adiscrete choice modelling approach on Royal Danish Theatre booking data , Applied Economics , 545-558.[Fischetti and Luzzi (2009)] Fischetti, M. and I. Luzzi (2009), Mixed-integer programming models for nesting prob-lems , Journal of Heuristics , 201–226.[Germany (2020)] Zusammen gegen Corona , [Hojny and Pfetsch (2019)] Hojny, C and M. Pfetsch (2019), Polytopes associated with symmetry handling , Mathe-matical Programming , 197-240.[Jacobs (2020)] Jacobs, J. (2020),
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How German theaters are adapting to the coronavirus , [The Netherlands (2020)] Dutch measures against coronavirus , [Sweden (2020)] Restrictions and prohibitions , [UK (2020)] Working safely during coronavirus (COVID-19) , [USA (2020)] State Data and Policy Actions to Address Coronavirus , [Wang et al. (2018)] Wang, A., C. L. Hanselman, and C. E. Gounaris (2018), A customized branch-and-boundapproach for irregular shape nesting , Journal of Global Optimization , 935-955. lom, Pendavingh, and Spieksma: Filling a theatre in times of corona
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INFORMS Journal on Applied Analytics ; manuscript no. (Please, provide the manuscript number!) Appendix A: Solutions for mge1 (all rows)
The labels in the figures indicate the segments within the theatre rooms that are depicted. (a) Maximum number of occupiedseats (in green) of the Small Room fora single show with target profile mge1 . (b) Maximum number of occupiedseats (in green and blue) of the SmallRoom for two consecutive shows withtarget profile mge1 .(c) Maximum number of occupied seats (ingreen) of the Grand Room for a single showwith target profile mge1 . (d) Maximum number of occupied seats (ingreen and blue) of the Grand Room for twoconsecutive shows with target profile mge1 . lom, Pendavingh, and Spieksma: Filling a theatre in times of corona Article submitted to
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Appendix B: Solutions for mge1 (occupied rows are between two empty rows)
The labels in the figures indicate the segments within the theatre rooms that are depicted. (a) Maximum number of occupiedseats (in green) of the Small Roomfor a single show with target profile mge1 . (b) Maximum number of occupiedseats (in green and blue) of theSmall Room for two consecutiveshows with target profile mge1 .(c) Maximum number of occupied seats (ingreen) of the Grand Room using a single showwith the target profile mge1 . (d) Maximum number of occupied seats (ingreen and blue) of the Grand Room usingtwo consecutive shows with the target profile mge1mge1