Fair layout design
Patrick Schneuwly University of Fribourg, iiUF – Institute of Informatics
Marino Widmer University of Fribourg, iiUF – Institute of Informatics
Abstract This paper tackles the design of fairs which are subject to layout considerations. A fair means a large scale exhibition for goods and services like a trade fair or a regional fair. A method for designing fair layouts using a slicing structure and the horseshoe cut is presented, and numerical results for the layout of a real fair are reported. The selected criterias are evaluated using the obtained results. Keywords layout design; modelling; facility layout; optimization
Corresponding author Patrick Schneuwly University of Fribourg, iiUF – Institute of Informatics Rue Faucigny 2, CH-1700 Fribourg
[email protected]
Fair layout design : Introduction
1-1
Fair layout design 1
Introduction
Making locational decisions corresponds to the choice of locations within a spatial context. Examples of such choices include locating of factories, warehouses, schools, hospitals and emergency services. In contrast to locational decisions, the facility layout problem is concerned with finding the most efficient arrangement of departments or machines with area requirements within a facility. By most efficient is meant optimal for a given criterion, such as material handling costs, space utilization, flexibility of layout, safety and working conditions [Dom97]. For years, this problem has received considerable attention from firms engaged in manufacturing activities. Within manufacturing, researchers usually reduce the cost function to the material handling cost, which is the most important cost factor [Mel96]. The material handling cost function is based on the interaction between the departments or machines (i.e. material flow) and on the inter-department respective inter-machine distances. The facility layout problem has first been mathematically formulated as a quadratic assignment problem [Koo57]. Since this initial work a variety of contributions has been published (see the surveys [Her87, Fra92, Dom97]). The topic of this paper is another, less known case that is also subject to layout considerations : the fair layout design. When talking about a fair, a large-scale exhibition of goods and services like a trade fair, a regional fair, or a sample fair is meant. Figure 1 gives an example of a fair ground plan. It is a regional fair in the small swiss town Romont, where about 120 exhibitors expose theirs goods and services during 8 days on a exhibition surface of 10’000 square meters that is roofed over by a tent. Every two years, around 50’000 visitors attend this fair.
Figure 1. Fair layout Like in the facility layout problem, the fair ground designer needs to place the exhibition areas in a non-overlapping manner on the fair ground. Usually he has to consider space requirements for the different exhibition areas. Apart from the exhibition space, a fair ground plan demands also space for the visitor corridor. It is
Fair layout design : Problem description
2-2
obvious, that an acceptable fair layout must arrange the visitor space in such a way that each exhibition area can be reached by the visitors. The fair layout designer has to deal with many constraints that can affect the fair ground plan. Building geometry can serve as a restriction on the layout because the layout has to fit in the shape of the building. When there exists a building within which the layout has to be designed, it can impose a large number of restrictions on the solution. For example, the layout solution will be affected by the present location of walls and columns, windows, lights, ventilation equipment, power lines, and emergency exits. A given fair layout solution can have dramatic impact on many issues like visitor convenience, space utilization, attraction of an exhibition area, shape of an exhibition area, safety, and heating/cooling requirements. Very often, tradeoffs have to be made between conflicting factors to produce a feasible layout. For example, the higher the space utilization, the lower the ground cost for the fair organization. However, a small amount of space dedicated to the exhibitors and visitors is not preferred by both the exhibitors (i.e. area becomes less attractive) and the visitors (i.e. fair is less convenient). Furthermore, if the layout designer dedicates more space to the visitor corridor, then the higher is the visitors safety in case of an emergency. In opposition, the fair organization will be faced with increasing ground costs. As seen above, the design process of a fair layout is difficult and time-consuming, and many open questions arise in the field of fair layout design : • Can we solve the fair layout problem with an analytical approach ? • How to characterize a “good” fair layout ? • How can we perform the design task better and faster ? To partially answer these questions a model to represent a fair layout was developed and two quantitative criterias to measure if a layout solution is efficient were selected. In the next section, a more precise description of the fair layout problem is given based on the ”Romont regional fair” mentioned above. Section 3 presents a resolution method for the problem described in Section 2. Its application to real data and some computational results are reported in Section 4. Section 5 concludes the paper with a discussion on this method’s limitations, extensions, and potential for future work.
2
Problem description
This section describes in a first part the problem more precisely according to the ”Romont regional fair“ mentioned in Section 1. The second part looks at some appropriate criterias to measure the efficiency of a layout solution. 2.1
The regional fair in Romont
Every two years, the fair layout designer of the fair in Romont has to place about 120 exhibition areas on a surface of 10’000 square meters. He has basically two approaches to accomplish his task. One approach is to start from the list of exhibitors and theirs surface requests, and to construct the layout by placing one area at a time. Another approach is to start from a predefined layout, and then assign each exhibitor a surface that fulfills the area requirements. In Romont, the fair layout is typically constructed using the first approach. Another important property of the Romont fair is
Fair layout design : Problem description
2-3
that the layout is redrawn each time from scratch. So the layout of one fair never looks like the layout of the preceding one. The layout design process consists of two major problems. On the one hand, the layout designer has to allocate space to the exhibitors. On the other hand, he has to define the visitors corridor. Exhibition space and area requirements In a fair layout, exhibition areas with unequal area requirements are placed within a facility in a way that they do not overlap. The area requirements of the exhibition areas includes requested area size, aspect ratio, shape, and favourite location. The most important is in fact the area size requirement. Most of the exhibition areas have the shape of a rectangle, whereas a few look like a L-shape and are favorably placed in a corner (e.g. area 49 in Figure 2).
Figure 2. Round tour The facility where the exhibition areas has to fit in introduces additional restrictions. The areas have to be placed in a tent of rectangular shape. The short side is 52 meters long, the long side 141 meters, and it is possible to expand the long side. Inside the tent, we have to pay attention to fix walls, columns, aisle space to access the toilets, and corridors for emergency exits. Some exhibition areas are grouped together if the presented services or goods are similar. For example, all car sellers (e.g. exhibition areas 70 to 86 in Figure 1) are placed side by side. Visitor’s space and round tour The fair layout designer needs moreover to allocate space for the visitors. In the case of Romont, a one-way path is imposed. It is like a round tour with an entrance and an exit. That way, the fair organization achieves that each visitor has to walk through the whole fair and that each exhibition area is accessible by the visitors. As an example, take a look at Figure 1. The entrance of the fair is situated in the lower right corner of the building. Then the visitor will first pass exhibition area 1 and 2 on the left, then exhibition area 3 and 4 on the right, and so on. Throughout the layout, the round tour is generally 3 meters wide. The minimum width is 2 meters.
Fair layout design : Problem description
2.2
2-4
Selection of appropriate criterias
As already discussed in Section 1, a given fair layout solution can have a dramatic impact on many issues. The factors which would be influenced in the Romont fair are described below. Visitor’s convenience It is obvious, that the visitor’s convenience is increased by a generous allocation of visitor space. The importance of this factor increases with the number of visitors. If the fair is crowded, then the visitors will edge one’s way through the narrower corridors. Attraction value of an exhibition area Each exhibition area has some potential attraction value to the visitors with respect to the relative position of the area in the round tour. See the exhibition area 60 in the Figure 2. This exhibition area has the shape of a rectangle, and the round tour passes around the area on three sides. A visitor is certainly more attracted by this area than by the area with number 58, which has only one of the longer sides adjacent to the corridor. Also the order in which the exhibition areas are placed on the visitors way has an influence on the attraction value of an exhibition area. These are more psychological factors : after visiting several exhibitors presenting the same services or goods, a visitor may get tired. Also, the later a visitor gets into an exhibition area, the less attentive he may be. To resume, the later an exhibition area is placed on the visitor’s one-way round tour, the smaller becomes the potential attraction of the area. Space utilization Every fair layout produces dead space that is allocated neither as exhibition areas nor as visitor corridor. As the fair ground rent is proportional to the layout surface required, the fair organization tends to cut down the dead space. The three presented factors are conflicting and tradeoffs have to be made. Usually it is the experience and knowledge of the fair layout designer that influences the design process in such a way, that each factor is considered as with a first attention. On the basis of the discussion above, we propose two quantitative criterias to compare and evaluate alternative layout solutions. Both criterias are based on the adjacency between the exhibition areas and the visitor’s round tour. Adjacency index Let us focus on a single exhibition area, and let bin = adjacency length between round tour i and exhibition area n, pn = perimeter of the surface of exhibition area n. Then, the adjacency index ain is defined as
ain =
bin pn
The adjacency index ain relates the adjacency of an exhibition area n to the perimeter of its surface according to a round tour i. Look at the examples in Figure 3. The surface of area 1 corresponds to a square that is 4 units long and 4 units wide, whereas
Fair layout design : Problem description
2-5
the surface of area 2 corresponds to a rectangle that is 8 units long and 2 units wide. Furthermore, two round tours are defined (i.e. first and second row in Figure 3), the bold line denotes adjacency of the round tour and the area, and can be seen as a potential place for the visitor to enter the exhibition area. The upper round tour goes along one of the longer sides of the area. This means, that the adjacency length b1n is equal to the long side of the area. The lower round tour goes all around the area. In this case, b2n is equal to the perimeter of the area. ain is calculated for each of the four configurations, and it can be seen that ain is nothing else than a value between 0 and 1 that indicates in percent how much of the perimeter is adjacent to the round tour1. To conclude then, the higher the index ain, the higher is the potential attraction of the area, because the visitor is faced with more opportunities to enter the exhibition area. area 1
area 2
round tour 1
a11 =
b11 4 = = 0.25 p1 16
a12 =
b12 8 = = 0.4 p2 20
round tour 2
a21 =
b21 16 = =1 p1 16
a22 =
b22 20 = =1 p2 20
Figure 3. Adjacency index On the one hand, the adjacency index can be used to compare the exhibition areas among each other and to find area locations that are more attractive. On the other hand, the whole fair can be evaluated by summing up the adjacency indexes of all exhibition areas. The fair organization is naturally interested in a high overall adjacency, which is achieved by m
Maximize
∑a n =1
in
where m is the number of exhibition areas to be placed. One may say, that the attraction of an exhibition area depends more on marketing capabilities than on the adjacency index. This is perhaps true for an individual exhibitor, but to reveal from the point of view of the fair organization, it is important
1
Another adjacency index is imaginable, where bin is related to the surface of the exhibition area. However, it behaves like ain, except that it decreases declining for increasing surfaces, whereas ain remains constant.
Fair layout design : Resolution method
3-6
to
Maximize Min(ain ) This way, the minimum of the indexes is maximized, and an equality between the areas with respect to their adjacency index is guaranteed. By improving the area with the weakest adjacency index, a layout where all exhibitors are satisfied with the location of their area in the tent is obtained. This Max-Min strategy tries to give each exhibitor the same intial conditions. Then, it is up to the exhibitor to attract as many visitors as possible by choosing adequate advertising techniques. For this reason the term “potential visitor attraction” is used when talking about the adjacency index. Length of the round tour Beneath the adjacency index, we can look at another interesting value in a fair layout. Let li = length of the visitor round tour i. The determination of li depends on the representation of the round tour in a given layout. For example, a possible round tour for the fair in Romont is drawn as a bold line in Figure 2. The round tour is lying somewhere in the corridor space allocated for the visitors. A property about li is m
2li ≥ ∑ bin n =1
In other words, twice the length of the round tour contains the sum of all adjacency lengths between each exhibition area and the round tour i. The remaining difference is due to the neighborship of the round tour with dead space and fix walls. By
Minimize 2l i −
m
∑b n =1
in
the round tour is prohibited to pass along dead space or fix walls, and the visitors do not need to walk along uninteresting places like empty corridors and walls.
3
Resolution method
The chosen resolution method tries to enforce geometric constraints representing the fair ground plan, the exhibition areas, and the round tour. It is a simple coding for the fair layout problem described in Section 2. The so called slicing structure is used to represent the fair ground plan and the exhibition areas. This approach was introduced by [Tam92] to model a facility layout problem. The adaptation of this method to the fair layout problem is discussed in the first two parts of this section. To take into account the round tour in a fair layout, the method is then extended by introducing the horseshoe cut. This is the content of the third part.
Fair layout design : Resolution method
3.1
3-7
Slicing structure
The layout of the exhibition areas is represented as a slicing structure constructed by recursively partitioning a rectangular block (i.e. the fair ground) in such a way that each rectangular partition in the slicing structure corresponds to the space allocated to an exhibition area. Table 1 lists six exhibition areas and theirs surface requirements. A layout of these six exhibition areas is represented as a slicing structure shown in Figure 4. We say that the area requirement is satisfied if the area allocated to each exhibition area is at least as large as the surface required. Notice that the area requirements are all satisfied, assuming that the dimension of the initial rectangular block is at least 220 square meters.
Figure 4. Slicing structure and slicing tree An equivalent representation of a slicing structure is a slicing tree. It is a binary tree which shows the recursive partitioning process that generates a slicing structure. An example of a slicing tree is shown in Figure 4. Each internal node represents the way a rectangular partition is cut. The kind of cut is denoted by a cut symbol assigned to each internal node (e.g. ‘NCbelow‘, ‘NCright‘). As shown in Figure 4, partitions reserved for exhibition areas correspond to the leaves of the tree. To each leaf is assigned a unique integer denoting the identifier of an exhibition area.
Identifier Exhibitor
Surface requirement
1
Area 1
50 square meters
2
Area 2
20 square meters
3
Area 3
30 square meters
4
Area 4
15 square meters
5
Area 5
45 square meters
6
Area 6
60 square meters
Total
220 square meters
Table 1. Exhibition areas
Fair layout design : Resolution method
3-8
Figure 5. Two basic cuts Suppose we fix the structure of a slicing tree and only change the cut of an internal node. It results in a creation of a different layout. Then the space of all layouts is defined as the set of all slicing trees S that can be generated by rearranging the cuts of a slicing tree. Note that each element s ∈ S has the same structure. They only differ in the cuts of the internal nodes. The two basic cuts are illustrated in Figure 5. The below cut ‘NCbelow’ signifies a horizontal partitioning, whereas ‘NCright’signifies a vertical one. 3.2
Constructing a layout from a slicing tree
Since the procedure to convert a slicing tree to a slicing structure must allocate enough space to each exhibition area, we will assume that the total usable fair ground is at least as large as the total surface of all the exhibition areas. To enforce the surface requirements, the point where a rectangular partition is cut, called the cut point, must be decided in such a way that the split partitions receive the requested surface. The cut point is determined in a top-down fashion starting from the root of the tree and going down to the children nodes recursively. Since the dimensions of a node (i.e. a partition) and the areas of its children are known, the cut point can be determined in a straightforward manner. With n internal nodes the time complexity is simply O(n). A consequence of this construction is that each exhibition area is assigned to a partitioned block of rectangular shape. L-shapes are not supported. 3.3
Horseshoe cut and the round tour
The slicing structure that we presented till now allows for the representation of the exhibition areas. In order to take into account the round tour the slicing structure is extended by introducing a special cut operation : the horseshoe cut. The name comes from the fact that the round tour associated to a basic horseshoe cut has the shape of a horseshoe. The basic idea behind the horseshoe is to cut a rectangular block three times with basic cuts, and to associate a round tour within the partitioned block, as showed in Figure 6. The associated round tour is represented as a bold line.
Figure 6. Equivalence of horsehoe and basic cuts
Fair layout design : Resolution method
3-9
Furthermore the corresponding slicing tree and an equivalent representation in terms of basic cuts is given. A different horseshoe cut arises depending on which side the round tour enters the rectangular block. Figure 7 summarizes the different horseshoe cuts (e.g. ‘HS2up’, ‘HS2below’, ‘HS2left’, ‘HS2right’). The cut ‘HS2left’ for example means that the round tour enters the block from the left side. The four possible configurations of this horseshoes are in fact generated by a 90 degree rotation of the slicing structure and its associated round tour.
Figure 7. Four basic horseshoe cuts Further, four different types of horseshoe cuts are distinguished with respect to how the rectangular block is cut three times. Figure 8 overviews the four possibilities ‘HS1up’, ‘HS2up’, ‘HS3up’, and ‘HS4up’ in the case where the round tour enters from the upper side. They only differ in the choice of the basic cuts applied. The following naming convention is adopted for the cut symbol 2 : ‘HSxyyyy’. ‘HS’ stands for horseshoe cut, x determines the horseshoe type 1 to 4, and yyyy denotes from which side the round tour is entering the partitioned block.
1
2
3
4
Figure 8. Four horseshoe cut types
The definition of a slicing structure can be reformulated after the introduction of the horseshoe cut. A slicing structure is a representation of the fair layout and its round tour, constructed by recursively applying basic and horseshoe cuts. Figure 9 shows an
2
Analogically, cut symbols ‘NCyyyy’were introduced for the basic cuts. ‘NC’stands for normal cut.
Fair layout design : Resolution method
3-10
Figure 9. Horseshoe cuts and round tour artificial example of a fair layout with 15 exhibition areas with associated surface requirements. Again, all area requirements are satisfied because the surface of the fair ground is equal to the sum of exhibition surfaces. The bold line indicates the associated round tour, and the slicing tree is an equivalent representation that shows the partitioning process. In the example, the horseshoe cut is applied four times. The adaptation of the round tour is crucial when applying the horseshoe cut recursively. Any recursive horseshoe cut separates the before constructed round tour and enlarges it with the round tour associated to the nested horseshoe. The separation points are determined by the entrance and exit of the round tour in the nested cut (see recursive construction in the Appendix). Generally, the construction of a valid layout and the associated round tour is possible from a given slicing tree if the following conditions are fulfilled : • The root node must represent a horseshoe cut. This cut determines the fair entrance and the fair exit in the layout. • Recursive horseshoe cuts are only allowed if the continuity of the round tour can be guaranteed. In the layout in Figure 9 for example, it would not be allowed to cut the partition assigned to area 2 by applying a ‘HS1left’ cut, because this would result in a separated round tour. • Basic cuts are only allowed if the two new partitions are adjacent to the round tour. The exhibition area 14 in Figure 9 for instance is nearly not adjacent to the round tour. The presented method offers a simple coding scheme of fair layouts. Furthermore, it becomes easy to calculate the adjacency index for each exhibition area and the length of the round tour by looking at the cut points computed for each partitioned block.
Fair layout design : Computational result
4
4-11
Computational result
This section reports computational results of the fair layout in Romont. In the first part the presented method is applied to real data, whereas in the second part, an evaluation of the obtained layout solution is made based on the criterias introduced in Section 2. 4.1
Layout solution for the fair in Romont
The objective was to find one layout solution for the fair in Romont using the method presented in the previous section. Starting from a given fair layout (e.g. Figure 1), horseshoe and basic cuts were gradually detected to model the layout with the method. This way a slicing tree was generated containing 12 nodes representing horseshoe cuts, 81 nodes representing basic cuts, and 118 leaves for the partitions reserved to the exhibition areas.
Figure 10. Layout solution for Romont
Model
Height
Width
Surface
real layout
52 meters
142 meters
7’384 square meters
slicing structure
41 meters
112 meters
4’588 square meters
Table 2. Fair ground parameters By constructing the layout from the generated slicing tree the layout solution as shown in Figure 10 was obtained. Since the presented resolution method neither does allocate space for the round tour (i.e. visitor corridor) nor does produce any dead space, some adjustment had to be made to the initial parameters of the method. Table 2 summarizes the parameters for the fair ground. As shown in the first row, the fair ground in the real layout (e.g. Figure 1) is 52 meters large and 142 meters long3. This
3
The exhibition area 66 (in the upper left corner of the layout) and the entrance space (in the lower right corner) were neglected to obtain a rectangular fair ground shape.
Fair layout design : Computational result
4-12
results in a total surface of 7’384 square meters. As pointed out in the previous section, the slicing structure demands that the surface of the fair ground is equal to the sum of all exhibition surfaces to be placed. In the case of Romont, this sum is 4’588 square meters. With respect to the rectangular proportions of the real fair ground, the usable fair ground for the slicing structure thus is 41 meters large and 112 meters long. It seems that the resolution method is adequate to map real fair layouts. The layout solution generated with the slicing structure and horseshoe cuts is similar to the real layout, and the arrangement of the exhibition areas and the disposition of the round tour could be easily reproduced4. area n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
surf. 50 20 10 15 77 60 35 15 88 3 32 28 16 33 60 27 20 6 6 9 16 30 32 15 15 70 72 30 12 12 12 24 12 18 21 20 100 4 80 72
real aon 0.50 0.22 0.64 0.31 0.31 0.31 0.29 0.69 0.34 0.38 0.50 0.32 0.25 0.61 0.35 0.38 0.44 0.30 0.30 0.17 0.40 0.73 0.33 0.31 0.31 0.59 0.39 0.50 0.29 0.29 0.29 0.32 0.29 0.33 0.65 0.28 0.40 0.50 0.39 0.10
adj. aon 0.42 0.28 0.62 0.82 0.47 0.96 0.38 0.63 0.63 0.00 0.38 0.23 0.63 0.63 0.14 0.44 0.25 0.10 0.10 0.00 0.46 0.86 0.13 0.32 0.32 0.63 0.20 0.63 0.20 0.20 0.20 0.14 0.28 0.34 0.80 0.46 0.19 0.30 0.33 0.34
area n 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 67 66 68 69 70 71 72 73 74 75 76 77 78 79 80
surf. 20 15 15 64 12 12 24 30 90 24 24 12 18 24 15 18 24 24 48 108 16 20 30 8 6 21 31 12 12 100 24 15 15 110 88 110 88 108 96 40
real aon 0.72 0.31 0.50 0.38 0.29 0.50 0.30 0.55 0.15 0.36 0.36 0.29 0.33 0.32 0.31 0.33 0.25 0.30 0.64 0.71 0.25 0.11 0.38 0.33 0.50 0.40 0.29 0.29 0.42 0.45 0.31 0.38 0.24 0.26 0.24 0.29 0.29 0.30 -
adj. aon 0.74 0.09 0.63 0.37 0.13 0.53 0.32 0.63 0.38 0.28 0.13 0.63 0.34 0.38 0.06 0.19 0.25 0.63 0.63 0.65 0.71 0.25 0.42 0.20 0.63 0.19 0.37 0.37 0.77 0.63 0.24 0.63 0.47 0.25 0.32 0.28 0.29 0.29 -
area n 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
surf. 99 99 99 110 99 64 12 12 50 15 24 120 100 40 32 18 6 60 12 9 18 9 21 12 36 15 32 24 24 12 15 12 12 15 8 14 8 36 8 400
real aon 0.28 0.28 0.25 0.23 0.23 0.41 0.29 0.29 0.50 0.50 0.36 0.43 0.75 0.36 0.09 0.33 0.50 0.10 0.29 0.50 0.50 0.25 0.35 0.50 0.09 0.50 0.33 0.30 0.50 0.29 0.31 0.29 0.29 0.31 0.33 0.39 0.33 0.42 0.33 0.35
adj. aon 0.28 0.29 0.28 0.32 0.28 0.46 0.23 0.00 0.52 1.08 0.49 0.18 1.12 0.80 0.51 0.55 0.03 0.05 0.22 0.63 0.08 0.74 0.49 0.29 0.28 0.63 0.39 0.34 0.63 0.39 0.43 0.39 0.39 0.23 0.36 0.43 0.63 0.32 0.11 0.34
Table 3. Adjacency indexes of the fair Romont
4
Except of exhibition area 80 that was omitted. Because of its location according to the round tour, it is not possible to represent this area with a horseshoe cut.
Fair layout design : Computational result
4.2
4-13
Evaluation of the layout solution
This part justifies the two criterias introduced in Section 2, namely the adjacency index and the length of the round tour. The values of the criterias computed for the generated solution are compared to the real values obtained from the architecture fair plan of Romont. Adjacency index Table 3 lists two adjacency indexes for each of the 118 exhibition areas in the third and fourth column of the table. The first index is computed from the variables of the real layout. Those can directly be obtained from the architecture plan (i.e. layout in Figure 1). By adopting the round tour as a line centered in the visitor corridor5, it becomes easy to measure the adjacency lengths, and to calculate the indexes. The second index mentioned in Table 3 gives the adjacency variables for the generated layout. The construction of the slicing structure and the already computed cut points of each partition allow the calculation of the indexes. These indexes were then adjusted to make them comparable with those for the real layout. The adjustment factor is 1.268 and is obtained from Table 1 by dividing the fair ground width of the real layout through the initial partition width of the slicing structure. The comparison of the two columns illustrates the significance of the adjacency index. In the real layout, area 62 produces the lowest adjacency index with 0.11, whereas in the generated layout six areas with an index smaller than 0.1 can be found. And there are even three areas with an index equal to zero (i.e. no adjacency to the round tour at all). On the other hand in the real layout, area 93 produces the highest index with 0.75, whereas in the generated layout eight areas with an index higher than 0.75 can be found. In fact, the variance of the adjacency index in the generated layout is higher than in the real fair layout. That is not surprising, because the fair layout designer uses his common sense (hopefully) to design the layout and therefore tries to allocate the exhibition areas equally, whereas our resolution method does not yet accomplish any optimization at all, and the geometric constraints enforcing horseshoe cuts lead sometimes to exceptional adjacency indexes, as reported above. To measure the overall adjacency of the fair layout solutions let m
Bi =
∑b n =1
in
m
and m
Ai =
∑a n =1
in
m
where m is the number of exhibition areas to be placed.
5
Aisle space between two neighbouring areas does not belong to the round tour (e.g. space between area 5 and 7 in Figure 1).
Fair layout design : Computational result
4-14
Table 4 lists these two averages according to the real, the generated, and the adjusted fair layout solution, where the round tour 0 is the round tour of each layout. Both B0 and A0 are significant lower for the generated layout comparing to the same variables for the real layout. This is due to the fact that no space is allocated for the visitor corridor by the slicing structure in the generated layout. Thus, the whole layout is much more compressed. To obtain an equality between both layouts, the generated layout is rescaled by multiplying the variables by the adjustment factor. The adjusted variables are then slightly higher than the real variables, as shown in Table 4. Keeping in mind that no optimization is performed, this result is even more surprising. Layout
B0
A0
l0
real
8.79 meters 0.36
664.00 meters
generated
7.81 meters 0.31
460.69 meters
adjusted
9.90 meters 0.39
584.29 meters
Table 4. Overall adjacency
What are the reasons of this improvement of the averaged adjacency variables ? The comparison is relativized by the following reasonings : • The adjustment factor produces an approximation of the generated model to the real model. Furthermore, the computation of the adjacency variables for the real layout is arbitrary regarding to the treatment of aisle space that is not valuated as belonging to the round tour (e.g. aisle space between area 5 and 7 in Figure 1), but can be seen as a potential place for a visitor to enter an area. • The adjusted model favours the adjacency index at the expense of the exhibition area shapes. The slicing structure does not support L-shapes for instance, and produces in some cases undesired area shapes (e.g. exhibition area 42 in Figure 10). • As already seen, the variance of the adjacency variable is higher for the generated layout than for the real layout. Thus, the index is unequally distributed, and the layout may produce several insufficient adjacencies. Length of the round tour Analogically, the length of the round tour was computed for the real, generated and adjusted model, as shown in Table 4. The behaviour resembles that of the adjacency variables, except that l0 of the adjusted layout is not able to outperform l0 of the real fair. This is due to some properties of the round tour construction not considered by the adjustment factor. In fact, the round tour in the real layout is approximated as a line centered in the visitor corridor, and its length is made artificially longer when the round tour goes along fix walls or turns around columns.
Fair layout design : Discussion and conclusion
5
5-15
Discussion and conclusion
The presented method is suitable to redesign the fair layout of an existing fair. The symbolic layout representation takes into account the exhibitor surface requirement of individual areas using a slicing structure, and associates a round tour to the layout corresponding to the horseshoe cuts. By comparing the obtained layout solutions, the properties of the adjacency index and the length of the round tour as criterias are discussed. Until now, no optimization is performed at all. An interesting extension would be the application of an algorithm that performs exchanges in the slicing tree. By 2-way exchanges of the cut symbols on the internal nodes, or by exchanging the exhibition areas assigned to leaves, several slicing trees would be generated. This leads to different layout solutions, constructed from the slicing trees. The discussed criterias could serve as objective functions to evaluate the different solutions generated by a heuristic. A second potential for future work is the introduction of new cut symbols. On the one hand, more sophisticated basic cuts could allow the generation of arbitrary area shapes. On the other hand, new horseshoe cuts to represent more exhaustive round tour courses are imaginable. The problem of constructing an initial slicing tree is not adressed in this paper. But it is important to understand that the structure of a given slicing tree is crucial regarding to the layout solution deduced from the slicing tree. Consequently, future research on fair layouts must analyse the construction of slicing tree in more detail. Except the surface requirement of an exhibition area, the method does not meet practical constraints. Yet, in practice a lot of different constraints can arise depending on the exhibition areas or the fair ground, such as preferred location of exhibition areas, special area shapes, grouping of exhibitors, and fix walls, columns or installations (e.g. power supply) of the fair building. To really assist the fair layout designer in his task, a computerized layout procedure must support these kind of constraints.
Acknowledgements The authors would like to thank Mr. François Clerc, President of the Romont fair organization’s committee, for his collaboration.
Fair layout design : Appendix. Recursive construction
5-16
Appendix. Recursive construction
Slicing tree
Step 0 : Initial block
Step 1 : HS1right
Step 2 : HS2below
Step 3 : HS1right
Step 4 : HS1up
Fair layout design : References
5-17
References [Dom97]
Domschke W., Krispin G. (1997) Location and Layout Planning, A survey. OR Spektrum Vol. 19 p. 181-194
[Fra92]
Francis R. L., McGinnis L. F., White J. A. (1992) Facility Layout and Location, An analytical approach, 2nd edn. Prentice-Hall, Englewood Cliffs
[Her87]
Heragu S., Kusiak A. (1987) The facility layout problem. European Journal of Operational Research Vol. 29 p. 229-251
[Koo57]
Koopmans T.C., Beckmann J.C. (1957) Assignment problems and the location of economic activities. Econometrica Vol. 25 p. 53-76
[Mel96]
Meller R.D., Gau K.-Y. (1996) Facility Layout objective functions and robust layouts. International Journal of Production Research Vol. 34 No. 10 p. 2727-2742
[Tam92]
Tam K. Y. (1992) Genetic algorithms, function optimization, and facility layout design. European Journal of Operational Research Vol. 63 p. 322-346
