Aug 15, 2012 - The problem of discrete optimization is solved regarding the choice of menu structure minimizing the average search time. It is demonstrated ...
ISSN 0005-1179, Automation and Remote Control, 2012, Vol. 73, No. 8, pp. 1410–1423. © Pleiades Publishing, Ltd., 2012. Original Russian Text © M.V. Goubko, A.I. Danilenko, 2010, published in Problemy Upravleniya, 2010, No. 4, pp. 49–58.
CONTROL SCIENCES
Mathematical Model of Hierarchical Menu Structure Optimization M. V. Goubko and A. I. Danilenko Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia Received March 10, 2010
Abstract—A mathematical model is proposed to optimize the structure of hierarchical menus and directories. The model considers each element popularity. The problem of discrete optimization is solved regarding the choice of menu structure minimizing the average search time. It is demonstrated that optimal menu panels should provide the user with identical number of options having popularity levels split in the same proportion. It is indicated that the model allows for comparing the types of menu, as well as for choosing the best one. A certain algorithm is developed to design optimal menu, taking into account both semantic constraints and results of optimization. Application of the algorithm is illustrated using mobile phone menu optimization as an example. DOI: 10.1134/S0005117912080140
1. INTRODUCTION Hierarchical menus represent a popular method of providing access to commands and data. The corresponding examples are command menus in computer programs (Fig. 1), directories of websites (Fig. 2), mobile phone menus, interactive telephone systems, etc. As a rule, menu is (not always the only) complete directory of commands or links within the system; consequently, easy-to-use menu is an important quality factor of the whole interface. The
Fig. 1. A fragment of Apple Keynote command menu. 1410
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Fig. 2. A fragment of Google™ Directory two-level menu.
appearance and user properties of menu may vary depending on the location and tasks fulfilled by the menu. For instance, mobile phone menu is designed under small screen size of the device; voice menus possess no visual interface, resulting in additional constraints imposed on their structure and content. The menu of computer programs (see Fig. 1) should have standard appearance being familiar to users of the corresponding operating system. Designing websites is subject to less rules, thus giving full play to creative development of the menus and directories. Two-level hierarchy is widely adopted for, e.g., large directories of websites (illustrated in Fig. 2); the same panel serves to display both the categories and underlying subcategories. Generally, menu efficiency is defined by the time to access required element. Note for specific set of elements there exist numerous ways to group them in hierarchy of categories. The problem of menu design consists in finding a certain hierarchy of categories ensuring the minimum access time (taking into consideration the choice of the suitable view for every panel). 2. RELATED WORK Before computer systems were actually developed, similar problems had been treated within the theory of questionnaires [1]. In addition to detailed description of the theory, paper [1] draws an analogy with the theory of organizational hierarchy optimization (we emphasize the latter involves the same mathematical techniques as the present paper). Wide adoption of computer graphical interfaces in the 1980s initiated intensive development of formal methods to optimize menu structure (rich in content, yet somewhat out-of-date overview could be found in [2]). Today, most attention is focused on improving the user properties of websites, mobile phones and consumer electronics [3–5]. A series of publications are dedicated to studying various principles of menu organization. Menu items can be located in alphabetical order [6], in random order [7, 8] or sorted with respect to subcategories [9, 10]. Structure and display of menu items may vary depending on their usage [11, 12]. An independent direction of research is the impact of breadth and depth of hierarchy on search time. The central issues is, first, how many options should be suggested to the user in a single panel of menu (this number is known as the breadth of hierarchy) and, second, how many levels should exist within the hierarchy (what depth should it have). There are different approaches to the stated issues. Classical model [8] (later extended in paper [10]) analyzes the design of symmetric menu with fixed breadth over the elements possessing identical popularity; it is based on the assumption that selecting any item of the menu requires the same time. The suggested linear model gives the expression for optimal breadth of hierarchy (notably, 6–8 items on the average). The results derived there are far from experimental ones. An alternative model is suggested in paper [6] under similar constraints, yet with logarithmic relation between search time and breadth of hierarchy. AUTOMATION AND REMOTE CONTROL
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This model claims optimality of wider hierarchies, see also [11]. The present paper could be treated as a continuation of these works, since the current model generalizes the approaches described in [6, 8, 10] and eliminates several constraints. A prominent role in analysis of menu structures is played by numerous experimental investigations [7, 9, 13–17] based on different models and prerequisites. Notwithstanding general principles that have been identified during the research, distinctions among the approaches and artificial character of experiments make it impossible to reach a common opinion regarding the impact of breadth and width of hierarchy on usability of a menu. Some works deal with variable breadth of hierarchy within the menu. For instance, papers [17, 18] demonstrate that broader menus at last levels are more efficient against the menus getting narrower at the end. However, e.g., in [16, 19] the authors discover no substantial distinctions between such structures. In addition, there is a direction of research concentrated on studying the features of specific menus (voice menus [20], mobile phone menus [3, 5] and others). Therefore, the existing investigations of hierarchical menu structure possess significant differences both in assumptions regarding properties of the system and user behavior and in the methods involved. Moreover, it appears difficult to apply the obtained results (theoretical and experimental ones) in practice. Notably, theoretical results are subject to strong constraints being not valid for real menus. On the other hand, experimental methods consider specific field, and the corresponding results not always allow for extension. Such situation has led to a series of works (e.g., [21]) that compare various methods and estimate their applicability depending on conditions and user strategies. Still, up to now there has been no unified technique to describe the principles of menu design in different cases. Such technique should provide elements of menu structure optimization, as well as should enable comparing various types and structures of menu in the sense of user friendliness. This paper analyzes a certain mathematical model answering the formulated questions via the results of mathematical theory of hierarchical structure optimization [22, 23]. Being adopted within the model under consideration, general approach makes it possible to weaken several assumptions of the existing models; thus, the obtained experimental results can be used to develop computer-aided design (CAD) tools for menus. 3. PROBLEM STATEMENT Building hierarchical menu is based on a set of elements, N = {1, . . . , n} (e.g., commands or links in web directories) that should be accessed through the menu. Denote by μ(w) the popularity level of the element w ∈ N , i.e., probability of the event that the user would need this element. When designing the menu, the set N of elements is divided into categories; next, the elements within each category are divided into subcategories, and so on. We will call by the menu panel for the category s ⊆ N the totality composed of the category s and the set of underlying options (subcategories or terminal elements). For instance, Fig. 1 illustrates four opened menu panels. The upper-level categories (Slide, Format, etc) are located in the main panel of the menu, while next level categories belong to the embedded panels. Such multilevel division of elements into categories forms tree-like structure of the menu. Tree leafs are given by the elements of the set N , and the rest nodes appear responsible for the categories. Each category is uniquely characterized by the subset s ⊆ N of the corresponding lower-level elements. Note probability of the event that the user enters the menu panel that relates to the μ (w).1 category s constitutes μs := w∈s
1
This formula holds when the user makes no mistakes during selection of the menu options. AUTOMATION AND REMOTE CONTROL
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Imagine the user enters the menu panel of the category s ⊆ N which includes k options with popularity levels μ1 , . . . , μk ; in this case μ1 +. . .+μk = μs . Hence, yi = μi /μs defines the conditional probability of choosing the option i provided the user staying in the menu panel of the category s. In other words, the vector y1 , . . . , yk determines the proportion (note y1 + . . . + yk = 1 according to the definition) used to divide the popularity levels of the options included in the category s. As a performance criterion for the menu, let us involve the average search time consumed by the user to find a required element (this is the average time of a single user session). It is often supposed (see [6–10, 14–17]) that the user seeks to reach one lower-level element (e.g., a single record in the access menu to the databases or a single command in command menus). Denoting by t(w) the user search time of the element w ∈ N , one has to minimize the average access time to a single element T =
μ (w) t (w).
w∈N
On the other hand, the total time is spent by the user to find proper options within the categories; consequently, the average session time could be represented as the sum of average time periods of staying in each panel of the menu. In the first place, the latter time depends on the set of provided options within specific panel. Moreover, selection time is essentially affected by user properties of the menu. For instance, menu options may have been sorted alphabetically or by the content, items in the menu panel may have been located in horizontal or vertical way (or even embedded in the cells of rectangular matrix); categories may have been represented by their names, icons or by the both, etc. Hereinafter, the type of menu will be understood as the totality of all factors influencing the appearance2 and user properties of the menu; except for the number and composition of the menu items themselves. Let Ω be the set of all possible types of menus that could be employed in the current context. Given the particular type θ ∈ Ω of the menu, denote by ti (k, θ) the average time required to find and select the ith option3 in the menu panel which includes k options. This time includes the period to understand and analyze the options, as well as the period to select the necessary one (e.g., to move the cursor or press the button). In addition to the set of options and type of the menu, several factors have an impact on the selection time; personal properties of the users (qualification, age), technical constraints of the hardware and others are among them, that may not be affected by the menu designer. If information on potential users, hardware delays and other external parameters appears available at the design stage, it should be accounted for in the parameter ti (k, θ). Such detailed consideration allows for deriving more precise (yet, less universal) recommendations regarding the menu structure. In the absence of the mentioned information, one should utilize the average values4 making it possible to obtain universal (but “optimal in the mean”) parameters of the menu. Experimental investigations are the primary way to evaluate ti (k, θ). Suppose the user gets into the menu panel of the category s ⊆ N with k(s) subcategories s1 , . . . , sk . Multiplying the selection time ti (k, θ) of every option by the corresponding conditional probability yi (s) := μsi /μs (the event that this option is selected), one obtains the average time required to the user for choosing the option in the menu panel:
k(s)
t y1 (s), . . . , yk(s) (s), θ =
yi (s)ti (k(s), θ).
i=1 2 3 4
For convenience in the sequel we use “visual” terms, but the same results are valid for other types of menus including voice ones. For each combination of k and θ, we believe a certain numbering of menu items from 1 to k is specified. Below we show that time linearly enters the optimization criterion; hence, averaged parameters can be used. AUTOMATION AND REMOTE CONTROL
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In general case, specific type of menu may be employed for every panel; the type has to be selected during optimization. This could be performed independently for each panel through minimizing the time of stay in the panel: t
∗
k(s)
y1 (s), . . . , yk(s) (s) = min θ∈Ω
yi (s)ti (k(s), θ).
(1)
i=1
The average time of a single user session in hierarchical menu H is comprised of the following. This is the sum of average time periods of the user’s stay in every panel of the menu s (belonging to the hierarchy H) being multiplied by the probabilities μs of entering the panel in question: T (H) =
μs t∗ (y1 (s), . . . , yk(s) (s)).
(2)
s∈H
Therefore, the problem lies in finding a certain hierarchy minimizing session time (2) over the set of feasible tree-like hierarchies superstructured over the set N . 4. THE THEORY OF HIERARCHICAL STRUCTURE OPTIMIZATION: SOME ELEMENTS The problem of average search time minimization in hierarchical menu belongs to wide range of problems related to optimal hierarchical structures [22, 23]. The problems of evaluating optimal hierarchies arise in numerous fields; despite various interpretations of these problems, they possess common mathematical framework enabling the application of general approach to solve them. The underlying ideas of the approach are provided in books [22, 23]. In particular, it is demonstrated in [23] that the so-called “homogeneous tree” appears optimal for optimization criterion (2) being representable as the sum of uniform functions. In such tree every category has the same number of subcategories with popularity levels distributed in the identical proportion y1 , . . . , yk (y1 + . . . + yk = 1). An example of homogeneous tree having the breadth 2 and proportion (1/3, 2/3) is given by Fig. 3.
Fig. 3. An example of homogeneous tree.
The average search time within the hierarchy H possessing the breadth k with proportion y1 , . . . , yk could be defined by T (H) = −
w∈N
μ (w) ln μ (w)
t∗ (y1 , . . . , yk ) −
k i=1
.
(3)
yi ln yi
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It is shown in [23] that the problem of optimal tree-like structure is reduced to evaluation of tree parameters minimizing expression (3) with respect to all k = 2, . . . , n and proportions y1 , . . . , yk satisfying the condition yi ≥ min μ (w), i.e., w∈N
TL (N ) = −
μ (w) ln μ (w) min min k
w∈N
t∗ (y1 , . . . , yk )
y1 ,...,yk
−
k i=1
.
(4)
yi ln yi
Due to discrete parameters, it is often impossible to build the best homogeneous tree in precise way. However, the average search time (4) of the tree can be always calculated; note it provides rather accurate lower estimate for the average search time in optimal hierarchy. Several efficient algorithms [24] are developed to build many suboptimal trees based on various groupings of lowerlevel elements.5 5. OPTIMAL HIERARCHICAL MENU 5.1. General Conclusions Thus, within the framework of the model under consideration, optimal structure of the menu should be a homogeneous hierarchy; in other words, each panel of the menu should include the same (if possible) number of options with popularity levels divided in the identical proportion. Obviously, this makes inefficient the application of several types of menu within the same hierarchy. Indeed, formula (1) implies that the best menu type θ ∈ Ω for each panel s ⊆ N is evaluated through minimization θ ∗ (s) = arg min θ∈Ω
ks
yi (s)ti (k(s), θ).
(5)
i=1
Due to homogeneity of optimal hierarchy, the menu breadth k(s) and proportion y1 (s), . . . , y k(s) (s) are identical for all the panels. Consequently, the minimum of (5) would be attained at the same point for all the panels. Reformulating this statement, all panels of optimal menu possess the same type irrespective of their location in the hierarchy. Hence, one may first find parameters of the best homogeneous tree for each of the menu types θ ∈ Ω considered. Second, recommend as the optimal type of menu the one with the homogeneous tree having the minimum session time. Many factors should be taken into account when designing an easy-to-use menu (type of the menu panels, user strategy, qualification and personal properties of the users, etc.). The actions of different users may vary even in the same menu. For instance, the menu options can be viewed sequentially or (under known sorting of options) using the bisection method. The users who often address the menu may remember the location of the required option and choose it directly (the so-called “expert” behavior). In addition, personal properties of users (e.g., their age [17]) exert an impact on performance and number of mistakes being made. Users with nonstandard behavior are described by other formulas defining the time of viewing the menu panel. Note due to linear character of the model these time costs may be averaged over the whole group of users; thus, optimal in the mean menu for the whole group is given by the menu of the user with average time costs. Hereinafter we proceed from utilization of averaged estimates of time delay parameters. In the following paragraphs the problem of evaluating optimal menu structure is solved for classical models of user behavior; moreover, practically relevant complications of the model are either studied. 5
This allows for involving the present algorithm to construct suboptimal hierarchical menus under semantic constraints, i.e., requirements regarding the meaning of every category (see Section 6). AUTOMATION AND REMOTE CONTROL
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5.2. Exhaustive Search The strategy of exhaustive search [21] implies the user first runs over all options in the menu panel and then makes the choice. It is assumed that the user makes no mistake when choosing the necessary option. In this case, the selection time for a certain option in the panel appears independent of location of the option. Hence, it equals to the average time of the user’s staying in the menu panel t(k) = tresp + tload k + tread k + tclick .
(6)
Here tresp stands for server response time, tload means loading time for a single item of the menu, tread is the time needed to read and understand a single item of the menu, and tclick is the time required for making the choice. Optimal parameters of homogeneous tree (i.e., the number k of options and proportion y1 , . . . , yk ) ensure minimum to function (3). Having in mind expression (6), one obtains this is equivalent to minimization of the function −
(tresp + tload k + tclick ) + tread k k i=1
.
yi ln yi
It could be easily verified that for any fixed k symmetric hierarchy is optimal, i.e., yi = 1/k, i = 1, . . . , k. Therefore, optimal number of options k in every panel of the menu is evaluated through optimization (with respect to all k = 2, 3, . . .) of the following function: [(tresp + tload k + tclick ) + tread k] / ln k.
(7)
Example 1. Consider parameters being typical for web directories: tresp = 2 s, tload = 0.02 s, tread = 1 s, and tclick = 1 s. Substitute them in function (7) and perform numerical optimization to obtain that in optimal menu every panel should include k = 5 options. Note the options in the menu panel should have identical popularity. ♦ The stated results agree with the formula of optimal menu breadth derived in classical paper [8]; the principal difference lies in that consideration in [8] is initially limited to homogeneous symmetric trees. In contrast, we have demonstrated above that exactly homogeneous symmetric tree with the breadth defined by (7) is optimal over the set of all possible tree-like structures (including nonhomogeneous and asymmetric ones). Moreover, in paper [8] the results are developed under the condition of identical popularity of all the elements belonging to the set N ; we do not involve such constraint in the present paper. 5.3. Sequential Search The strategy of sequential search [21] implies the user runs over the options in the menu panel in specific sequence and selects the necessary option as soon as finds it (the rest options are then not examined). Under such model of behavior, selection time for the menu option depends on serial number of the option. The average time of the user’s staying in the menu panel is defined via the averaging with respect to all options (taking into account their relative popularity levels): t (y1 , . . . , yk ) = tresp + tload k + tread
k
iyi + tclick .
(8)
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Let A(k) = (tresp + tload k + tclick )/tread , then the average time of stay in the menu panel makes
t (y1 , . . . , yk ) = tread A(k) +
k
iyi .
(9)
i=1
Search time within the best homogeneous hierarchy is determined through minimization of function (3). According to expression (9), this is equivalent to minimizing the function A (k) + −tread
k i=1
k i=1
iyi (10)
yi ln yi
over all k’s and proportions y1 , . . . , yk . For fixed value of k, first-order conditions show that optimal proportion satisfies yi = a (k)A(k)+i , i = 1, . . . , k, where a(k) represents solution to the equation
k aA(k)+i = 1. Substitute optimal
i=1
proportion in function (10) to obtain the following formula of optimal number of options k in the menu panel:
k = arg min 1/ ln 1 a k k =2,3,...
.
(11)
Example 2. Recall typical parameters of menu given in Example 1. Substituting them in expression (11) yields the broader hierarchy with k = 13 options is optimal provided sequential strategy. Popularity levels in optimal homogeneous tree are distributed in proportion y1 ≈ 0.27, y2 ≈ 0.20, y3 ≈ 0.15, . . . , y13 ≈ 0.007. That is, the first option (among the total thirteen ones being viewed by the user) should have over a quarter popularity, while the last option should have less than 0.01. For the set N of elements, average time (3) of a single user session is approximately μ (w) ln μ (w). ♦ 3.25 × w∈N
Therefore, with the reader sequentially viewing the options in the menu panel, the most popular elements or categories should be located in the beginning; we emphasize the popularity levels of the options decrease in geometrical progression. In paper [8] the results of exhaustive search are generalized to the case of sequential strategy, viz., symmetric menu is suggested as the corresponding solution (the parameters of sequential strategy differ from that of exhaustive one). In this section such approach has been illustrated nonoptimal. For instance, under parameters used in Examples 1 and 2, the gain of optimal asymmetric menu constitutes more than 11% as against the structure proposed in paper [8]. Note asymmetric menus are almost not investigated in the literature, although there exist numerous fields (voice menus of telephone queuing systems, menus of mobile phones) where sequential search is a direct technological demand of menu implementation. The analysis of the current section appears relevant to these fields. Similarly, one may evaluate parameters of optimal menu for complicated strategies of search and types of menu (e.g., two-level menus, see Fig. 2). 5.4. Modelling of User Mistakes Sometimes the user is unable to correctly identify the category of the required element. This leads to mistakes during menu navigation. If the user selects wrong category, usually he may get back to the upper level (to the previous panel of the menu) and modify the choice. Let us assume that the names of terminal elements of the menu are formulated explicitly; thus, the user makes AUTOMATION AND REMOTE CONTROL
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mistakes with categories only. Two types of mistakes are possible as follows. The user may either choose wrong subcategory in the menu panel or miss the required subcategory in the correct panel and get back to the upper level. Suppose that, after having made the mistake in a certain panel, the user acts carefully and does not repeat the mistake during the second view of the panel in question. Behavior of the user making the first-type mistake can be modeled in the following way. The user enters the menu panel, views the options and makes (the wrong) choice according to his strategy (sequential or exhaustive search). In the new panel of the menu, the user has to examine all the options to understand the required one is absent among them. Then the user gets back to the parent panel of the menu and makes proper choice according to his strategy (as if he has committed no mistake). In the case of the second-type mistake, the sequence of actions performed by the user remains the same; the only exception is that the user consumes additional time to completely examine and exit from specific menu (finally, he returns there). For simplicity the description below is in terms of the first-type mistakes. Having made the mistake, the user spends additional time that could be divided into two components. The first component includes time required for wrong choice in the parent panel of the menu (making the mistake); the second one corresponds to viewing all the options within subcategory and leaving it (getting back to the previous panel). Time costs related to mistakes are defined by behavioral strategy of the user. For instance, if the user wrongly selects the option j in the menu panel with k options, then time costs due to the mistake under sequential search are specified by tjmistake = tresp + tload k + tread j + tclick .
(12)
In situation with exhaustive search, these costs appear independent of the chosen option: tjmistake = tresp + tload k + tread k + tclick .
(13)
For getting back to correct panel, the user has to view all the items in the subcategory and select the command “back to the parent level;” this procedure requires the time being equal to trecover = tresp + tload ksub + tread ksub + tclick ,
(14)
where ksub indicates the number of options in the wrong panel of the menu. Suppose in the menu panel the user makes mistake with probability p. Then the event of the category i being wrongly selected has the probability pyi ; the underlying interpretation in practice appears the following. The higher is the popularity level of the category, the greater is the corresponding probability of wrong selection. In this case, during a single session the user enters the category s by mistake with probability pμs . Therefore, the average time of staying in the panel is supplemented with specific periods caused, first, by possible mistake of the user and, second, by necessity to view the whole panel completely and exit from it (when the user wrongly enters the panel). Being summed up, these periods define a “makeweight” to the average time of the user’s stay in the panel s: μs ptrecover + μs
k
yi ptimistake .
i=1
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any choice in the panel, see formula (6). Hence, taking into account the probability of mistake, the average time of the user’s stay in the menu panel (provided exhaustive search) constitutes ts (k) = (1 + 2p) (tresp + tload k + tread k + tclick ) .
(15)
Substitute time (15) in expression representing the average search time (3) in the menu. Evidently, the time possesses linear growth with respect to the probability of mistake; nevertheless, the minimum points does not depend on p. In other words, under exhaustive search the probability of making mistakes has no impact on the structure of optimal menu. Consider situation when the users follow the strategy of sequential search. Add to time (8) supplementary periods (12) and (14) (having in mind the probability of mistake). After certain transformations, we obtain that the average time of the user’s stay in the menu panel is given by ts (y1 , . . . , yk ) = (1 + 2p) (tresp + tload k + tclick ) + ptread k + (1 + p) tread
k
iyi .
(16)
i=1
Similarly to Subsection 5.3, introduce notation B (k) =
(1 + 2p) (tresp + tload k + tclick ) + ptread k (1 + p) tread p p k = 1+ A (k) + 1+p 1+p
and rewrite time (16) as
ts (y1 , . . . , yk ) = (1 + p) tread B (k) +
k
iyi .
i=1
When minimizing (3), first-order conditions imply optimal proportion satisfies yi = b (k)B(k)+i , i = 1, . . . , k; here b(k) is a solution to the equation
k bB(k)+i = 1. Optimal number of options in
i=1
the menu is determined through minimization of the function 1/ ln(1/b(k )) over all k = 2, 3, . . . . Example 3. Assume in the menu panel the user makes mistake with probability p = 0.2. Under the same parameters as in Examples 1 and 2, optimal menu has k = 8 options for each panel. Note the proportion for popularity levels of subcategories are y1 ≈ 0.25, y2 ≈ 0.19, y3 ≈ 0.15, . . . , y8 ≈ 0.05. Compare the results with the ones of Example 2. Obviously, the user’s mistake being taken into account leads to narrower menus (enabling faster viewing the wrong menu and getting back to the correct one). ♦ 6. MENU DESIGN UNDER SEMANTIC CONSTRAINTS Within the described optimization problem, semantic aspects are not treated; notably, any groupings of the elements are feasible, and each of them is characterized only by total popularity level. In practice, the set of feasible menu structures appears narrow due to semantic constraints. In the first place, this is the requirement of intelligent filling the categories in the menu. Semantic quality of the categories’ names exerts a major impact on the selection time [25]. However, solving the problem without semantic constraints allows for drawing general conclusions regarding the structure and type of optimal menu, as well as for developing certain tools for (partial or complete) automation of menu filling process. The lower estimate (4) of time costs is still relevant to the problem under semantic constraints. Hence, one may first compare the expected time costs in the current structure of menu with the AUTOMATION AND REMOTE CONTROL
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lower estimate (4). The resulting difference being slight (not exceeding 5%), the current menu could be acceptable. When building new structure of the menu, the designer may involve optimal tree as a certain template and fill it with sensible content. It has been mentioned above that there exist large number of hierarchies (probability to find proper hierarchy with sensible grouping of elements into categories appears rather great). Using the algorithms [24] as the basis, it is easy to construct a certain algorithm of evaluating suboptimal tree (composed of sensible categories) for the problem with semantic constraints. Such algorithm presupposes the existence of elements classification with respect to several criteria (alternatively, the presence of a certain function to define semantic closeness of the elements). Unfortunately, in the majority of cases today, the process of menu building could not be totally automated. Verifying the meaningfulness of (2n − n − 1) nontrivial categories of elements turns out extremely time-consuming for the designer. Thus, it seems better (and more realistic) not to formalize semantic aspects during menu design, but to use the opportunities and experience of menu designer in this field. Example 4. Consider the following example of menu design for real system to illustrate the presented ideas. Mobile phones belong to those platforms having the problem of menu optimization as a major one. Since their screen is small, hierarchical menu appears almost the only alternative to provide access to numerous commands of the devices. In addition, usually dimensions of the screen do not allow for covering the whole menu; thus, the user has to view the options sequentially. (a)
(b)
Fig. 4. Mobile phone menu structures: (a) original and (b) optimized.
Consider the menu of sending and receiving the messages in Nokia 7510 mobile phone. Figure 4a briefly demonstrates the structure of original menu. Popularity levels of commands and categories (see the figure) are based on analysis of mobile phone usage and statistical investigations [3, 5]. It AUTOMATION AND REMOTE CONTROL
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should be emphasized that the menu structure is strongly nonhomogeneous. Notably, 14 options are provided at the top level, while just 2–5 ones exist at the second level (the lower levels are not illustrated). As it has been shown in Section 5, such structure could be optimized. Direction of optimization is defined by specific parameters of the menu. Experiments make it possible to evaluate the average time. In particular, switching time to new panel is independent of the number of options in the menu and constitutes tresp ≈ 1 s. The average time of reading a single item is tread ≈ 1 s (including the time of looking through the menu items in the screen). The average time of pressing the selection button makes tclick ≈ 0.5 s. Suppose the users follow the strategy of sequential search (inconvenient control tools confine behavior of the user) and make mistake in the panel with probability p = 0.05. Hence, the average time of the user’s staying in the panel is given by formula (16). Under the stated parameters, numerical computations imply that the average time of a single user session in the original menu is T ≈ 8.36 s. The results of Subsection 4.3 indicate that optimal menu has k = 8 options in each panel, while optimal proportion for the popularity levels is y ≈ (0.33, 0.23, 0.16, 0.11, 0.07, 0.05, 0.03, 0.02); in other words, the structure of optimal menu appears strongly asymmetric. Provided the given set of commands, theoretical minimum of the average access time equals to 6.74 s. However, building optimal tree is impossible due to actual constraints (menu commands could not be organized arbitrarily) and discreteness of the problem. Using the algorithms of suboptimal trees [24] and having in mind the actual constraints, one may reorganize the menu commands in the following structure (see Fig. 4b). According to numerical computations, the average time of the user’s staying in such menu approximates 7.22 s. Therefore, modifying the menu structure under the evaluated optimal parameters increases the speed of command access by 13.5% (in the mean). Compare the analytical and numerical results to observe that the constructed menu (even not being the only possible one) is close to theoretical minimum in time costs. At the same time, the optimized menu does not violate logical structure of commands in the category. ♦ 7. MENU DESIGN AUTOMATION Proceeding from arguments stated in Section 6, it seems possible to suggest a certain algorithm of menu design automation based on the presented theory. Let us describe the key steps of designing the menu structure using CAD systems. Step 1 . Menu designer loads the set of elements N and their popularity levels in the system (note popularity levels are obtained during experiments or based on expert appraisals). These elements will be included in the menu. Step 2 . The designer chooses the types of menu being applicable in current environment and defines the time required to perform elementary actions (taking into account the planned usage mode of the menu and qualification level of potential users). Step 3 . The system computes the lower bounds for the average time of a single user session (for different types of menu) and suggests the best type. Step 4 . The designer selects the type of menu being the most efficient and easy-to-use in current environment; thus, he fixes the optimal number k of items in the panel and the proportion y1 , . . . , yk of their popularity levels. Step 5 . The designer divides the set N of elements into intelligent categories s1 , . . . , sk , seeking to reach the optimal number of categories and distribution of their popularity levels. The system assists the designer through estimating the quality of current division, e.g., using the formula c(μ(s1 ), . . . , μ(sk )) − μ(N )c(y1 , . . . , yk ) . μ(N )c(y1 , . . . , yk ) AUTOMATION AND REMOTE CONTROL
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Step 6 . As soon as acceptable division has been obtained, the procedure is repeated within each category, and so on (until the level of individual elements is achieved). The system simultaneously evaluates the variation between the actual average time of a single user session and the optimal theoretical one. Step 7 . If necessary, the designer manually improves the resulting structure, modifying the categories with worse quality rates (17). The process of dividing the elements into categories could be either automated provided there is a certain classification of elements of the set N (for instance, keywords for websites or publications). Such classification may be employed for automatic distribution of elements within hierarchy with optimal (alternatively, near optimal) parameters. 8. CONCLUSION Analysis of the proposed model indicates that optimal structure of hierarchical menu represents a homogeneous tree. In general case it seems impossible to build absolutely homogeneous tree. However, there always exist numerous hierarchies being near optimal, and a certain menu with intelligent division of elements into categories could be found among them. The presented results allow for evaluating optimal parameters of menu; striving for them, one may design efficient menu (see Example 4). The described technique is applicable to wide range of menu types and behavioral strategies of the users. For instance, the ways of model complexification (in particular, taking into account the user mistakes) have been studied under basic strategies of the users. The general model makes it possible to show that involving a single type for all panels appears beneficial for a single hierarchy. There are many experimental investigations of user behavior in different-type menus. Having been provided in the paper, the general model fixes relevant parameters for the average search time; thus, it allows for concentrating on them the research efforts during experiments for further forecasting and optimization of menu structure. The derived theoretical results could be applied in CAD systems to build efficient menu. Based on parameters of external environment being selected by the user, the system computes optimal properties and type of the menu, as well as gives easy-to-use tools to fill the menu according to efficient template. During the filling process, certain tools are provided to compare the average access time in the obtained menu with the theoretical minimum time; this is to estimate the efficiency of the current structure. Further automation presupposes automatic filling the menu structure based on classification of elements. REFERENCES 1. Parkhomenko, P.P., Questionnaires and Organizational Hierarchies, Autom. Remote Control, 2010, no. 6, pp. 1124–1134. 2. Norman, K.L., The Psychology of Menu Selection: Designing Cognitive Control at the Human/Computer Interface, Norwood: Ablex, 1991. 3. Thimbleby, H., Analysis and Simulation of User Interfaces, in Proc. BCS Human Computer Interaction, 2000, pp. 221–237. 4. Thimbleby, H., Press On: Principles of Interaction Programming, Boston: MIT Press, 2007. 5. Andersson, E. and Isaksson, I.-M., Exploring Alternatives to the Hierarchical Menu Structure Used in Mobile Phones, Umea: Umea Univ., Dept. of Computing Sci., 2007. 6. Landauer, T.K. and Nachbar, D.W., Selection from Alphabetic and Numeric Menu Trees Using a Touch Screen: Depth, Breadth and Width, Proc. SIGCHI Conf. on Human Factors in Computing Systems, 1985, pp. 73–78. AUTOMATION AND REMOTE CONTROL
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7. Miller, D.P., The Depth/Breadth Tradeoff in Hierarchical Computer Menus, Proc. Human Factors Society’s 25th Annual Meeting, 1981, pp. 296–300. 8. Lee, E. and MacGregor, J., Minimizing User Search Time in Menu Retrieval Systems, Human Factors, 1985, vol. 27, no. 2, pp. 157–162. 9. Snowberry, K., Parkinson, S., and Sisson, N., Computer Display Menus, Ergonomics, 1983, vol. 26, pp. 699–712. 10. Paap, K.R. and Roske-Hofstrand, R.J., The Optimal Number of Menu Options per Panel, Human Factors, 1986, vol. 28, no. 4, pp. 377–385. 11. Cockburn, A., Gutwin, C., and Greenberg, S.A., Predictive Model of Menu Performance, Proc. ACM CHI’07 , 2007, pp. 627–636. 12. Sears, A. and Shneiderman, B., Split Menus: Effectively Using Selection Frequency to Organize Menus, ACM ToCHI , 2004, vol. 1, no. 1, pp. 27–51. 13. Shneiderman, B., Software Psychology: Human Factors in Computer and Information Systems, Cambridge: Winthrop, 1980. 14. Kiger, J.I., The Depth/Breadth Tradeoff in the Design of Menu-driven Interfaces, Int. J. Man-Machine Stud., 1984, vol. 20, pp. 201–213. 15. Jacko, J.A. and Salvendy, G., Hierarchical Menu Design: Breadth, Depth, and Task Complexity, Percept. Motor Skills, 1996, vol. 82, pp. 1187–1201. 16. Larson, K. and Czerwinski, M., Web Page Design: Implications of Memory, Structure and Scent from Information Retrieval, Proc. ACM CHI’98 , 1998, pp. 18–23. 17. Zaphiris, P., Depth vs. Breadth in the Arrangement of Web Links, Proc. 44th Annual Meeting of the Human Factors and Ergonomics Society, 2000, pp. 139–144. 18. Norman, K.L. and Chin, J.P., The Effect of Tree Structure on Search in a Hierarchical Menu Selection System, Behav. Inf. Techn., 1988, vol. 7, pp. 51–65. 19. Bernard, M.L., Examining a Metric for Predicting the Accessibility of Information within Hypertext Structures, PhD Dissertation, Wichita State Univ., 2002. 20. Roberts, T.L. and Engelbeck, G., The Effects of Device Technology on the Usability of Advanced Telephone Functions, Int. Conf. on Human Factors in Computing Systems CHI , 1989, pp. 331–338. 21. Hollink, V., Van Someren, M., and Wielinga, B., Navigation Behavior Models for Link Structure Optimization, User Model. User Adap. Inter., 2007, vol. 17, no. 4, pp. 339–377. 22. Voronin, A.A. and Mishin, S.P., Optimal’nye ierarkhicheskie struktury (Optimal Hierarchical Structures), Moscow: Inst. Probl. Upravlen., 2003. 23. Gubko, M.V., Matematicheskie modeli optimizatsii ierarkhicheskikh struktur (Mathematical Models of Hierarchical Structures Optimization), Moscow: Lenard, 2006. 24. Gubko, M.V., Algorithms to Construct Suboptimal Organization Hierarchies, Autom. Remote Control , 2009, no. 1, pp. 147–162. 25. Mehlenbacher, B., Duffy, T.M, and Palmer, J., Finding Information on a Menu: Linking Menu Organization to the User’s Goals, J. Human-Computer Interaction, 1989, vol. 4, pp. 231–251.
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