We give an introduction to possible applications of MaxSAT solvers in the area of automotive (re-)configuration. Where a SAT solver merely produces the answer “unsatisfiable” when given an inconsistent set of constraints, a MaxSAT solver computes the maximum subset which can be satisfied. Hence, a MaxSAT solver can compute repair suggestions, e.g. for non-constructible vehicle orders or for inconsistent configuration constraints. We implemented different state-of-the-art MaxSAT algorithms in a uniform setting within a logic framework. We evaluate the different algorithms on (re-)configuration benchmarks generated from problem instances of the automotive industry from our collaboration with German car manufacturer BMW.
The well-known NP-complete SAT problem of propositional logic—is a given propositional formula satisfiable— has many practical applications; see [Marques-Silva, 2008] for an overview. Ku¨chlin and Sinz [Ku¨chlin and Sinz, 2000] pioneered the application of SAT solving for the verification of the configuration constraints and the bill-of-materials in the product documentation of the automotive industry on the example of Mercedes-Benz. A standard problem to be solved there is the following: Given a (sub-)set O = {o1, . . . , on} of equipment options and a set C = {c1, . . . , cm} of configuration constraints whose variables are all options, is it possible to configure a car with the options in O such that C is satisfied? This gives us the SAT problem SAT(C ∪ O), where the options form unit clauses. If the answer is true, then the partial configurationO is valid and can be extended to a full valid configurationF which satisfiesC, and F can be readily obtained from the SAT solver.
For the unsatisfiable case, two main questions arise: (1) Which constraints (or clauses for a CNF formula C) of the input formula caused the unsatisfiability? (2) How many (and which) clauses can be maximally satisfied? WSI Informatics, Uniwww-sr.informatik.
The first question can be answered with proof tracing techniques [Zhang and Malik, 2003; As´ın et al., 2010]. Here a CDCL SAT solver records a trace while solving the formula. From this trace, a resolution based proof can be deduced, which shows the clauses involved in the unsatisfiable core. An unsatisfiable core is also called conflict.
The answer to the second question can be of important
practical use, too. For example, a customer may want to know
a maximal valid subset of an invalid O. Similarly, the car
manufacturer may want to know which maximal subset of C
is still satisfied by a currently invalid, but frequently desired
option set. This optimization problem can be answered with
MaxSAT, a generalization of the SAT problem
Some modern MaxSAT algorithms use SAT solvers as sub-routines by reducing the problem to several SAT solver calls [Fu and Malik, 2006; Marques-Silva and Planes, 2008; Anso´tegui et al., 2009]. With this approach, we can make use of all modern techniques (such as clause learning, nonchronological backtracking, or watched literals) of state-ofthe-art SAT solvers, which are not generally applicable to MaxSAT solvers.
MaxSAT can be used to answer further questions of practical use. For example: (1) After choosing components with priorities, what is the maximum sum of priorities that can be achieved for a valid configuration? (2) When considering the price of each component, how much is the minimal cost of a valid configuration?
Reconfiguration is of high practical relevance in the automotive industry [Manhart, 2005]. The after-sales business asks for extensions, replacements, or removal of components of a valid configuration with minimal effort. For example, when replacing the alarm system with a newer one, or when moving a vehicle from the U.S. to Europe, we would like to keep the maximal number of already installed components. One approach for reconfiguration uses answer set programming (ASP), which is a decidable fragment of first-order logic [Friedrich et al., 2011]. In this paper, we will describe a MaxSAT based approach for reconfiguration.
This paper is organized as follows. Section 2 defines the MaxSAT variants and notations. In Section 3 we give a short introduction to automotive configuration based on SAT, followed by a complete example. In Section 4 we describe our approach to use MaxSAT for automotive configuration to solve the above questions followed by detailed complete examples. Section 6 shows experimental proof-of-concept results based on different modern MaxSAT solvers. Section 8 concludes the paper. 2
A Boolean assignment v is a mapping from a set of Boolean variables X to {0, 1}. If a propositional formula ϕ evaluates to true under an assignment v (denoted as v |= ϕ), we call v a satisfying assignment or model for ϕ, otherwise an unsatisfying assignment. The SAT problem of propositional logic is the question whether such a satisfying assignment v exists for a given formula ϕ or not.
A literal is a variable or its negation. A clause is a disjunction of literals. Given a propositional formula ϕ = Vm i=1 ψi in conjunctive normal form (CNF) over n variables, where ψi is a clause for all 1 ≤ i ≤ m and m ∈ N≥0, the solution to the Maximum Satisfiability problem (MaxSAT) is the maximal number of clauses which can be satisfied by an assignmentv. Equation (1) shows a formal definition.
m MaxSAT(ϕ) := max X j=1 kψikv v ∈ {0, 1}n Where kψikv = 1, if v |= ψi, otherwise kψik = 0.
We notice that for the corresponding MinUNSAT problem whose solution is the minimum number of unsatisfied clauses, equation (2) holds.
MaxSAT(ϕ) + MinUNSAT(ϕ) = m Equation (2) also holds for the same resulting model. As a consequence, we only have to compute one problem to directly get the optimum and the corresponding model for both problems.
There are two extensions of the MaxSAT problem, called Weighted MaxSAT (WMaxSAT) and Partial MaxSAT (PMaxSAT). As the name suggests, in a weighted MaxSAT instance each clause ψi has a weight wi ∈ N≥0 (denoted by the tuple (ψi, wi)). The Weighted MaxSAT problem then asks for the maximal sum of weights of satisfied clauses. Furthermore, in a partial MaxSAT instance, the clauses are divided into disjoint hard and soft clauses sets: Hard ∪˙ Soft. An optimal solution satisfies all hard clauses and a maximal number of soft clauses. Both extensions can be combined to Partial Weighted MaxSAT (PWMaxSAT).
The relationship of equation (2) also holds for each MaxSAT variant. 3
Automotive configuration can be represented as a constraint
satisfaction problem
We will give a simplified and short introduction into this representation: (1) Each component (option) c is represented by a separate variable xc; the component will be used in the final configuration assignmentv if and only if v(xc) = 1; (2) components of a family (e.g. different steering wheels) will be restricted by cardinality constraints [Sinz, 2005; Bailleux et al., 2009] to choose exactly one (or at most one, if the component is an optional feature); (3) dependencies between components are expressed as clauses (e.g. the implication (xa ∧ xb) → (xc ∨ xd) means “If components a and b are chosen, then component c or d has to be chosen (or both)”; in clause form (¬xa ∨ ¬xb ∨ xc ∨ xd)).
The resulting formula in CNF is: ϕcar := ϕcc ∧ ϕdep (3) Where ϕcc are the clauses of the families’ cardinality constraints and ϕdep are the clauses of the dependencies between the components. With this representation, we can answer the following questions using a SAT solver:
2. Forced component: A component, which is used in every valid configuration. 3. Redundant component: A component, which can never be used in any valid configuration. 3.1
We consider the families of components with their limitations listed in Table 1.
Furthermore, we consider the dependencies between the components listed in Table 2.
case 1 by attaching priorities to the components: A customer chooses components c1, . . . , cn which lead to an invalid configuration. Additionally, the customer has priorities p1, . . . , pn, pi ∈ N>0, for each component. We can answer the question, which sum of priorities can be maximally reached for a valid configuration by solving Formula (6) with a Partial Weighted MaxSAT solver.
ϕcar har|d{czla}uses ∧ (xc1 , p1) ∧ . . . ∧ (xcn , pn) | } soft {clzauses 3. (Reconfiguration) We can use the introduced techniques in the use cases 1 and 2 for reconfiguration. Let us assume a customer wants to add, replace, or remove components of her existing car. She chooses the components c1, . . . , ck with priorities p1, . . . , pk ∈ N>0. If the priority or partial state (hard or soft) of a clause of an originally chosen component has changed, the original clause will be replaced by the new partial weighted clause. Otherwise, the clause will be kept. We solve Formula (7) with a Partial Weighted MaxSAT solver to reach the maximal sum of priorities.
For example, the implication “ G1 → E1 ∨ E2” means “If gearbox G1 is chosen, then engine E1 or E2 has to be chosen”.
With the resulting formula ϕcar from the above specifications, we consider two customer cases: 1. A customer chooses engine E1 and control unit C1 for the car. But she does not want the air conditioner AC2. We test Formula (4) for satisfiability.
ϕcar ∧ xG1 ∧ xC1 ∧ ¬xAC2 (4) The result is true. Derived from the resulting model, we can choose the components D1, C1, G1, E1 to get a complete valid configuration assignment. 2. A customer chooses the components E1, G2, C2, D3 and N2, AC1, AS1, R2. The result is false.
The question now is, which maximal subset of the original choice will lead to a valid configuration?
With the SAT based configuration, two main problems arise. First, if the configuration is not valid, it is not possible to know which components cause the conflict. Second, even if we know the components causing the conflict, we do not know, which components to omit to get a valid configuration with a maximal number of components we wanted originally. The example 2 of Subsection 3.1 shows such a case.
As mentioned in the introduction, the first problem can be handled with proof tracing to explain a conflict for an invalid configuration. The second problem can be handled with MaxSAT and its extensions. We explain this approach in the next section in detail. 4
For the representation of automotive configuration as a MaxSAT instance we consider the Partial MaxSAT problem. We use the SAT based specification ϕcar of Section 3 and divide the clauses into hard and soft ones. First, all cardinality constraints are marked as hard clauses, because they have to be satisfied (e.g. it is not possible to configure a car with more than one steering wheel). Second, it is possible that the dependencies between components do not necessarily have to be satisfied (e.g. a dependency could have been created due to marketing reasons; “No black seats for all Japanese cars”). On the other hand, technical dependencies have to be satisfied (e.g. a conflict between an engine and a gearbox). For simplicity reasons, we also mark all dependencies as hard clauses.
With the representation above, we can consider the following advanced use cases and answer the new arising questions with the help of a Partial (Weighted) MaxSAT solver:
chooses components c1, . . . , cn which lead to an invalid configuration. We can answer the question, what the maximal number of the chosen components for a valid configuration is, by solving Formula (5) with a Partial MaxSAT solver.
ϕcar har|d{czla}uses ∧ xc1 ∧ . . . ∧ xcn | } soft {clzauses (5) (6) (7) ϕcar har|d{czla}uses ∧ (xc1 , p1) ∧ . . . ∧ (xcn , pn) | } soft {clzauses To force certain new components to be installed or old components to be kept, we can designate the corresponding clauses as hard clauses.
To reach a valid reconfiguration for the customer, a reconfiguration scenario can be considered as a process in different steps: • Check for validation after the customer chooses new components with priorities as previously described. • If the hard clauses are unsatisfiable, check for validation after the sales division sets additional dependencies as soft clauses (with priorities). • If the hard clauses are unsatisfiable, check for validation after the engineering divison sets additional dependencies as soft clauses (with priorites).
If the hard clauses are unsatisfiable after all steps, there is no valid configuration, because technical limitations are reached which can not be set as soft clauses. Otherwise, if the hard clauses are satisfiable in one step, we can compute the maximal sum of priorities of the soft clauses while satisfying the hard clauses.
have prices p1, . . . , pn, pi ∈ N>0. We want to know which components have to be chosen, to get a valid configuration with minimal cost. We can answer the question by solving Formula (8) with a Partial Weighted MinUNSAT solver.
ϕcar har|d{czla}uses ∧ (¬xc1 , p1) ∧ . . . ∧ (¬xcn , pn) | } soft {clzauses (8) Instead of finding the minimal costs of a valid configuration, we could also compute a valid configuration of minimal weights, CO2 emissions, or other interesting targets.
In all situations above, the resulting model of the solver tells us which components to choose to get the optimum.
Additionally, we can add arbitrary hard clauses to enforce certain constraints: (1) Unit clauses to enforce the in- or exclusion of a component; (2) Additional dependencies between components (e.g. “When engine E1 is chosen, then choose gearbox G2”; (xE1 → xG2 )); (3) Additional cardinality constraints (e.g. xD1 ∨ xD2 to ensure that one of the dashboards D1 or D2 will be chosen).
For example, in Situation 4 (minimization of costs), we could add unit clauses to enforce the inclusion of certain components and then compute the minimal costs of the configuration. The result is a valid configuration with minimal costs which includes our chosen components. 4.1
We reconsider the example in Subsection 3.1.
1. In the second case, the choice of the customer was unsatisfiable. With the MaxSAT based approach of configuration we can find an assignment of a valid configuration where a maximum number of components is included. After solving Formula (5) with a Partial MaxSAT solver, we obtain the results shown in Table 3. We can reach a valid configuration by changing two of the choices (bold rows in the table) and therefore, we can keep 6 of our 8 original components at most. For the alarm system, the resulting model did not set another alarm system variable to true, because this is an optional feature.
In general, the result obtained from the solver may not be the only optimum. There can be other different assignments with the same number of satisfied clauses. xD4 ). 2. We consider another case, where the customer chooses the components with priorities as shown in Table 4. Additionally, she wants dashboard D2, D3, or D4. To enforce this constraint, we add the hard clause (xD2 ∨xD3 ∨ R2 3. After the previous configuration, the customer wants to reconfigure her existing car. Table 5 shows her choice. We can imagine that for technical or financial reasons, the engine E1 and gearbox G2 can not be replaced. We set them as hard clauses. However, control unit C2 and dashboard D4 can possibly be replaced and therefore are set as soft clauses. The results show that dashboard D4 was replaced by dashboard D2 and radio R2 has to be removed in favor of other components. 4. Now we associate the components with prices (as shown in Table 6) and we want to know a valid configuration with a minimal total price. For the minimal costs we solve Formula (8) with a Partial Weighted MinUNSAT solver. For the maximal costs we solve Formula (6) with a Partial Weighted MaxSAT solver by considering the prices as priorities. The results are: • Minimal cost: e 3,900 • Maximal cost: e 8,625 Table 7 lists the components to choose to reach the minimal and maximal costs.
In order to give the reader an impression of how MaxSAT can be computed, we present a short incomplete overview of some algorithmic techniques.
Branch-and-Bound The general branch and bound approach to explore the search tree of optimization problems can also be used for solving MaxSAT and its extensions. Each node of the tree represents a variable of the instance and has two children for the two values the variable can be assigned to. Tree pruning is used as soon as a partial solution becomes worse than the best solution found elsewhere in the tree. Heuristics have been developed e.g. by Wallace and Freuder to narrow the search space predicting the final value of partial solutions [Wallace and Freuder, 1993]. Basic SAT-based Given an unsatisfiable SAT problemϕ = {C1, . . . , Cm}, we may iteratively try to remove individual clauses Ci until the subproblem ϕ0 becomes satisfiable. ϕ0 will then be maximal in the sense that adding another clause will make it unsatisfiable, but another, larger, subproblem may exist which could be found by removing clauses from ϕ in a different order.
In SAT solving, clause removal can be simulated by augmenting each clause Ci with a fresh blocking variable bi. As long as bi is set to false, the solver needs to satisfy Ci, but the constraint Ci can effectively be blocked by setting bi to true instead. Now, in order to remove as few clauses as possible, we add m blocking variables to ϕ as above and restrict the use of the bi by an additional cardinality constraint CC(k), which is a formula that prevents more than k of the bi to be set to true. Iterating over k from below until ϕ(k) becomes satisfiable, or from above untilϕ(k) becomes unsatisfiable, gives us the MaxSAT result m − k, and the subset of clauses whose bi are set to false forms one satisfiable subset of maximum cardinality.
Algorithm 1 reflects basic approach. One improvement of this approach is the use of binary search.
Algorithm 1: Basic SAT-based approach Input: ϕ = {C1, . . . , Cm} Output: Minimal number of unsatisfied clauses ϕ ← {C1 ∨ b1, . . . , Cm ∨ bm} cost ← m while SAT(ϕ ∪ CNF(Pm
i=1 bi < cost)) do cost ← cost − 1 return cost Core-guided SAT-based Modern proof-tracing SAT solvers return an unsatisfiable subset (unsat core) μ ⊆ ϕ when given an unsatisfiableϕ. It is then clear that at least one clause of μ has to be blocked before ϕ can become satisfiable, and thus the search can be narrowed compared to the basic approach. An algorithm based on this idea was proposed by Fu and Malik for partial MaxSAT [Fu and Malik, 2006]. In every iteration where the instance is unsatisfiable, we add a new blocking variable to all soft clauses of the unsatisfiable core and a new cardinality constraint to achieve that exactly one of the currently added blocking variables has to be satisfied. We can not just iterate over the unsat cores and count them, because they may not be disjoint.
This idea can also be extended for partial weighted MaxSAT [Anso´tegui et al., 2009]. 6
For our benchmarks we used product configuration formulas of a current (2013) product line of the German car manufacturer BMW. We added unit clauses to create unsatisfiable customer orders. We defined the following three categories for hard and soft clauses: • Order: Soft clauses are unit clauses of the customer’s order. All other clauses are hard. This asks, wich of the
customer’s wishes can be maximally satisfied. • Packages: Soft clauses are clauses which represent packages, e.g. a sports package, which triggers all relevant sports components. The unit clauses of the customer’s order and all other clauses are hard. This asks, which of the package restrictions can be maximally satisfied w.r.t. the customer’s wishes. • Packages & more: Soft clauses are package clauses and additional other sales relevant conditions. The unit clauses of the customer’s order and all other clauses are hard. This asks, which of the package restrictions and additional restrictions can be maximally satisfied w.r.t. the customer’s wishes.
The upper half of Table 8 shows detailed statistics about each category. The second half of the table shows how many instances have an optimum. No optimum means that there is at least one conflict involving only hard clauses. The average optimum is the average of the result of the minimal number of unsatisfiable clauses. For example, the average optimum of 2.127 within the ‘Order’ category means that on average 2.127 of the customer’s choices can not be satisfied.
We applied our benchmarks to three different state-of-theart MaxSAT solvers, namely: • akmaxsat [Ku¨gel, 2012]: A partial weighted MaxSAT solver based on a branch-and-bound approach. One of the best performing solvers in last year’s MaxSAT competition1. • Fu & Malik [Fu and Malik, 2006]: A partial MaxSAT solver based on exploiting unsatisfiable cores and adding blocking variables to each soft clause of each found unsatisfiable core. • PM2 [Anso´tegui et al., 2009]: A partial MaxSAT solver based on exploiting unsatisfiable cores. But unlike the Fu & Malik solver this approach only uses exactly one blocking variable to each clause.
For akmaxsat we used the implementation of Adrian Ku¨gel2. We implemented the Fu & Malik and PM2 algorithms on top of our own Java SAT solver, which is optimized for our industrial collaborations. The cardinality constraints in the Fu & Malik approach are only of the form Pn i=1 xi = 1 for given 1http://maxsat.ia.udl.cat:81/12 2http://www.uni-ulm.de/in/theo/m/alumni/ kuegel.html variables x1, . . . , xn. We encode this constraint through the constraints (Win=1 xi) and Vin=1 Vjn=i+1(¬xi ∨ ¬xj ) . The cardinality constraints in the PM2 approach uses general limitations, which we implemented with the encoding proposed in [Bailleux et al., 2009].
All our benchmarks were run on the same environment: Operating System: Ubuntu 12.04 64 Bit; Processor: Intel Core i7-3520M, 2,90 GHz; Main memory: 8 GB; JVM 1.7.0 (for Fu & Malik and PM2).
Table 9 shows the results of our time measurements of each solver in each category. The listed times are the average times a solver needed to solve an instance of a category. We listed the average time in each category Solver akmaxsat has an average time of remarkable less than 0.6 seconds in each category. Our implementation of Fu & Malik has a reasonable average time of less than 6 seconds in each category. Our implementation of PM2 has a reasonable average time for the first category ‘Order’, but exceeded our time limit of 3, 600 seconds per instance on too many instances of categories ‘Packages’ and ‘Packages & more’ to get a reasonable average time.
Figures 1, 2 and 3 show the performance of each solver in the first category ‘Order’. These figures show the relation between the optimum and the response time of the instances. Especially for Fu & Malik and PM2 the response time seems to grow linearly with increasing optimum.
4 Optimum (quantity) 5 1 2 6 7 3
4 Optimum (quantity) 5 7
In [Junker, 2004] general satisfaction problems are considered, where we have a knowlegde base of constraints which have to be satisfied and customer requirements, which we would like to satisfy. In the context of MaxSAT, the knowledge base can be considered as hard clauses, whereas the customer requirements can be considered as soft clauses. In the case of inconsistency, the proposed algorithm QuickXplain delivers preferred explanations, which are based on a given total ordering of the constraints.
The work of [Reiter, 1987] proposes an algorithm for computing minimal diagnoses using a conflict detection algorithm. A diagnosis is a minimal subset Δ of the customer requirements, such that the constraints without Δ is consistent. In [Felfernig et al., 2012] another algorithm is proposed, called FastDiag, which computes a preferred minimal diagnosis without calculating the corresponding conflicts. 8
In this paper we showed detailed examples of how MaxSAT and its extensions can be applied in automotive configuration. With this approach we are able to repair an unsatisfiable customer order by computing the optimal solution which satisfies as many of the customer’s choices as possible. Furthermore, we showed how MaxSAT also can be used in reconfiguration scenarios. From an already configured car we can compute the minimal number of components to change when adding, changing, or removing components.
We created realistic benchmarks for our MaxSAT applications out of the product formulas of our commercial collaboration with BMW. Our time measurements of these benchmarks against the state-of-the-art MaxSAT solvers akmaxsat, Fu & Malik, and PM2, showed that we have a reasonable response time, except for PM2 in two categories. These results suggest that MaxSAT can be applied for industrial automotive (re-)configuration problems. 1 2 6 7 3
4 Optimum (quantity) 5
In Figure 4 we can also recognize the linear growing response time with increasing optimum. Also note the lower line of quickly solvable instances.
0 2 4 6
8 10 12 Optimum (quantity) 14 16 18 20 [Reiter, 1987] Raymond Reiter. A theory of diagnosis from first principles. Artificial Intelligence , 32(1):57 – 95, April 1987.