Showing papers in "Annals of Operations Research in 1998"

Stowage planning for container ships to reduce the number of shifts

Abstract: This paper deals with a stowage plan for containers in a container ship. Containers on board a container ship are placed in vertical stacks, located in many bays. Since the access to the containers is only from the top of the stack, a common situation is that containers designated for port J must be unloaded and reloaded at port I (before J) in order to access containers below them, designated for port I. This operation is called "shifting". A container ship calling at many ports may encounter a large number of shifting operations, some of which can be avoided by efficient stowage planning. In general, the stowage plan must also take into account stability and strength requirements, as well as several other constraints on the placement of containers. In this paper, we only deal with stowage planning in order to minimize the number of shiftings, without considering stability and several other constraints. First, we briefly present a 0-1 binary linear programming formulation that can find the optimal solution for stowage planning. However, finding the optimal solution using this model is quite limited because of the large number of binary variables and constraints needed for the formulation. Moreover, in [3] the stowage planning problem is shown to be NP-complete. For these reasons, the Suspensory Heuristic Procedure was developed.

Design and operational issues in AGV-served manufacturing systems

Abstract: Automated Guided Vehicle (AGV) systems are already in widespread use and their importance for material handling is expected to grow rapidly. The advantages that such systems can offer include increased flexibility, better space utilization, improved factory floor safety, reduction in overall operating cost, and easier interface with other automated systems. This survey paper focuses on design and operational issues that arise in AGV systems. The objectives of the paper are to unify various lines of research related to AGVs and to suggest directions for future study. We consider problems arising in flowpath design, fleet sizing, job and vehicle scheduling, dispatching and conflict-free routing. Flowpath design problems address computationally intractable issues in the physical layout of a single loop and complex networks. Transportation and related models, waiting line analysis and simulation approaches are used to address fleet sizing questions. Scheduling issues focus on three flowpath layouts. In line layouts, the most important issues include finding an efficient job sequencing algorithm and identifying optimal AGV launch times. In loop layouts, issues such as joint scheduling of the job and AGV schedules, interface with a larger manufacturing system, dynamic job arrivals, and the location of the AGV parking area, are important. For complex network layouts, joint scheduling, heuristic dispatching rules, and conflict-free routing of AGVs, are considered. We identify the inefficiencies that result from addressing these issues in isolation, suggesting the need for integration. We also provide a summary of the most important open research issues related to all the above topics.

Assignment and sequencing models for thescheduling of process systems

Abstract: This paper presents an overview of assignment and sequencing models that are used inthe scheduling of process operations with mathematical programming techniques Althoughscheduling models are problem specific, there are common features which translate intosimilar types of constraints Two major categories of scheduling models are identified:single-unit assignment models in which the assignment of tasks to units is known a priori,and multiple-unit assignment models in which several machines compete for the processingof products The most critical modeling issues are the time domain representation and networkstructure of the processing plant Furthermore, a summary of the major features of thescheduling model is presented along with computational experience, as well as a discussionon their strengths and limitations

Dynamic stochastic programmingfor asset-liability management

Abstract: Multistage stochastic programming - in contrast to stochastic control - has found wideapplication in the formulation and solution of financial problems characterized by a largenumber of state variables and a generally low number of possible decision stages. Theliterature on the use of multistage recourse modelling to formalize complex portfolio optimizationproblems dates back to the early seventies, when the technique was first adopted tosolve a fixed income security portfolio problem. We present here the CALM model, whichhas been designed to deal with uncertainty affecting both assets (in either the portfolio orthe market) and liabilities (in the form of scenario dependent payments or borrowing costs).We consider as an instance a pension fund problem in which portfolio rebalancing is allowedover a long-term horizon at discrete time points and where liabilities refer to five differentclasses of pension contracts. The portfolio manager, given an initial wealth, seeks the maximizationof terminal wealth at the horizon, with investment returns modelled as discretestate random vectors. Decision vectors represent possible investments in the market andholding or selling assets in the portfolio, as well as borrowing decisions from a credit lineor deposits with a bank. Computational results are presented for a set of 10-stage portfolioproblems using different solution methods and libraries (OSL, CPLEX, OB1). The portfolioproblem, with an underlying vector data process which allows up to 2688 realizations at the10-year horizon, is solved on an IBM RS6000y590 for a set of twenty-four large-scale testproblems using the simplex and barrier methods provided by CPLEX (the latter for eitherlinear or quadratic objective), the predictorycorrector interior point method provided in OB1,the simplex method of OSL, the MSLiP-OSL code instantiating nested Benders decompositionwith subproblem solution using OSL simplex, and the current version of MSLiP.

Multicriteria filtering methods based onconcordance and non-discordance principles

Abstract: This paper introduces multicriteria decision-aid methods for assigning alternatives topre-defined categories and develops a unified framework for preference aggregation methodsthat are based on concordance and non-discordance principles. Within this framework,we propose new multicriteria classification procedures based on non-totally compensatorymeasures of preference and similarity. We assume that the categories are given a priori andare characterised by fictitious alternatives serving as reference points, each representing atypical element of a category, or each representing a boundary between two categories.Several assignment methods are presented, all based on a filtering process exploiting binaryrelations constructed following the concordance and non-discordance principles. We firstconsider the case of ordered categories and a filtering procedure exploiting valued preferencerelations is proposed for assessing the membership of alternatives in categories. Then weconsider the case of non-ordered categories and filtering methods exploiting valued in-difference relations are introduced. Finally, a small example is given.

A class of stochastic programs withdecision dependent random elements

Abstract: In the “standard” formulation of a stochastic program with recourse, the distribution ofthe random parameters is independent of the decisions. When this is not the case, the problemis significantly more difficult to solve. This paper identifies a class of problems that are“manageable” and proposes an algorithmic procedure for solving problems of this type. Wegive bounds and algorithms for the case where the distributions and the variables controllinginformation discovery are discrete. Computational experience is reported.

An application of genetic and tabu searches to the freight railroad operating plan problem

Abstract: This paper addresses the joint train-scheduling and demand-flow problem for a major US freight railroad. No efficient optimization techniques are known to solve the NP-hard combinatorial optimization problem. Genetic search is used to find acceptable solutions; however, its performance is found to deteriorate as the problem size grows. A "tabu-enhanced" genetic search algorithm is proposed to improve the genetic search performance. The searches are applied to test problems with known optima to gauge them for solution speed and nearness to optimality. The tabu-enhanced genetic search is found to take on average only 6% of the iterations required by genetic search, consistently achieves better approximations to the optimum and maintains its performance as the problem size grows. The tabu-enhanced search is then applied to the full-scale operating plan problem. Model results reveal a potential for 4% cost savings over the current railroad operating plan coupled with a 6% reduction in late service.

A method for solving ship routing problemswith inventory constraints

Abstract: We present an optimisation-based solution approach for a real ship planning problem,which is a combination of a variant of the multi-vehicle pickup and delivery problemwith time windows (m-PDPTW), and a multi-inventory model This problem involves thedesign of a set of minimum cost routes for a fleet of heterogeneous ships servicing a set ofproduction and consumption harbours with a single product (ammonia) The production andinventory information at each harbour, together with the ship capacities and the location ofthe harbours, determine the number of possible arrivals at each harbour during the planningperiod, the time windows for start of service and the load quantity intervals at each arrivalWe call this problem the inventory pickup and delivery problem with time windows -IPDPTW In the mathematical programming model, we duplicate some of the variables anduse a Dantzig - Wolfe decomposition approach Then the IPDPTW decomposes into a sub-problemfor each harbour and each ship By synchronising the solutions from both types ofsubproblems, we get extra constraints in the master problem as compared to the masterproblem for the m-PDPTW discussed in the literature The LP-relaxation of the masterproblem is solved by column generation, where the columns represent ship routes or harbourvisit sequences Finally, this iterative solution process is embedded in a branch-and-boundsearch to make the solution integer optimal Our computational results indicate that theproposed method works for the real planning problem

A mathematical programming model for system reconfiguration in a dynamic cellular manufacturing environment

Abstract: In a dynamic manufacturing environment, manufacturing cell configurations based on current part mix and production process may need to be revised once the part mix or the production process has changed. However, machine and equipment moving costs make frequent reconfiguration uneconomical and sometimes impossible. Designing a sustainable cellular manufacturing system in a dynamic environment is studied in this paper. An integer programming model is developed to minimize material handling and machine costs as well as cell reconfiguration cost for a planning horizon of multiple time periods. Solving this integer programming problem is NP-complete. A decomposition approach is developed so that the decomposed subproblems can be solved with less computational effort. Dynamic programming is then employed to find a solution of the original problem. Numerical examples are presented to illustrate the model and the solution technique developed in this paper.

Characterizable fuzzy preference structures

Abstract: In this paper, we study the existence, construction and reconstruction of fuzzy preferencestructures. Starting from the definition of a classical preference structure, we propose anatural definition of a fuzzy preference structure, merely requiring the fuzzification of theset operations involved. Upon evaluating the existence of these structures, we discover thatthe idea of fuzzy preferences is best captured when fuzzy preference structures are definedusing a L U ukasiewicz triplet. We then proceed to investigate the role of the completenesscondition in these structures. This rather extensive investigation leads to the proposal of astrongest completeness condition, and results in the definition of a one-parameter class offuzzy preference structures. Invoking earlier results by Fodor and Roubens, the constructionof these structures from a reflexive binary fuzzy relation is then easily obtained. Thereconstruction of such a structure from its fuzzy large preference relation inevitable toobtain a full characterization of these structures in analogy to the classical case is morecumbersome. The main result of this paper is the discovery of a non-trivial characterizingcondition that enables us to fully characterize the members of a two-parameter class offuzzy preference structures in terms of their fuzzy large preference relation. As a remarkableside-result, we discover three limit classes of characterizable fuzzy preference structures,traces of which are found throughout the preference modelling literature.

Mathematical models in airline schedule planning: A survey

Abstract: The schedule is an airline's primary product, having the most influence (along with price) on a passenger's choice of an airline. Once an airline decides (at least tentatively) on a schedule, a host of related problems have to be resolved before it can consider the schedule feasible, and can proceed to market the schedule. Among these problems are traffic forecasting and allocation that forecasts traffic on each flight leg for use in the fleet assignment model, fleet assignment that decides the fleet type of the aircraft flying the legs in the schedule, equipment swapping to change an assigned equipment type on a leg if and when necessary, through flight selection for determining which pairs of flights to market as one-stops (without any aircraft change), maintenance routing that develops aircraft rotations to provide adequate opportunities for overnight maintenance, and flight numbering to number flights as consistently as possible with a prior schedule. Considerable methodological and computational advances have been made in the recent past in developing models and solution methods for almost all of the problems mentioned above. In this paper we survey these various models and solution techniques.

DEA and benchmarks – an application to Nordic banks

Abstract: In this paper, Data Envelopment Analysis (DEA) is developed to analyze the efficiencyof a single bank. The inputs are given in terms of cost of personnel, cost of material andexpected cost of credit losses. Outputs concern lending, deposits and gross revenues (interestmargins and non-interest income). The data covers 48 large Nordic banks during the twoyears 1992 and 1993. Fourteen banks are from Denmark, thirteen from Finland, twelve fromNorway and nine from Sweden. For each of these banks, the DEA method is used to form a“reference bank”, which is a convex combination of the best competing banks (those at theefficiency frontier). The three inputs and the three outputs of the reference bank will beused as benchmarks. This procedure implies that one can only say that one single bank isless efficient than its reference bank, not less efficient than another bank. The results showthat 4-7 Nordic banks were situated at the efficiency frontier for those two years. Thesebanks should then be used to form reference banks for other banks, and to set benchmarksfor them. Such benchmarks would have been slightly different, dependent on the “window”to be used, 1992, 1993 or 1992 + 1993.

Solution approaches to hub location problems

Abstract: The hub location problem involves a network of origins and destinations over which transport takes place. Any distribution system falls into this type of category. In this paper, we present a new quadratic integer formulation for the Uncapacitated Hub Location Problem (UHP), which is based on the idea of multi-commodity flows in networks. This new formulation lends itself well for using a branch-and-bound procedure to find optimal solutions. The branch-and-bound procedure is not implemented in a traditional fashion, where bounds are obtained by linearizing the objective function and relaxing the integrality constraints. Instead, a more sophisticated approach is used where bounds are obtained by employing the underlying network structure of the problem. In addition, an artificial intelligence-based technique (Genetic Search) is designed to find solutions quickly and efficiently. The two solution approaches assume that the number of hubs is a variable, each spoke is assigned to a single hub, and all hubs are interconnected. The model and the algorithm can be applied even when all the hubs are not directly linked.

The application of a tabu search metaheuristic to the assembly line balancing problem

Abstract: This paper describes the application of tabu search, a metaheuristic technique for optimization problems, to assembly line balancing problems. Four different versions of algorithms are developed. They all share the same tabu search strategy except that the first one uses the best improvement with task aggregation, the second one uses best improvement without task aggregation, the third one uses the first improvement with task aggregation, and the last one uses the first improvement without task aggregation. Computational experiments with these different search strategies have been performed for some assembly line problems from the open literature. The results show that tabu search performs extremely well. Except for a few cases, tabu search always finds optimal solutions.

Characterizations of scoring methods for preference aggregation

Abstract: The paper surveys more than forty characterizations of scoring methods for preferenceaggregation and contains one new result. A general scoring operator is self-consistent ifalternative i is assigned a greater score than j whenever i gets no worse (better) results ofcomparisons and its “opponents” are assigned, respectively, greater (no smaller) scores thanthose of j. We prove that self-consistency is satisfied if and only if the application of ascoring operator reduces to the solution of a homogeneous system of algebraic equationswith a monotone function on the left-hand side.

Routing helicopters for crew exchanges on off-shore locations

Abstract: This paper deals with a vehicle routing problem with split demands, namely the problem of determining a flight schedule for helicopters to off-shore platform locations for exchanging crew people employed on these platforms The problem is formulated as an LP model and solved by means of a column-generation technique including solving TSP problems Since the final solution needs to be integral, we have chosen a rounding procedure to obtain an integer solution Since the LP approach needs a considerable amount of computer time, it is only suitable for long-term planning practices For the usual short-term planning, we have designed the so-called Cluster-and-Route Heuristic together with a number of improvement heuristics The Cluster-and-Route procedure constructs a suitable clustering of the platforms and simultaneously forms the routes of the helicopter flights associated with the clusters This approach is different from the usual heuristics, in which the clusters are constructed first, and the routes for each cluster are made afterwards Simulations with various data sets show that the new heuristic outperforms the usual heuristics for vehicle routing problems Even better results are obtained when improvement heuristics are applied We use four improvement heuristics, including, so-called 1-opt and 2-opt procedures

Practical job shop scheduling

Abstract: The Shifting Bottleneck procedure is an intuitive and reasonably good approximation algorithm for the notoriously difficult classical job shop scheduling problem. The principle of decomposing a classical job shop problem into a series of single-machine problems can also easily be applied to job shop problems with practical features, such as transportation times, simultaneous resource requirements, setup times, and many minor but important other characteristics. We report on the continuous research in the area of extending the Shifting Bottleneck procedure to deal with those practical features. We call job shops with such additional features practical job shops. We discuss experiences with the Shifting Bottleneck procedure in a number of practical cases

Scheduling large robotic cells without buffers

Abstract: A robotic cell is a manufacturing system that is widely used in industry. A robotic cell contains two or more robot-served machines, repetitively producing a family of similar parts, in a steady state. There are no buffers at or between the machines. Both the robot move cycle and the sequence of parts to produce are chosen in order to minimize the cycle time needed to produce a given set of parts. This objective is also equivalent to throughput rate maximization. In practice, simple robot move cycles that produce one unit are preferred by industry. In an m machine cell for m >= 2, there are m! such cycles that are potentially optimal. Choosing any one of these cycles reduces the cycle time minimization problem to a unique part sequencing problem. We prove the following results in an m machine cell, for any m >= 2. The part sequencing problems associated with these robot move cycles are classified into the following categories: (i) sequence independent; (ii) capable of formulation as a traveling salesman problem (TSP), but polynomially solvable; (iii) capable of formulation as a TSP and unary NP-hard; and (iv) unary NP-hard, but not having TSP structure. As a consequence of this classification, we prove that the part sequencing problems associated with exactly 2m-2 of the m! available robot cycles are polynomially solvable. The remaining cycles have associated part sequencing problems which are unary NP-hard.

Frequency assignment in cellular phone networks

Abstract: We present a graph-theoretic model for the frequency assignment problem in cellular phone networks. Obeying several technical and legal restrictions, frequencies have to be assigned to transceivers so that interference is as small as possible. This optimization problem is NP-hard. Good approximation cannot be guaranteed unless P = NP. We describe several assignment heuristics. These heuristics are simple and not too hard to implement. We give an assessment of the heuristics' efficiency and practical usefulness. For this purpose, typical instances of frequency assignment problems with up to 4240 transceivers and 75 frequencies of a German cellular phone network operator are used. The results are satisfying from a practitioner's point of view. The best performing heuristics were integrated into a network planning system used in practice.

Modelling path flows for a combined ship routingand inventory management problem

Abstract: We consider a combined time constrained ship routing and inventory managementproblem. A fleet of ships transports a single product between production and consumptionharbours. The transporter has the responsibility for keeping the stock level within its limitsat all actual harbours, and there should be no need to stop the production at any harboursdue to missing transportation possibilities. The number of arrivals to each harbour and thequantities loaded and discharged at each arrival are determined by the continuous productionrates at the harbours, the stock limits and the actual ships visiting the harbours. We use apath flow formulation for this planning problem, and generate paths for each ship includinginformation about the geographical route, the load quantity and start time at each harbourarrival. In addition, we generate paths for each harbour including information about thenumber of arrivals to the harbour, the load quantity and start time at each harbour arrival.We emphasise the formulation of the path generation problems which are subproblems inthe total planning problem. The generated paths appear as columns in a path flow problemwhich corresponds to a master problem. We use a column generation approach to solve thecontinuous problem. The solution is made integer optimal by branch-and-bound. Computationalresults indicate that a path flow formulation and an optimisation based solutionapproach work for real instances of the planning problem.

Spectral expansion solution for some finite capacity queues

Abstract: The spectral expansion method is successfully extended for the steady-state solution of a class of Markov processes of finite state space These processes are of QBD structure, with boundary The solution is exact and efficient Some qualitative comparisons are made with some of the existing methods In this framework, a finite capacity multiserver system, alternating in a Markovian environment, serving a Poisson stream of jobs, is analysed Another system considered is a two-stage queueing network with feedback and a finite intermediate waiting room This system is modelled and solved by spectral expansion Typical numerical results are presented

Solution procedures for the dynamic facility layout problem

Abstract: Existing optimal solution procedures for the dynamic facility layout problem require the repeated solution of quadratic assignment problems within the framework of a dynamic program. The computational requirements for this problem necessitate the development of efficient algorithms for special cases of the problem and strong bounds for the general case of the problem. The concept of incomplete dynamic programming is applied to the dynamic facility layout problem to find the optimal solution to the problem with fixed rearrangement costs at exceptionally reduced solution times. Heuristics are also developed to provide a solution methodology for larger problems. A lower bound is developed for the general problem that dominates all existing bounds, and it is shown how the bound can be used as an initial test for optimality before the dynamic program is solved. Finally, the incomplete dynamic programming methodology is extended to find an upper bound for the general problem that dominates all previous bounds.

Modeling Norwegian petroleum productionand transportation

Abstract: In the continental shelf off the coast of Norway, there are several petroleum fields containinga mixture of oil and gas. A multiperiod mixed integer programming model for investmentplanning for these fields has been used by The Norwegian Petroleum Directorate for morethan fifteen years. In practical use, the production from each field has mostly been declaredto follow profiles given by the user, but the user may also declare that the production canvary from the given profile. This paper describes the model and comments on some of thereal problems the model has been used to analyze and the modeling process involved.

Nash equilibrium in multiparty competitionwith “stochastic” voters

Abstract: Theoretical spatial models of electoral voting tend to predict either convergence to an electoral mean (when voting is probabilistic) or chaos (when voting is deterministic). Here, we construct an empirical model of voting for the Israeli Knesset in 1992 (based on a large electoral sample and on analysis of party declarations). The probabilistic voting model so estimated fits the known election results. We then use the same model to simulate the effect of expected vote maximization by the parties. Contrary to the usual results, there is no unique convergent Nash equilibrium under this objective function. We do infer, however, that the two large parties are “Downsian”, in the sense that they maximize expected vote (up to the margin of error of the model). We suggest that the empirical results are compatible with a hybrid model of utility maximization, where each party computes the effects of its policy declaration both in terms of electoral response and of post-election coalition negotations.

Reformulation of capacitated facility location problems:How redundant information can help

Abstract: When solving hard combinatorial optimization problems by branch-and-bound, obtaininga good lower bound (considering a minimization problem) from the linear relaxation iscrucial for the performance of the algorithm. On the other hand, we want to avoid an initialformulation that is too large. This requires careful modeling of the problem. One way ofobtaining a good linear formulation is by applying a cutting plane algorithm where strongcutting planes are added if they violate the current fractional solution. By “strong” cuttingplanes, we mean linear inequalities that define high-dimensional faces of the convex hull offeasible solutions. For some classes of inequalities, effective algorithms for identifyingviolated inequalities belonging to these classes have been implemented as standard featuresin commercial branch-and-bound packages. Such classes are for instance the knapsack coverinequalities and the flow cover inequalities that were originally developed for the knapsackproblem and the single-node flow problem. These problems form relaxations of severalcapacitated combinatorial optimization problems such as various capacitated facility locationproblems. If, however, we consider traditional models for location problems, then theknapsack and single-node flow relaxations are not explicitly stated in the models, andunless we modify the models, the mentioned classes of inequalities will not be generated“automatically” by the systems. The extra variables and constraints that we need to add tothe traditional models in order to make the various relaxations explicit are redundant, notonly to the integer formulation but also to the linear relaxation. Computational experimentsdo, however, indicate that the inequalities that are generated based on the relaxations arevery effective and that the gain from the stronger linear relaxation far outweighs the drawbackof expanding the traditional models.

Calculating the exact bounds of optimal valuesin LP with interval coefficients

Abstract: The paper deals with computing the exact upper and lower bounds of optimal values forlinear programming problems whose coefficients vary in given intervals. The theoreticalbackground for calculating these bounds is described and corresponding algorithms aregiven. A comparison with other approaches, some applications and a software package arementioned.

Objectivity and subjectivity in thedecision making process

Abstract: Classical Operations Research assumed objectivity. Operations researchers hardly botheredto ask the decision maker about his or her preferences, assuming that a well-defined singleobjective function was an adequate representation of the decision problem. Many multi-criteriadecision methods began in response to this failure of Operations Research. Othermethods took a totally different and more subjective point of view. In this paper, we discussobjective and subjective descriptions, their interpretation and use in decision making. In thecenter of the ability to distinguish between these objective and subjective components standscientific methods and scientists. We argue that it is possible and necessary to understandwhere and how objective measures should be replaced by subjective measures and judgementin the decision making process.

The traveling salesman game: An application ofcost allocation in a gas and oil company

Abstract: In this article, a cost allocation problem that arises in a distribution planning situation atthe Logistics Department at Norsk Hydro Olje AB is studied. A specific tour is considered,for which the total distribution cost is to be divided among the customers that are visited.This problem is formulated as a traveling salesman game, and cost allocation methods basedon different concepts from cooperative game theory, such as the nucleolus, the Shapleyvalue and the t-value, are discussed. Additionally, a new concept is introduced: the demandnucleolus. Computational results for the Norsk Hydro case are presented and discussed.

Jackson's Pseudo Preemptive Schedule for the Pm / r i , q i / C max scheduling problem

Abstract: The aim of this paper is to introduce Jackson's Pseudo Preemptive Schedule (JPPS) for the m parallel and identical processor scheduling problem Pm/r i , q i /C max . JPPS generalizes Jackson's Preemptive Schedule (JPS) which was introduced for the one-processor sequencing problem 1/r i , q i /C max . JPS can be computed in O(nlog n) time and plays a central role in solving NP-hard disjunctive scheduling problems such as the job shop problem. The make-span of JPPS can be computed in O(nlog n + nmlog m) time, and is a tight lower bound for the Pm/r i , q i /C max . So JPPS could also play a central role in solving the Resource Constrained Project Scheduling Problem.

Bounds and error bounds for queueing networks

Abstract: Queueing networks are an important means to model and evaluate a variety of practical systems. Unfortunately, analytic results are often not available. Numerical computation may then have to be employed. Or, system modifications might be suggested to obtain simple bounds or computationally easy approximations. Formal analytic support for the accuaracy or nature of such modifications or approximations then becomes of interest. To this end, the Markov reward approach is surveyed and illustrated as a technique to conclude a priori error bounds as well as to formally prove bounds when comparing two related systems. More precisely, the technique can be applied to: perturbations, finite truncations, infinite approximations, system modifications, or system simplifications (bounds).