The stochastic opportunistic replacement problem, part III: improved bounding procedures

We consider the problem to find a schedule for component replacement in a multi-component system, whose components possess stochastic lives and economic dependencies, such that the expected costs for maintenance during a pre-defined time period are minimized. The problem was considered in Patriksson et al. (Ann Oper Res 224:51–75, 2015), in which a two-stage approximation of the problem was optimized through decomposition (denoted the optimization policy). The current paper improves the effectiveness of the decomposition approach by establishing a tighter bound on the value of the recourse function (i.e., the second stage in the approximation). A general lower bound on the expected maintenance cost is also established. Numerical experiments with 100 simulation scenarios for each of four test instances show that the tighter bound yields a decomposition generating fewer optimality cuts. They also illustrate the quality of the lower bound. Contrary to results presented earlier, an age-based policy performs on par with the optimization policy, although most simple policies perform worse than the optimization policy

Verklighetsnära matematik : en studie av infärgning och autenticitet av uppgifter i läroböcker i matematik

Läroplaner i matematik för gymnasiet tar upp att matematik ska kopplas till karaktärsämnen. Särskilt tydligt är det i läroplanen för Matematik 1a, som läses av alla elever på yrkesprogram. Skolverket har i en modul med titeln "Undervisa matematik på yrkesprogram" förtydligat vad som önskas i fråga om att koppla samman matematik med karaktärsämnen. Speciellt finns ett fokus på att uppgifter bör vara autentiska. Lärare låter ofta undervisningen utgå från en lärobok i matematik, snarare än från läroplanen. Det är därför viktigt att läroböcker har god överensstämmelse med läroplanen. Det föreliggande arbetet är en undersökning av huruvida läroböcker för Matematik 1A erbjuder stöd för att koppla samman matematik med karaktärsämnen, samt av i vilken utsträckning sådant stöd är autentiskt. Undersökningen består av analys av totalt fyra uppgifter hämtade från två olika läroböcker i matematik. Bedömningen av uppgifternas autenticitet görs med avseende på en modifierad version av ett ramverk som använts av Skolverket för att skapa instruktioner till hur man konstruerar autentiska uppgifter. Arbetets omfattning är mycket begränsat, varför inga säkra besked kan ges. Undersökningen tyder på att det i läroböcker för Matematik 1A görs försök att sammankoppla matematiken med karaktärsämnen, men att det finns brister i fråga om autenticitet av sådan sammankoppling

Verklighetsnära matematik : en studie av infärgning och autenticitet av uppgifter i läroböcker i matematik

Läroplaner i matematik för gymnasiet tar upp att matematik ska kopplas till karaktärsämnen. Särskilt tydligt är det i läroplanen för Matematik 1a, som läses av alla elever på yrkesprogram. Skolverket har i en modul med titeln "Undervisa matematik på yrkesprogram" förtydligat vad som önskas i fråga om att koppla samman matematik med karaktärsämnen. Speciellt finns ett fokus på att uppgifter bör vara autentiska. Lärare låter ofta undervisningen utgå från en lärobok i matematik, snarare än från läroplanen. Det är därför viktigt att läroböcker har god överensstämmelse med läroplanen. Det föreliggande arbetet är en undersökning av huruvida läroböcker för Matematik 1A erbjuder stöd för att koppla samman matematik med karaktärsämnen, samt av i vilken utsträckning sådant stöd är autentiskt. Undersökningen består av analys av totalt fyra uppgifter hämtade från två olika läroböcker i matematik. Bedömningen av uppgifternas autenticitet görs med avseende på en modifierad version av ett ramverk som använts av Skolverket för att skapa instruktioner till hur man konstruerar autentiska uppgifter. Arbetets omfattning är mycket begränsat, varför inga säkra besked kan ges. Undersökningen tyder på att det i läroböcker för Matematik 1A görs försök att sammankoppla matematiken med karaktärsämnen, men att det finns brister i fråga om autenticitet av sådan sammankoppling

Combinatorial Optimization : Three Applications

Combinatorial optimization is a diverse area of mathematics. It concerns optimization on feasible regions defined by discrete sets, graphs, hypergraphs, matroids, etc. . . which all have a large number of applications. They occur in virtually all domains of human activity since humans always want to do things easier, faster, consume less resources, etc. . . This thesis concerns three applied problems within combinatorial optimization. The first paper generalizes previous optimal upper bounds on the minimum Euclidean distance for phase-shift keying (PSK) block codes, that are explicit in the parameters alphabet size, block length and code size. There is a strong connection between high minimum Euclidean distance and good error-correcting capabilities. The bounds are generalized in several respects, such as from codes on symmetric PSK to codes on asymmetric PSK. They are also generalized to other types of noise than Gaussian, allowing more efficient block codes when noise is non-Gaussian. We provide examples of codes on asymmetric PSK that have higher minimum Euclidean distance than any comparable codes on symmetric PSK.Several classes of codes are shown to be optimal among codes on symmetric PSK since their Euclidean distance coincides with the bound. The second paper considers a parallel computer system with m identical computers,where we study optimal performance precaution for one possible computer crash. We anticipate that some computer may crash, and restrict the cost in such a situation. How costly is such a precaution when no crash occurs? We set a restriction that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use an optimal allocation with m - 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time defines a function f(r,m). In the paper we establish upper and lower bounds of the worst-case cost function f(r,m) and characterize worst-case programs. The third paper considers the problem of Map Matching (MM), i.e. matching time and location measurements of a vehicle to a route in a road network. The paper presents a probabilistic algorithm for MM based on a second order hidden Markov model (HMM), as opposed to first order HMMs which are usually used. This allows a more detailed analysis of the data while preserving algorithmic complexity O(n). Both measurement densities and transition probabilities are determined with respect to Kolmogorov's third axiom, which in this context implies that the probabilities are additive over a partition of a road segment

Combinatorial Optimization - Three Applications

Combinatorial optimization is a diverse area of mathematics. It concerns optimization on feasible regions defined by discrete sets, graphs, hypergraphs, matroids, etc. . . which all have a large number of applications. They occur in virtually all domains of human activity since humans always want to do things easier, faster, consume less resources, etc. . . This thesis concerns three applied problems within combinatorial optimization. The first paper generalizes previous optimal upper bounds on the minimum Euclidean distance for phase-shift keying (PSK) block codes, that are explicit in the parameters alphabet size, block length and code size. There is a strong connection between high minimum Euclidean distance and good error-correcting capabilities. The bounds are generalized in several respects, such as from codes on symmetric PSK to codes on asymmetric PSK. They are also generalized to other types of noise than Gaussian, allowing more efficient block codes when noise is non-Gaussian. We provide examples of codes on asymmetric PSK that have higher minimum Euclidean distance than any comparable codes on symmetric PSK.Several classes of codes are shown to be optimal among codes on symmetric PSK since their Euclidean distance coincides with the bound. The second paper considers a parallel computer system with m identical computers,where we study optimal performance precaution for one possible computer crash. We anticipate that some computer may crash, and restrict the cost in such a situation. How costly is such a precaution when no crash occurs? We set a restriction that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use an optimal allocation with m - 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time defines a function f(r,m). In the paper we establish upper and lower bounds of the worst-case cost function f(r,m) and characterize worst-case programs. The third paper considers the problem of Map Matching (MM), i.e. matching time and location measurements of a vehicle to a route in a road network. The paper presents a probabilistic algorithm for MM based on a second order hidden Markov model (HMM), as opposed to first order HMMs which are usually used. This allows a more detailed analysis of the data while preserving algorithmic complexity O(n). Both measurement densities and transition probabilities are determined with respect to Kolmogorov's third axiom, which in this context implies that the probabilities are additive over a partition of a road segment

Bounds on the Performance of PSK Block Codes

In wireless communication, the minimum Euclidean distance between codewords is a major factor for the ability to correct errors in messages, and it is of interest to maximize the minimum Euclidean distance. The thesis improves previously established general upper bounds on minimum Euclidean distance of phase shift keying block codes. There are no requirements on structure of codes, as the bound depends only on alphabet size, word length and code size. Prior to this thesis, bounds found by use of a method of Elias, had been improved by generalization of Elias' method. The method used here is an attempt to optimize that generalization

Optimal Computer Crash Performance Precaution

Distributed Computing and NetworkingFor a parallel computer system with m identical computers, we study optimal performance precaution for one possible computer crash. We want to calculate the cost of crash precaution in the case of no crash. We thus define a tolerance level r meaning that we only tolerate that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use optimal allocation on m - 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then, what is the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time of the same parallel program? We denote the maximal ratio of completion times f(r, m) - i.e., the ratio for worst-case programs. In the paper we establish upper and lower bounds of the worst-case cost function f (r, m) and characterize worst-case programs

Optimal Computer Crash Performance Precaution

Distributed Computing and Networkin

In press: Generalized upper bounds on the minimum distance of PSK block codes

This paper generalizes previous optimal upper bounds on the minimum Euclidean distance for phase shift keying (PSK) block codes, that are explicit in three parameters: alphabet size, block length and code size. The bounds are primarily generalized from codes over symmetric PSK to codes over asymmetric PSK and also to general alphabet size. Furthermore, block codes are optimized in the presence of other types of noise than Gaussian, which induces also non-Euclidean distance measures. In some instances, codes over asymmetric PSK prove to give higher Euclidean distance than any code over symmetric PSK with the same parameters. We also provide certain classes of codes that are optimal among codes over symmetric PSK

Optimal computer crash performance precaution

For a parallel computer system withmidentical computers, we study optimal performance precaution for one possible computer crash. We want to calculate the cost of crash precaution in the case of no crash. We thus define a tolerance level r meaning that we only tolerate that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use optimal allocation on m - 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then, what is the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time of the same parallel program? We denote the maximal ratio of completion times f(r, m) - i.e., the ratio for worst-case programs. In the paper we establish upper and lower bounds of the worst-case cost function f(r, m) and characterize worst-case programs