I detta examensarbete inom tillämpad matematik och industriell ekonomi undersöks bemanningsstrategin för Mat AB för att sänka den totala leveranstiden. Mat AB är ett startup som arbetar med leverans av mat- och apoteksvaror. Precis som många andra företag i branschen fick Mat AB ett uppsving under våren 2020 och möter precis som många andra snabbt växande företag organisatoriska utmaningar. Arbetet undersöker leverans- och försäljningsdata från våren 2021. Utifrån detta fastställs om dagens bemanningsstrategi är optimal eller om det finns bättre sätt att organisera personalen på för att sänka leveranstiden och på så sätt öka värdeskapandet och sänka personalkostnader. Modelleringen och undersökningen av verksamheten genomfördes med multipel linjär regressionsanalys och köteori. Genom att modellera verksamheten som olika kösystem kunde olika bemanningsstrategier teoretiskt prövas för att fastställa om mer optimala strategier finns att tillgå. Arbetet presenterar utifrån datan den befintliga leveranstiden och ger förslag på alternativa bemanningsstrategier genom modellering. Resultatet visar att det finns alternativa bemanningsstrategier att tillgå som skulle sänka den totala leveranstiden för Mat AB. I rapporten diskuteras även faktorer som inte finns med i modelleringen som kan påverka leveranstiden för vidare analys. / This thesis combines applied mathematics with industrial economics to investigate the staffing strategy for Mat AB in order to reduce the total delivery time. Mat AB is a startup that works with delivery of food and pharmacy goods. Like many other companies in the industry, Mat AB received a boost in the spring of 2020 and, like many other fast-growing companies, faces organizational challenges. The thesis examines delivery and sales data from the spring of 2021. Based on this, it is determined whether the current staffing strategy is optimal or whether there are better ways to organize the staff to reduce the delivery time and thus increase value creation and reduce staff costs. The modeling and investigation of the operation was carried out with multiple linear regression analysis and queue theory. By modeling the business as a queuing system, different staffing strategies could theoretically be tested to determine whether more optimal strategies were available. Based on the data, the thesis presents the existing delivery time and suggests alternative staffing strategies through modeling. The results show that there are alternative staffing strategies that would reduce the total delivery time for Mat AB. The report also discusses factors that are not included in the modeling that may affect the delivery time for further analyses.

With a high increase of users in the world's ever expanding sharing economy, Airbnb has become a customary solution in short term rentals of accommodations. In this market, it is the host's job to choose a pricing which sufficiently corresponds to what tenants are willing to pay. There can be multiple methods of choosing the price but this study aims to determine and evaluate which factors have a significant impact on short term rental pricing of housing and to what degree. By modelling this issue, the reader can make a better understanding of what to pay or charge for an accommodation. This study also serves as ground work for further investigations exploring nested and multi-leveled factors. The study is limited to the Spanish short term rental market, taking a more in-depth look at the cities of Barcelona, Madrid and Palma. Moreover, listings between 2015 and 2017 are considered in the study. In the end, factors identified as significant on accommodation pricing were Entire Home, Accommodates, Bathrooms, Review Scores Rating etc.. Some of the factors are interchangeable as they have a miniscule effect on the accommodation pricing. Conversely, Entire Home and Accommodates is seen as absolute necessities for the model as they, together, explain two-thirds of the variations in price. / Med en ständigt ökande användarskara, i världens ständigt expanderande delningsekonomi, har Airbnb blivit en allt vanligare lösning för korttidsuthyrning av boenden. På denna marknad är det värdens uppgift att välja en prissättning som tillräckligt motsvarar vad hyresgästerna är villiga att betala. Det kan finnas flera metoder för prissättning, men denna studie syftar till att bestämma och utvärdera vilka faktorer som har en betydande inverkan på prissättningen av bostäder tillgängliga för korttidsuthyrning. Genom modellering av detta problem kan läsaren få en bättre förståelse för vad man ska betala eller hur mycket man skall ta betalt för ett boende. Denna studie fungerar också som grundarbete för vidare undersökningar som utforskar inbäddade samt flernivåfaktorer. Studien är begränsad till den spanska korttidsuthyrningsmarknaden, med fokus riktat mot städerna Barcelona, Madrid och Palma. Endast annonser mellan 2015 och 2017 tas i beaktning i studien. I slutändan identifierades de faktorer som har en betydande effekt på prissättningen av boenden som Entire Home, Accommodates, Bathrooms, Review Scores Rating med mera. Vissa av faktorerna är utbytbara eftersom de har en minimal effekt på prissättningen av boenden. Däremot anses Entire Home och Accommodates vara absolut nödvändiga för modellen eftersom de tillsammans förklarar två tredjedelar av variationerna i priset.

Detta arbete presenterar en matematisk modell vilken optimerar byggvaruaktören Nordströms artikelallokering med avseende på transportsträcka vid utleverans. Det linjära flödesproblemet kan således nyttjas för att undersöka hur Nordströms lagerhållning kan uppdateras med syftet att reducera antalet suboptimala leveranser från mer avlägsna filialer. Vidare presenteras en kvalitativ analys kring lagerhållningsstrategier och dess implementerbarhet.  Lösningen genererades med hjälp av iterativa metoder inom heltalsprogrammering. Den optimala allokeringslösningen jämfördes med Nordströms aktuella lagersortiment för att identifiera potentiella lagerbrister vilka orsakat suboptimala leveranser. Slutligen rekommenderas Nordströms att utöka dess lagerförda kvantitet av 500 specificerade artiklar för att undvika suboptimala leveranser från mer avlägsna filialer. / This report considers the application of a mathematical model using integer linear programming for optimal allocation of articles. The model aims at establishing the optimal inventory levels at Nordström’s, a construction material manufacturer, in order to minimize the delivery transportation distances. The network flow problem can thus be used to investigate how Nordström's inventory can be updated in order to reduce sub-optimal deliveries from more remote facilities. Furthermore, a qualitative analysis of inventory management strategies and their implementability is presented. The optimal solution was generated using iterative methods of integer programming. Identification of inventory shortages was enabled by comparing the optimal allocation to Nordström’s current inventory levels. Finally, Nordströms is recommended to increase its stocked levels of 500 specified articles in order to avoid sub-optimal deliveries from more remote facilities.

Data assimilation arises in a vast array of different topics: traditionally in meteorological and oceanographic modelling, wind tunnel or water tunnel experiments and recently from biomedical engineering. Data assimilation is a process for combine measured or observed data with a mathematical model, to obtain estimates of the expected data. The measured data usually contains inaccuracies and is given with low spatial and/or temporal resolution. In this thesis data assimilation for time dependent fluid flow is considered. The flow is assumed to satisfy a given partial differential equation, representing the mathematical model. The problem is to determine the initial state which leads to a flow field which satisfies the flow equation and is close to the given data. In the first part we consider one-dimensional flow governed by Burgers’ equation. We analyze two iterative methods for data assimilation problem for this equation. One of them so called adjoint optimization method, is based on minimization in L2-norm. We show that this minimization problem is ill-posed but the adjoint optimization iterative method is regularizing, and represents the well-known Landweber method in inverse problems. The second method is based on L2-minimization of the gradient. We prove that this problem always has a solution. We present numerical comparisons of these two methods. In the second part three-dimensional inviscid compressible flow represented by the Euler equations is considered. Adjoint technique is used to obtain an explicit formula for the gradient to the optimization problem. The gradient is used in combination with a quasi-Newton method to obtain a solution. The main focus regards the derivation of the adjoint equations with boundary conditions. An existing flow solver EDGE has been modified to solve the adjoint Euler equations and the gradient computations are validated numerically. The proposed iteration method are applied to a test problem where the initial pressure state is reconstructed, for exact data as well as when disturbances in data are present. The numerical convergence and the result are satisfying.

<p>In this thesis we have created a computer program in Java language which calculates European call- and put options with four different models based on the article The Pricing of Options on Assets with Stochastic Volatilities by John Hull and Alan White. Two of the models use stochastic volatility as an input. The paper describes the foundations of stochastic volatility option pricing and compares the output of the models. The model which better estimates the real option price is dependent on further research of the model parameters involved.</p>

<p>This thesis concerns the identifiability issue; which, if any, parameters can be deduced from the input and output behavior of a model? The two types of identifiability concepts, a priori and practical, will be addressed and explained. Two methods for identifiability testing are evaluated and the result shows that the two methods work well if they are combined. The first method is for a priori identifiability analysis and it can determine the a priori identifiability of a system in polynomial time. The result from the method is probabilistic with a high probability of correct answer. The other method takes the simulation approach to determine whether the model is practically identifiable. Non-identifiable parameters manifest themselves as a functional relationship between the parameters and the method uses transformations of the parameter estimates to conclude if the parameters are linked. The two methods are verified on models with known identifiability properties and then tested on some examples from systems biology. Although the output from one of the methods is cumbersome to interpret, the results show that the number of parameters that can be determined in practice (practical identifiability) are far fewer than the ones that can be determined in theory (a priori identifiability). The reason for this is the lack of quality, noise and lack of excitation, of the measurements.</p> / <p>Fokus i denna rapport är på identifierbarhetsproblemet. Vilka parametrar kan unikt bestämmas från en modell? Det existerar två typer av identifierbarhetsbegrepp, a priori och praktisk identifierbarhet, som kommer att förklaras. Två metoder för identifierbarhetstestning är utvärderade och resultaten visar på att de två metoderna fungerar bra om de kombineras med varandra. Den första metoden är för a priori identifierbarhetsanalys och den kan avgöra identifierbarheten för ett system i polynomiell tid. Resultaten från metoden är slumpmässigt med hög sannolikhet för ett korrekt svar. Den andra metoden använder sig av simuleringar för att avgöra om modellen är praktiskt identifierbar. Icke-identifierbara parametrar yttrar sig som funktionella kopplingar mellan parametrar och metoden använder sig av transformationer av parameterskattningarna för att avgöra om parametrarna är kopplade. De två metoderna är verifierade på modeller där identifierbarheten är känd och är därefter testade på några exempel från systembiologi. Trots att resultaten från den ena metoden är besvärliga att tolka visar resultaten på att antalet parametrar som går att bestämma i verkligheten (praktiskt identifierbara) är betydligt färre än de parametrar som kan bestämmas i teorin (a priori identifierbara). Anledningen beror på brist på kvalitet, både brus och brist på excitation, i mätningarna.</p>

<p>This thesis is about a generalization of permutation theory. The concept of pattern avoidance in permutation matrices is investigated in a larger class of matrices - the alternating sign matrices. The main result is that the set of alternating sign matrices avoiding the pattern 132, is counted by the large Schröder numbers. An algebraic and a bijective proof is presented. Another class is shown to be counted by every second Fibonacci number. Further research in this new area of combinatorics is discussed.</p>

<p>In this work we consider the nonlinear case of Black-Scholes equation and apply it to American options. Also, method of Liao and Khaliq of high order was applied to nonlinear Black-Scholes equation in case of American options. Here, we use this method oh fourth order in time and space to raise American option price accuracy.</p>

<p>This master project is dedicated to the analysis of one of the nancialmarket models in an illiquid market. This is a nonlinear model. Using analytical methods we studied the symmetry properties of theequation which described the given model. We called this equation aSchonbucher-Wilmott equation or the main equation. We have foundinnitesimal generators of the Lie algebra, containing the informationabout the symmetry group admitted by the main equation. We foundthat there could be dierent types of the unknown function g, whichwas located in the main equation, in particular four types which admitsricher symmetry group. According to the type of the function gthe equation was split up into four PDEs with the dierent Lie algebrasin each case. Using the generators we studied the structure ofthe Lie algebras and found optimal systems of subalgebras. Then weused the optimal systems for dierent reductions of the PDE equationsto some ODEs. Obtained ODEs were easier to solve than the correspondingPDE. Thereafter we proceeded to the solution of the desiredSchonbucher-Wilmott equation. In the project we were guided by thepapers of Bank, Baum [1] and Schonbucher, Wilmott [2]. In these twopapers authors introduced distinct approaches of the analysis of thenonlinear model - stochastic and dierential ones. Both approaches leadunder some additional assumptions to the same nonlinear equation - the main equation.</p>

<p>New methods for global shape description of three-dimensional digital objects are presented. The shape of an object is first represented by a digital surface where the faces are either triangles or quadrilaterals. Techniques for computing a high-quality parameterization of the surface are developed and this parameterization is used to approximate the shape of the object. Spherical harmonics are used as basis functions for approximations of the coordinate functions. Information about the global shape is then captured by the coefficients in the spherical harmonics expansions.</p><p>For a starshaped object it is shown how a parameterization can be computed by a projection from its surface onto the unit sphere. An algorithm for computing the position at which the centre of the sphere should be placed, is presented. This algorithm is suited for digital voxel objects. Most of the work is concerned with digital objects whose surfaces are homeomorphic to the sphere. The standard method for computing parameterizations of such surfaces is shown to fail on many objects. This is due to the large distortions of the geometric properties of the surface that often occur with this method. Algorithms to handle this problem are suggested. Non-linear optimization methods are used to find a mapping between a surface and the sphere that minimizes geometric distortion and is useful as a parameterization of the surface. </p><p>The methods can be applied, for example, in medical imaging for shape recognition, detection of shape deformations and shape comparisons of three-dimensional objects.</p>