Finding Multiple Solutions of Multimodal Optimization Using Spiral Optimization Algorithm with Clustering

Multimodal optimization is one of the interesting problems in optimization which arises frequently in a widerange of engineering and practical applications. The goal of this problem is to find all of optimum solutions in a single run. Some algorithms fail to find all solutions that have been proven their existence analytically. In our paper [1], a method is proposed to find the roots of a system of non-linear equations using a clustering technique that combine with Spiral Optimization algorithm and Sobol sequence of points. An interesting benefit using this method is that the same inputs will give the same results. Most of the time this does not happen in meta-heuristic algorithms using random factors. Now the method is modified to find solutions of multimodal optimization problems. Generally in an optimization problem, the differential form of the objective function is needed. In this paper, the proposed method is to find optimum points of general multimodal functions that its differential form is not required. Several problems with benchmark functions have been examined using our method and they give good result

Determination of Gas Pressure Distribution in a Pipeline Network using the Broyden Method

A potential problem in natural gas pipeline networks is bottlenecks occurring in the flow system due to unexpected high pressure at the pipeline network junctions resulting in inaccurate quantity and quality (pressure) at the end user outlets. The gas operator should be able to measure the pressure distribution in its network so the consumers can expect adequate gas quality and quantity obtained at their outlets. In this paper, a new approach to determine the gas pressure distribution in a pipeline network is proposed. A practical and user-friendly software application was developed. The network was modeled as a collection of node pressures and edge flows. The steady state gas flow equations Panhandle A, Panhandle B and Weymouth to represent flow in pipes of different sizes and a valve and regulator equation were considered. The obtained system consists of a set of nonlinear equations of node pressures and edge flowrates. Application in a network in the field involving a large number of outlets will result in a large system of nonlinear equations to be solved. In this study, the Broyden method was used for solving the system of equations. It showed satisfactory performance when implemented with field data

Addressing challenges of real-world lot sizing problems with interactive multiobjective optimization

Monissa tosielämän ongelmissa on useita optimoitavia tavoitefunktioita, jotka ovat ristiriidassa keskenään. Esimerkiksi tuotantoyritysten varaston ohjauksessa on toimituseriä mitoitettaessa minimoitava kustannuksia ja samalla varmistettava nimikkeiden riittävyys sekä tarpeiden mukainen varastotaso. Monitavoiteoptimointiongelmilla on monia ns. Pareto-optimaalisia ratkaisuja, jotka kuvastavat tavoitefunktioiden välisiä vaihtosuhteita. Tarvitaan päätöksentekijä valitsemaan yksi niistä käytäntöön vietäväksi päätökseksi, joka parhaiten kuvasta hänen mieltymyksiään. Iteratiivisesti mieltymykset huomioon ottavat interaktiiviset menetelmät tukevat päätöksentekijää tehokkaasti. Siksi tässä väitöskirjassa tuetaan interaktiivisten menetelmien avulla toimituserän kokoon ja ajoitukseen liittyvää päätöksentekoa varaston ohjauksessa. Väitöskirja mallintaa ja ratkaisee käytännön haasteista nousevia toimituserän mitoitusongelmia. Ensin yhden nimikkeen toimituserää optimoidaan kysynnän vaihdellessa ja esitellään uusi käsite, varmuusaika. Sitten yhden nimikkeen toimituserää optimoidaan, kun kysyntää on vaikea hallita ja toimitusajat ovat epävarmat. Esiteltävä uusi kaava määrittää varmuusajan nimikkeen riittävyyden todennäköisyydelle. Kolmanneksi yhdistetään toimituserän mitoitus ja minimitoimituserän määrittely. Kaikkien näiden kolmen haasteen ratkaisemiseksi muotoillaan monitavoiteoptimointiongelmat ja ratkaistaan ne. Lopuksi esitellään uusi päätöksenteon tukimenetelmä DESMILS usean nimikkeen toimituserän optimointiin. Sen avulla mitä tahansa yhden nimikkeen monitavoitteinen toimituserän mitoitusmalli voidaan tehokkaasti laajentaa usealle nimikkeelle päätöksentekijän mieltymykset huomioiden. Mallien ja menetelmien soveltuvuutta havainnollistetaan tuotantoyrityksen datalla. Päätöksentekijänä toiminutta yrityksen toimitusketjun johtajaa tuettiin löytämään parhaat ratkaisut eri ongelmiin käyttäen interaktiivisia menetelmiä tai niiden yhdistelmiä. Lisäksi DESMILS auttoi päätöksentekijää mitoittamaan Paretooptimaaliset toimituserät 94 nimikkeelle niin, että hänen täytyi mitoittaa vain 10 huolella valitun nimikkeen tilausmäärät. Päätöksentekijästä kaikki käsitteet, mallit, interaktiiviset ratkaisuprosessit ja ratkaisut olivat hyödyllisiä ja tukivat päivittäisiä toimintoja. Onnistuneet tulokset havainnollistavat tutkimuksen käytännön arvoa ja hyötyä myös muiden toimituserän optimointiongelmien ratkaisemisessa.Many real-world problems involve multiple conflicting objective functions to be optimized simultaneously, including lot sizing problems, where we need to minimize costs while satisfying demand. A multiobjective optimization problem has many so-called Pareto optimal solutions reflecting different trade-offs. A decision maker (DM) is needed to select one of them to be applied in practice represent-ing best his/her preferences. Interactive methods, that iteratively incorporate the DM’s preferences, are beneficial in supporting the DM, and therefore, in this the-sis, we focus on solving lot sizing problems with interactive methods. This thesis tackles challenges in modeling and solving lot sizing problems inspired by real challenges. First, we consider a single-item lot sizing problem under demand uncertainty and propose a safety order time concept that can efficiently handle high fluctuations on demand. Second, we focus on a single-item lot sizing problem under demand and lead time uncertainties, and propose a probability of product availability formula to assess the quality of safety lead time. Third, we integrate a lot sizing problem and a minimum order quantity (MOQ) determination and propose a MOQ level formula to measure the quality of MOQ in satisfying demand. Besides, we also propose multiobjective optimization models to solve these problems. Last, we address a challenge in multi-item lot sizing problems by proposing a decision support approach, called DESMILS. DESMILS enables any single-item multiobjective lot sizing models to be applied in solving multi-item problems by accommodating different preferences from the DM. As a proof of concept, we utilized real data from a company to demonstrate the applicability of the proposed models and approaches. We supported the supply chain manager of the company, as the DM, to find his most preferred solutions by solving the proposed single-item lot sizing models, with interactive methods or the hybridization of methods that we propose. We then demonstrate that, with DESMILS, the DM found Pareto optimal lot sizes for 94 items by solving a single-item multiobjective lot sizing problem for only ten representative items. The DM found all concepts, models, interactive decision making processes, and results useful in his daily operations. These successful applications demonstrate the practical value of the research, which can also benefit others in lot sizing

DESMILS : a decision support approach for multi-item lot sizing using interactive multiobjective optimization

We propose a decision support approach, called DESMILS, to solve multi-item lot sizing problems with a large number of items by using single-item multiobjective lot sizing models. This approach for making lot sizing decisions considers multiple conflicting objective functions and incorporates a decision maker’s preferences to find the most preferred Pareto optimal solutions. DESMILS applies clustering, and items in one cluster are treated utilizing preferences that the decision maker has provided for a representative item of the cluster. Thus, the decision maker provides preferences to solve the single-item lot sizing problem for few items only and not for every item. The lot sizes are obtained by solving a multiobjective optimization problem with an interactive method, which iteratively incorporates preference information and supports the decision maker in learning about the trade-offs involved. As a proof of concept to demonstrate the behavior of DESMILS, we solve a multi-item lot sizing problem of a manufacturing company utilizing their real data. We describe how the supply chain manager as the decision maker found Pareto optimal lot sizes for 94 items by solving the single-item multiobjective lot sizing problem for only ten representative items. He found the solutions acceptable and the solution process convenient saving a significant amount of his time.peerReviewe

Interactive Multiobjective Optimization in Lot Sizing with Safety Stock and Safety Lead Time

In this paper, we integrate a lot sizing problem with the problem of determining optimal values of safety stock and safety lead time. We propose a probability of product availability formula to assess the quality of safety lead time and a multiobjective optimization model as an integrated lot sizing problem. In the proposed model, we optimize six objectives simultaneously: minimizing purchasing cost, ordering cost, holding cost and, at the same time, maximizing cycle service level, probability of product availability and inventory turnover. To present the applicability of the proposed model, we consider a real case study with data from a manufacturing company and apply the interactive NAUTILUS Navigator method to support the decision maker from the company to find his most preferred solution. In this way, we demonstrate how the decision maker navigates without having to trade-off among the conflicting objectives and could find a solution that reflects his preference well.peerReviewe

Integration of lot sizing and safety strategy placement using interactive multiobjective optimization

We address challenges of unpredicted demand and propose a multiobjective optimization model to integrate a lot sizing problem with safety strategy placement and optimize conflicting objectives simultaneously. The novel model is devoted to a single-item multi-period problem in periodic review policy. As a safety strategy, we use the traditional safety stock concept and a novel concept of safety order time, which uses a time period to determine the additional stock to handle demand uncertainty. The proposed model has four objective functions: purchasing and ordering cost, holding cost, cycle service level and inventory turnover. We bridge the gap between theory and a real industrial problem and solve the formulated problem by using an interactive trade-off-free multiobjective optimization method called E-NAUTILUS. It is well suited for computationally expensive problems. We also propose a novel user interface for the method. As a proof of concept for the model and the method, we use real data from a manufacturing company with the manager as the decision maker. We consider two types of items and demonstrate how a decision maker can find a most preferred solution with the best balance among the conflicting objectives and gain valuable insight.peerReviewe

Determination of Gas Pressure Distribution in a Pipeline Network using the Broyden Method

A potential problem in natural gas pipeline networks is bottlenecks occurring in the flow system due to unexpected high pressure at the pipeline network junctions resulting in inaccurate quantity and quality (pressure) at the end user outlets. The gas operator should be able to measure the pressure distribution in its network so the consumers can expect adequate gas quality and quantity obtained at their outlets. In this paper, a new approach to determine the gas pressure distribution in a pipeline network is proposed. A practical and user-friendly software application was developed. The network was modeled as a collection of node pressures and edge flows. The steady state gas flow equations Panhandle A, Panhandle B and Weymouth to represent flow in pipes of different sizes and a valve and regulator equation were considered. The obtained system consists of a set of nonlinear equations of node pressures and edge flowrates. Application in a network in the field involving a large number of outlets will result in a large system of nonlinear equations to be solved. In this study, the Broyden method was used for solving the system of equations. It showed satisfactory performance when implemented with field data