Nowadays in the world of mass consumption there is big demand for distributioncenters of bigger size. Managing such a center is a very complex and difficult taskregarding to the different processes and factors in a usual warehouse when we want tominimize the labor costs. Most of the workers’ working time is spent with travelingbetween source and destination points which cause deadheading. Even if a worker knowsthe structure of a warehouse well and because of that he or she can find the shortest pathbetween two points, it is still not guaranteed that there won’t be long traveling timebetween the locations of two consecutive tasks. We need optimal assignments betweentasks and workers.In the scientific literature Generalized Assignment Problem (GAP) is a wellknownproblem which deals with the assignment of m workers to n tasks consideringseveral constraints. The primary purpose of my thesis project was to choose a heuristics(genetic algorithm, tabu search or ant colony optimization) to be implemented into SAPExtended Warehouse Management (SAP EWM) by with task assignment will be moreeffective between tasks and resources.After system analysis I had to realize that due different constraints and businessdemands only 1:1 assingments are allowed in SAP EWM. Because of that I had to use adifferent and simpler approach – instead of the introduced heuristics – which could gainbetter assignments during the test phase in several cases. In the thesis I described indetails what ware the most important questions and problems which emerged during theplanning of my optimized assignment method.

<p>In this thesis a method for solving the Generalized Assignment Problem (GAP) is described. It is based on a reformulation of the original problem into a Set Partitioning Problem (SPP), in which the columns represent partial solutions to the original problem. For solving this problem, column generation, with systematic overgeneration of columns, is used. Conditions that guarantee that an optimal solution to a restricted SPP is optimal also in the original problem are given. In order to satisfy these conditions, not only columns with the most negative Lagrangian reduced costs need to be generated, but also others; this observation leads to the use of overgeneration of columns.</p><p>The Generalized Assignment Problem has shown to be NP-hard and therefore efficient algorithms are needed, especially for large problems. The application of the proposed method decomposes GAP into several knapsack problems via Lagrangian relaxation, and enumerates solutions to each of these problems. The solutions obtained from the knapsack problems form a Set Partitioning Problem, which consists of combining one solution from each knapsack problem to obtain a solution to the original problem. The algorithm has been tested on problems with 10 agents and 60 jobs. This leads to 10 knapsack problems, each with 60 variables.</p>

In this thesis a method for solving the Generalized Assignment Problem (GAP) is described. It is based on a reformulation of the original problem into a Set Partitioning Problem (SPP), in which the columns represent partial solutions to the original problem. For solving this problem, column generation, with systematic overgeneration of columns, is used. Conditions that guarantee that an optimal solution to a restricted SPP is optimal also in the original problem are given. In order to satisfy these conditions, not only columns with the most negative Lagrangian reduced costs need to be generated, but also others; this observation leads to the use of overgeneration of columns. The Generalized Assignment Problem has shown to be NP-hard and therefore efficient algorithms are needed, especially for large problems. The application of the proposed method decomposes GAP into several knapsack problems via Lagrangian relaxation, and enumerates solutions to each of these problems. The solutions obtained from the knapsack problems form a Set Partitioning Problem, which consists of combining one solution from each knapsack problem to obtain a solution to the original problem. The algorithm has been tested on problems with 10 agents and 60 jobs. This leads to 10 knapsack problems, each with 60 variables.

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The research reported in this thesis considers the classical combinatorial optimization problem known as the Generalized Assignment Problem (GAP). Since the mid 1970's researchers have been developing solution approaches for this particular type of problem due to its importance both in practical and theoretical terms. Early attempts at solving GAP tended to use exact integer programming techniques such as Branch and Bound. Although these tended to be reasonably successful on small problem instances they struggle to cope with the increase in computational effort required to solve larger instances. The increase in available computing power during the 1980's and 1990's coincided with the development of some highly efficient heuristic approaches such as Tabu Search (TS), Genetic Algorithms (GA) and Simulated Annealing (SA). Heuristic approaches were subsequently developed that were able to obtain high quality solutions to larger and more complex instances of GAP. Most of these heuristic approaches were able to outperform highly sophisticated commercial mathematical programming software since the heuristics tend to be tailored to the problem and therefore exploit its structure. A new approach for solving GAP has been developed during this research that combines the exact Branch and Bound approach and the heuristic strategy of Tabu Search to produce a hybrid algorithm for solving GAP. This approach utilizes the mathematical programming software Xpress-MP as a Branch and Bound solver in order to solve sub-problems that are generated by the Tabu Search guiding heuristic. Tabu Search makes use of memory structures that record information about attributes of solutions visited during the search. This information is used to guide the search and in the case of the hybrid algorithm to generate sub problems to pass to the Branch and Bound solver. The new algorithm has been developed, imp lemented and tested on benchmark test problems that are extremely challenging and a comprehensive report and analysis of the experimentation is reported in this thesis.

A key challenge for manufacturing companies is to store parts in an efficient way atthe lowest cost possible. As the demand of differentiated products increases, togetherwith the fact that old products are not phased out at the same pace, the need of usingstorage space as dynamically as possible becomes vital.Scania’s engine assembly manufactures engines for various automotive vehicles andmarine &amp; industry applications. The variation in engine range in Scania’s offeringleads to the need of holding a vast, and increasing, assortment of parts in the produc-tion. As a consequence, this puts more pressure on the logistics and furnishing withinthe engine assembly.This master thesis aims to facilitate the process of assigning parts’ storage locationsin the most profitable manner through an optimization model, the Location Model, inExcel VBA. Together with the model, suggestions of work methods are presented.By implementing the Location Model at Scania’s engine assembly, 4,98 % of all keptparts are recommended location changes, while resulting in cost savings, for the chosen30-day period. These location changes result in a cost saving of 6,73 % of the totallogistic costs for the same time period.

In this thesis, we consider the Multi Resource Agent Bottleneck Generalized Assignment Problem. We aim to minimize the maximum load over all agents. We study the Linear Programming (LP) relaxation of the problem. We use the optimal LP relaxation solutions in our Branch and Bound algorithm while defining lower and upper bounds and branching schemes. We find that our Branch and Bound algorithm returns optimal solutions to the problems with up to 60 jobs when the number of agents is 5, and up to 30 jobs when the number of agents is 10, in less than 20 minutes. To find approximate solutions, we define a tabu search algorithm and an &amp / #945 / approximation algorithm. Our computational results have revealed that these procedures can find high quality solutions to large sized instances very quickly.

Cette thèse porte sur la planification du personnel en accordant une attention particulière à l'aspect humain et aux facteurs ergonomiques dans le domaine de la production. Un certain nombre de modèles mathématiques sont présentés pour formuler les problèmes d'ordonnancement et de planification du personnel étudié. Concernant les modèles de planification, la productivité du système de fabrication et le bien-être des travailleurs sont ciblés. De cette manière, une méthode d'affectation des travailleurs est présentée pour réduire le temps de production et une méthode d'ordonnancement pour la rotation des tâches est présentée afin d’équilibrer la charge de travail des opérateurs. À cet effet, une analyse ergonomique est effectuée sur les postes de travail du système de production étudié. Cette analyse aboutit à l'évaluation des postes du travail suivant la convention dite des feux de circulation, c'est-à-dire que les postes sont classés dans les niveaux de charge faible, moyen et élevé qui sont représentés respectivement par les couleurs verte, jaune et rouge. Une approche mathématique est développée pour convertir ces résultats en valeurs numériques, car les paramètres quantitatifs sont plus applicables pour l'optimisation de la planification. Une programmation multi-objectifs est proposée pour optimiser les deux objectifs mentionnés du problème d'ordonnancement de tournée du personnel étudié. Les méthodes d'agrégation linéaire et de ε-contrainte sont appliquées pour résoudre ce modèle d'optimisation. En outre, cette thèse présente une nouvelle variante du problème d'affectation appelé problème d'affectation généralisée par séquence qui est défini pour la planification du personnel dans un système combiné constitué des postes de travail en série et en parallèle. Il est prouvé que ce problème d'optimisation combinatoire est NP-difficile et les méthodes exactes ne sont pas capables de résoudre les instances de grande taille. Ainsi, trois méthodes approchées composées de deux approches matheuristiques et une heuristique hybride sont développées pour résoudre ce problème. Les méthodes matheuristiques sont basées sur la décomposition de la formulation pour simplifier le modèle principal en deux ou plusieurs modèles plus petits. La troisième méthode est une heuristique gloutonne combinée à une recherche locale. En outre, dans la dernière étape de cette thèse, la planification des ressources humaines pour un système de soins à domicile est formulée mathématiquement. Selon la structure du système, une intégration des problèmes d'affectation et de tournées de véhicules est présentée. Enfin, une approche matheuristique en trois étapes est proposée pour résoudre ce problème d'optimisation combinatoire. / This thesis concerns the human resource planning by paying a special attention to the human aspect and ergonomic factors in the manufacturing domain. A number of mathematical models are presented to formulate the studied workforce scheduling and planning problems. In the planning models, the productivity of the manufacturing system and the well-being of the workers are targeted. In this way, a worker assignment approach is presented to reduce the production time and a job rotation scheduling approach is presented to balance the workloads on the operators. For this purpose, an ergonomic analysis is carried out on the jobs of the studied production system. This analysis results in the traffic light evaluation for the jobs, i.e., the jobs are categorized into the low, medium and high workload levels which are presented respectively by the green, yellow and red colors. A mathematical approach is developed to convert these outputs to the numerical values, because the quantitative parameters are more applicable for the optimization of the planning. A multi-objective programming is proposed to optimize two mentioned objectives of the studied workforce scheduling problem. Both linear aggregation and epsilon-constraint methods are applied to solve this optimization model. Furthermore, this thesis presents a novel variant of the assignment problem called sequencing generalized assignment problem which is defined for workforce scheduling in a combined system consisting of the jobs in series and in parallel. It is proved that this combinatorial optimization problem is NP-hard and the exact methods are not able to solve the large-scale instances. Hence, three approximate methods consisting of two matheuristic and a hybrid heuristic approaches are developed to solve it. The matheuristic methods are based on the decomposition of the formulation to break down and simplify the main model into two or more smaller models. The third method is a greedy heuristic combined with a local search. The efficiency of the three mentioned methods is evaluated by various instances of different sizes. Moreover, in the last step of this thesis, the human resource planning for a home healthcare system is formulated mathematically. According to the structure of the system, an integration of the worker assignment and vehicle routing problems is presented. Finally, a three-steps matheuristic approach is proposed to solve this combinatorial optimization problem.

The work presented in this dissertation has been focused on exploiting the branch-and-price (BNP) method for the solution of various stochastic mixed integer programming problems (MIPs). In particular, we address the stochastic generalized assignment problem (SGAP), a hospital staff scheduling problem (HSSP), a stochastic hospital staff scheduling problem (SHSSP), and a stochastic short-term personnel planning problem (SSTPP). The BNP method has been developed in concert with the dual stabilization technique and other enhancements of this method for each of these problems. In view of an excessive number of scenarios that arise for these problems, we also implement the Monte Carlo method within the BNP scheme. The superiority of the BNP-based method over the branch-and-cut (BNC) method is demonstrated for all of these problems. The first problem that we address is the SGAP for which the processing time of a job on a machine is assumed to be stochastic. Even though the generalized assignment problem (GAP) has been solved using the BNP method, yet no study has been reported in the literature on the use of the BNP method for the solution of the SGAP. Our work has been motivated by the desire to fill this gap. We begin by showing that it is better to solve the SGAP as a stochastic program in contrast to solving it by using the expected values of the times required to process the jobs on the machines. Then, we show that the stochastic model of the SGAP is a complete recourse model — a useful property which permits the first stage decisions to produce feasible solutions for the recourse problems. We develop three BNP-based methods for the solution of the SGAP. The first of these is BNP-SGAP, which is a combination of branch-and-bound and column generation methods. The pricing problem of BNP-SGAP is separable with regard to each machine, and it is a multiple-constraint knapsack problem. The second method is BNP-SGAP implemented in concert with the dual stabilization technique (DST), and it is designated as BNPDST-SGAP. We have introduced a new DST by modifying the Boxstep method of Pigatti et al. [76]. We have shown that our method performs better than the method of Pigatti et al. [76] resulting in over two-fold savings in cpu times on average. The third method that we develop for the solution of the SGAP is BNPDST-SGAP implemented with an advanced start to obtain an initial feasible solution. We use a greedy heuristic to obtain this solution, and this heuristic is a modification of a similar method used for the knapsack problem. It relies on the information available at a node of the underlying branch-and-bound tree. We have shown that this procedure obtains an initial feasible solution, if it exists at that node. We designate this method as BNPDSTKP-SGAP. We have also developed a BNC method to solve the SGAP using CPLEX 9.0. We have compared the performances of the BNP and BNC methods on various problem instances obtained by varying the number of machines, the ratio of the number of machines to the number of jobs, the machine capacity, and the penalty cost per unit of extra resource required at each machine. Our results show that all BNP-based methods perform better than the BNC method, with the best performance obtained for BNPDSTKP-SGAP. An issue with the use of the scenario-based methods that we have employed for the solution of the SGAP is that the number of scenarios generally grows exponentially in problem parameters, which gives rise to a large-size problem. To overcome the complexity caused by the presence of a large number of scenarios for the solution of the SGAP, we introduce the use of the Monte Carlo method (MCM) within the BNP scheme. We designate this method as BNPDSTKP-SGAP with MCM. It affords the use of a small subset of scenarios at a time to estimate the "true" optimal objective function value. Replications of the subsets of scenarios are carried out until the objective function value satisfies a stopping criterion. We have established theoretical results for the use of the MCM. These pertain to determining unbiased estimates of: (i) lower and upper bounds of the "true" optimal objective function value, (ii) the "true" optimal solution, and (iii) the optimality gap. We have also provided the 100(1-ï ¡) confidence interval on the optimality gap. Our experimental investigation has shown the efficacy of using this method. It obtains almost optimal solutions, with the objective function value lying within 5% of the "true" optimal objective function value, while giving almost ten-fold savings in cpu time. Our experimentation has also revealed that an increment in the number of scenarios in each replication makes a greater impact on the quality of the solution obtained than an increment in the number of replications. We have also observed the impact of a change in the variance of a processing time distribution on cpu time. As expected, the optimal objective function value increases with increment in processing time variability. Also, by comparing the results with the expected value solution, it is observed that the greater the variability in the data, the better it is to use the stochastic program. The second problem that we study is the hospital staff scheduling problem. We address the following three versions of this problem: HSSP (General): Implementation of schedule incorporating the four principal elements, namely, surgeons, operations, operating rooms, and operation times; HSSP (Priority): Inclusion of priority for some surgeons over the other surgeons regarding the use of the facility in HSSP (General); HSSP (Pre-arranged): Implementation of a completely pre-fixed schedule for some surgeons. The consideration of priority among the surgeons mimics the reality. Our BNP method for the solution of these problems is similar to that for the SGAP except for the following: (i) a feasible solution at a node is obtained with no additional assignment, i.e., it consists of the assignments made in the preceding nodes of that node in the branch-and-bound tree; (ii) the columns with positive reduced cost are candidates for augmentation in the CGM; and (iii) a new branching variable selection strategy is introduced, which selects a fractional variable as a branching variable by fixing a value of which we enforce the largest number of variables to either 0 or 1. The priority problem is separable in surgeons. The results of our experimentation have shown the efficacy of using the BNP-based method for the solution of each HSSP as it takes advantage of the inherent structure of each of these problems. We have also compared their performances with that of the BNC method developed using CPLEX. For the formulations HSSP (General), HSSP (Priority), and HSSP (Pre-arranged), the BNP method gives better results for 22 out of 30, 29 out of 34, and 20 out 32 experiments over the BNC method, respectively. Furthermore, while the BNC method fails to obtain an optimal solution for 15 experiments, the BNP method obtains optimal solutions for all 96 experiments conducted. Thus, the BNP method consistently outperforms the BNC method for all of these problems. The third problem that we have investigated in this study is the stochastic version of the HSSP, designated as the Stochastic HSSP (SHSSP), in which the operation times are assumed to be stochastic. We have introduced a formulation for this formulation, designated as SHSSP2 (General), which allows for overlapping of schedules for surgeons and operating rooms, and also, allows for an assignment of a surgeon to perform an operation that takes less than a pre-arranged operation time, but all incurring appropriate penalty costs. A comparison of the solution of SHSSP2 (General) and its value with those obtained by using expected values (the corresponding problem is designated as Expected-SHSSP2 (General)) reveals that Expected-SHSSP2 (General) may end up with inferior and infeasible schedules. We show that the recourse model for SHSSP2 (General) is a relatively complete recourse model. Consequently, we use the Monte Carlo method (MCM) to reduce the complexity of solving SHSSP2 (General) by considering fewer scenarios. We employ the branch-and-cut (BNC) method in concert with the MCM for solving SHSSP2 (General). The solution obtained is evaluated using tolerance ratio, closeness to optimality, length of confidence interval, and cpu time. The MCM substantially reduces computational effort while producing almost optimal solutions and small confidence intervals. We have also considered a special case of SHSSP2 (General), which considers no overlapping schedules for surgeons and operating rooms and assigns exactly the same operation time for each assignment under each scenario, and designate it as SHSSP2 (Special). With this, we consider another formulation that relies on the longest operation time among all scenarios for each assignment of a surgeon to an operation in order to avoid scheduling conflicts, and we designate this problem as SHSSP (Longest). We show SHSSP (Longest) to be equivalent to deterministic HSSP, designated as HSSP (Equivalent), and we further prove it to be equivalent to SHSSP (General) in terms of the optimal objective function value and the optimal assignments of operations to surgeons. The schedule produced by HSSP (Equivalent) does not allow any overlap among the operations performed in an operating room. That is, a new operation cannot be performed if a previous operation scheduled in that room takes longer than expected. However, the schedule generated by HSSP (Equivalent) may turn out to be a conservative one, and may end up with voids due to unused resources in case an operation in an operating room is completed earlier than the longest time allowed. Nevertheless, the schedule is still a feasible one. In such a case, the schedule can be left-shifted, if possible, because the scenarios are now revealed. Moreover, such voids could be used to perform other procedures (e.g., emergency operations) that have not been considered within the scope of the SHSSP addressed here. Besides, such a schedule can provide useful guidelines to plan for resources ahead of time. The fourth problem that we have addressed in this dissertation is the stochastic short-term personnel planning problem, designated as Stochastic STPP (SSTPP). This problem arises due to the need for finding appropriate temporary contractors (workers) to perform requisite jobs. We incorporate uncertainty in processing time or amount of resource required by a contractor to perform a job. Contrary to the SGAP, the recourse model for this problem is not a relatively complete recourse model. As a result, we cannot employ a MCM method for the solution of this problem as it may give rise to an infeasible solution. The BNP method for the SSTPP employs the DST and the advanced start procedure developed for the SGAP, and due to extra constraints and presence of binary decision variables, we use the branching variable selection strategy developed for the HSSP models. Because of the distinctive properties of the SSTPP, we have introduced a new node selection strategy. We have compared the performances of the BNC-based and BNP-based methods based on the cpu time required. The BNP method outperforms the BNC method in 75% of the experiments conducted, and the BNP method is found to be quite stable with smaller variance in cpu times than those for the BNC method. It affords solution of difficult problems in smaller cpu times than those required for the BNC method. / Ph. D.