arXiv subjects

S. Armagan Tarim

Publications and source records attributed to S. Armagan Tarim.

At least 19 recordsLinked to original sources

arXiv2026

The optimality of an (s, S) hiring policy on a workforce planning problem with fixed recruitment costs and binomial turnover

We study a finite-horizon workforce planning problem in which staff turnover in each period follows a binomial distribution whose parameters depend on the post-hiring workforce level. The model incorporates a fixed hiring cost that is incurred whenever recruitment occurs, regardless of the number of employees hired. The objective is to minimise the expected total cost, including recruitment, salary, and shortage costs, where deviations below period-specific staffing requirements are penalised. To analyse this stochastic dynamic programme with decision-dependent transition probabilities, we establish the discrete convexity of the variable single-period cost (the sum of expected salary and penalty costs) and the K-convexity of the expected total cost. Specifically, we introduce the concept of Binomial-K-convexity to facilitate the proof that K-convexity is preserved under Binomial propagation in the Bellman function. We then show that the optimal hiring policy exhibits an (s, S)-type structure: when the workforce level in a given period falls below a threshold s, staff are hired up to level S; otherwise, no hiring occurs. Furthermore, we develop a piecewise approximation approach that yields a mixed-integer linear programming (MILP) formulation for solving the problem and computing the (s, S) parameters for each period. Numerical results demonstrate that the proposed method achieves fast computation with small optimality gaps.

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arXiv2026

Mixed-Integer Linear Programming Approximations for the Stochastic Knapsack

We develop mathematical programming approximations to tackle the stochastic knapsack problem. In this problem, the decision maker considers items for which either weights or values, or both, are random. The aim is to select a subset of these items to be included into their knapsack. We study both static and dynamic variants of this problem: in the static setting, the decision about which items should be included in the knapsack is taken at the outset, before any random item value or weight is revealed; in the dynamic setting, items are received sequentially, and the decision about a particular item is made by taking into account previously observed values and weights. The knapsack has a given capacity, and if the total realised weight exceeds this capacity then a penalty cost is incurred for each unit of excess capacity utilised. The goal is to maximise the expected net profit. We tackle the case of normally distributed item weights and we show that our approach extends to the case in which item weights are correlated and follow a multivariate normal distribution. In addition, we show our approach represents an effective heuristic for the case in which item weights follow generic probability distributions. In an extensive computational study we demonstrate that our models are near-optimal and more scalable than other state-of-the-art approaches.

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arXiv2023

On the stochastic inventory problem under order capacity constraints

We consider the single-item single-stocking location stochastic inventory system under a fixed ordering cost component. A long-standing problem is that of determining the structure of the optimal control policy when this system is subject to order quantity capacity constraints; to date, only partial characterisations of the optimal policy have been discussed. An open question is whether a policy with a single continuous interval over which ordering is prescribed is optimal for this problem. Under the so-called "continuous order property" conjecture, we show that the optimal policy takes the modified multi-$(s,S)$ form. Moreover, we provide a numerical counterexample in which the continuous order property is violated, and hence show that a modified multi-$(s,S)$ policy is not optimal in general. However, in an extensive computational study, we show that instances violating the continuous order property are extremely rare in practice, and that the plans generated by a modified multi-$(s,S)$ policy can therefore be considered, for all practical purposes, optimal. Finally, we show that a modified $(s,S)$ policy also performs well in practice.

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arXiv2022

Stochastic Dynamic Programming Heuristic for the (R, s, S) Policy Parameters Computation

The (R, s, S) is a stochastic inventory control policy widely used by practitioners. In an inventory system managed according to this policy, the inventory is reviewed at instant R; if the observed inventory position is lower than the reorder level s an order is placed. The order's quantity is set to raise the inventory position to the order-up-to-level S. This paper introduces a new stochastic dynamic program (SDP) based heuristic to compute the (R, s, S) policy parameters for the non-stationary stochastic lot-sizing problem with backlogging of the excessive demand, fixed order and review costs, and linear holding and penalty costs. In a recent work, Visentin et al. (2021) present an approach to compute optimal policy parameters under these assumptions. Our model combines a greedy relaxation of the problem with a modified version of Scarf's (s, S) SDP. A simple implementation of the model requires a prohibitive computational effort to compute the parameters. However, we can speed up the computations by using K-convexity property and memorisation techniques. The resulting algorithm is considerably faster than the state-of-the-art, extending its adoptability by practitioners. An extensive computational study compares our approach with the algorithms available in the literature.

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arXiv2022

A hybrid estimation of distribution algorithm for joint stratification and sample allocation

In this study we propose a hybrid estimation of distribution algorithm (HEDA) to solve the joint stratification and sample allocation problem. This is a complex problem in which each the quality of each stratification from the set of all possible stratifications is measured its optimal sample allocation. EDAs are stochastic black-box optimization algorithms which can be used to estimate, build and sample probability models in the search for an optimal stratification. In this paper we enhance the exploitation properties of the EDA by adding a simulated annealing algorithm to make it a hybrid EDA. Results of empirical comparisons for atomic and continuous strata show that the HEDA attains the bests results found so far when compared to benchmark tests on the same data using a grouping genetic algorithm, simulated annealing algorithm or hill-climbing algorithm. However, the execution times and total execution are, in general, higher for the HEDA.

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arXiv2021

A Simulated Annealing Algorithm for Joint Stratification and Sample Allocation Designs

This study combines simulated annealing with delta evaluation to solve the joint stratification and sample allocation problem. In this problem, atomic strata are partitioned into mutually exclusive and collectively exhaustive strata. Each partition of atomic strata is a possible solution to the stratification problem, the quality of which is measured by its cost. The Bell number of possible solutions is enormous, for even a moderate number of atomic strata, and an additional layer of complexity is added with the evaluation time of each solution. Many larger scale combinatorial optimisation problems cannot be solved to optimality, because the search for an optimum solution requires a prohibitive amount of computation time. A number of local search heuristic algorithms have been designed for this problem but these can become trapped in local minima preventing any further improvements. We add, to the existing suite of local search algorithms, a simulated annealing algorithm that allows for an escape from local minima and uses delta evaluation to exploit the similarity between consecutive solutions, and thereby reduces the evaluation time. We compared the simulated annealing algorithm with two recent algorithms. In both cases, the simulated annealing algorithm attained a solution of comparable quality in considerably less computation time.

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arXiv2021

Combining K-means type algorithms with Hill Climbing for Joint Stratification and Sample Allocation Designs

In this paper we combine the k-means and/or k-means type algorithms with a hill climbing algorithm in stages to solve the joint stratification and sample allocation problem. This is a combinatorial optimisation problem in which we search for the optimal stratification from the set of all possible stratifications of basic strata. Each stratification being a solution the quality of which is measured by its cost. This problem is intractable for larger sets. Furthermore evaluating the cost of each solution is expensive. A number of heuristic algorithms have already been developed to solve this problem with the aim of finding acceptable solutions in reasonable computation times. However, the heuristics for these algorithms need to be trained in order to optimise performance in each instance. We compare the above multi-stage combination of algorithms with three recent algorithms and report the solution costs, evaluation times and training times. The multi-stage combinations generally compare well with the recent algorithms both in the case of atomic and continuous strata and provide the survey designer with a greater choice of algorithms to choose from.

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arXiv2021

A mathematical programming-based solution method for the nonstationary inventory problem under correlated demand

This paper extends the single-item single-stocking location non-stationary stochastic inventory problem to relax the assumption of independent demand. We present a mathematical programming-based solution method that relaxes the assumption of demand independence between time periods in an existing piecewise linear approximation strategy under the receding horizon control framework. Our method can be solved via off-the-shelf mixed-integer linear programming solvers. It can tackle demand under various assumptions: the multivariate normal distribution, a collection of time-series processes, and the Martingale Model of Forecast Evolution. We compare against solutions via stochastic dynamic programming to demonstrate that our method leads to near-optimal solutions.

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arXiv2020

Computing Optimal (R, s, S) Policy Parameters by a Hybrid of Branch-and-Bound and Stochastic Dynamic Programming

A well-know control policy in stochastic inventory control is the (R, s, S) policy, in which inventory is raised to an order-up-to-level S at a review instant R whenever it falls below reorder-level s. To date, little or no work has been devoted to developing approaches for computing (R, s, S) policy parameters. In this work, we introduce a hybrid approach that exploits tree search to compute optimal replenishment cycles, and stochastic dynamic programming to compute (s, S) levels for a given cycle. Up to 99.8% of the search tree is pruned by a branch-and-bound technique with bounds generated by dynamic programming. A numerical study shows that the method can solve instances of realistic size in a reasonable time.

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arXiv2020

A computational study for the inventory routing problem

In this work we compare several new computational approaches to an inventory routing problem, in which a single product is shipped from a warehouse to retailers via an uncapacitated vehicle. We survey exact algorithms for the Traveling Salesman Problem (TSP) and its relaxations in the literature for the routing component. For the inventory control component we survey classical mixed integer linear programming and shortest path formulations for inventory models. We present a numerical study comparing combinations of the two empirically in terms of cost and solution time.

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arXiv2020

A recursion-free functional approximation for the dynamic inventory problem

We consider the dynamic inventory problem with non-stationary demands. It has long been known that non-stationary (s, S) policies are optimal for this problem. However, finding optimal policy parameters remains a computational challenge as it requires solving a large-scale stochastic dynamic program. To address this, we devise a recursion-free approximation for the optimal cost function of the problem. This enables us to compute policy parameters heuristically, without resorting to a stochastic dynamic program. The heuristic is easy-to-understand and -use since it follows by elementary methods of convex minimization and shortest paths, yet it is very effective and outperforms earlier heuristics.

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arXiv2019

An (R, S) Based Heuristic Model for the Stochastic Joint Replenishment Problem

This paper considers the periodic-review stochastic joint replenishment problem (JRP) under Bookbinder and Tan's static-dynamic uncertainty control policy. According to a static-dynamic uncertainty control rule, the decision maker fixes timing of replenishments once and for all at the beginning of the planning horizon, the inventory position is then raised to a predefined order-up-to-position at the beginning of each replenishment period. In this policy, freezing the replenishment times ameliorates the inherent difficulties pertinent to replenishment coordination of multiple products, whereas dynamic order quantities facilitate dealing with uncertain demands. We adapt and extend an earlier mixed integer linear programming (MILP) model for computing static-dynamic uncertainty policy parameters, and demonstrate that the same can be used to approximate the optimal control rule for the JRP, also known as $(σ, \vec{S})$ policy. An extensive computational study illustrates the effectiveness of our approach when compared to alternative approaches in the literature.

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arXiv2018

A Grouping Genetic Algorithm for Joint Stratification and Sample Allocation Designs

Predicting the cheapest sample size for the optimal stratification in multivariate survey design is a problem in cases where the population frame is large. A solution exists that iteratively searches for the minimum sample size necessary to meet accuracy constraints in partitions of atomic strata created by the Cartesian product of auxiliary variables into larger strata. The optimal stratification can be found by testing all possible partitions. However the number of possible partitions grows exponentially with the number of initial strata. There are alternative ways of modelling this problem, one of the most natural is using Genetic Algorithms (GA). These evolutionary algorithms use recombination, mutation and selection to search for optimal solutions. They often converge on optimal or near-optimal solution more quickly than exact methods. We propose a new GA approach to this problem using grouping genetic operators instead of traditional operators. The results show a significant improvement in solution quality for similar computational effort, corresponding to large monetary savings.

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arXiv2017

Declarative Statistics

In this work we introduce declarative statistics, a suite of declarative modelling tools for statistical analysis. Statistical constraints represent the key building block of declarative statistics. First, we introduce a range of relevant counting and matrix constraints and associated decompositions, some of which novel, that are instrumental in the design of statistical constraints. Second, we introduce a selection of novel statistical constraints and associated decompositions, which constitute a self-contained toolbox that can be used to tackle a wide range of problems typically encountered by statisticians. Finally, we deploy these statistical constraints to a wide range of application areas drawn from classical statistics and we contrast our framework against established practices.

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arXiv2017

Computing non-stationary $(s, S)$ policies using mixed integer linear programming

This paper addresses the single-item single-stocking location stochastic lot sizing problem under the $(s, S) $ policy. We first present a mixed integer non-linear programming (MINLP) formulation for determining near-optimal $(s, S)$ policy parameters. To tackle larger instances, we then combine the previously introduced MINLP model and a binary search approach. These models can be reformulated as mixed integer linear programming (MILP) models which can be easily implemented and solved by using off-the-shelf optimisation software. Computational experiments demonstrate that optimality gaps of these models are around $0.3\%$ of the optimal policy cost and computational times are reasonable.

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arXiv2016

A comparison of non-stationary stochastic lot-sizing strategies

We consider the non-stationary stochastic lot sizing problem with backorder costs and make a cost comparison among different lot-sizing strategies. We initially provide an overview of the strategies and some corresponding solution approaches in the literature. We then compare the cost performances of the lot-sizing strategies on a common test bed while taking into account the added value of realized demand information. The results of this numerical experience enable us to derive novel insights about the cost performance of different stochastic lot-sizing strategies under re-planning with respect to demand realization.

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arXiv2015

Confidence-based Reasoning in Stochastic Constraint Programming

In this work we introduce a novel approach, based on sampling, for finding assignments that are likely to be solutions to stochastic constraint satisfaction problems and constraint optimisation problems. Our approach reduces the size of the original problem being analysed; by solving this reduced problem, with a given confidence probability, we obtain assignments that satisfy the chance constraints in the original model within prescribed error tolerance thresholds. To achieve this, we blend concepts from stochastic constraint programming and statistics. We discuss both exact and approximate variants of our method. The framework we introduce can be immediately employed in concert with existing approaches for solving stochastic constraint programs. A thorough computational study on a number of stochastic combinatorial optimisation problems demonstrates the effectiveness of our approach.

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arXiv2014

Statistical Constraints

We introduce statistical constraints, a declarative modelling tool that links statistics and constraint programming. We discuss two statistical constraints and some associated filtering algorithms. Finally, we illustrate applications to standard problems encountered in statistics and to a novel inspection scheduling problem in which the aim is to find inspection plans with desirable statistical properties.

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