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This paper introduces a graph-based algorithm for solving single-item, single-location inventory lot-sizing problems under non-stationary stochastic demand using the $(R_t, S_t)$ policy and a penalty cost scheme. The proposed method relaxes the original mixed-integer linear programming (MILP) model by eliminating non-negative order quantity constraints and formulating it as a shortest-path problem on a weighted directed acyclic graph. A repetitive augmentation procedure is proposed to resolve any infeasibility in the solution. This procedure consists of three stages: (1) filtration, (2) repeated augmentation by redirecting, reconnecting, and duplicating between newly introduced and existing nodes to adjust the graph and eliminate negative replenishment orders, and (3) re-optimising. The effectiveness and computational efficiency of the proposed approach are assessed through extensive experiments on 1,620 test instances across various demand patterns and parameter settings. The results show that 195 instances required augmentation, mainly those with high penalty costs, low fixed ordering costs, large demand variability, and extended planning horizons. The efficiency of the algorithm for instances with extended planning horizon scenarios demonstrates its suitability for use in real-world scenarios.

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., Comment: 30 pages, submitted manuscript

Stochastic Programming is a framework for modelling and solving problems of decision making under uncertainty. Stochastic Dynamic Programming is a branch of Stochastic Programming that takes a "functional equation" approach to the discovery of optimal policies. By leveraging constructs - lambda expressions, functional interfaces, collections and aggregate operators - implemented in Java to operationalise the MapReduce framework, jsdp provides a general purpose library for modelling and solving Stochastic Dynamic Programs., Comment: 8 pages

This paper addresses the two-stocking locations single item non-stationary stochastic lot-sizing problem. The inventory level at each location is reviewed periodically. Items can be reordered and received from a common central warehouse and can also be transshipped laterally from the other location. Lateral transshipment is assumed to be proactive to re-distribute the stock between two stocking locations. Therefore, the order of action in each period is: transshipping (if necessary), reordering (if necessary) and satisfying the demand at each location and each installation. The costs are imposed on transshipping, ordering, holding, and back-ordering. The key issue in such systems is to determine the quantity of the lateral transshipment between depots and the order quantities from the warehouse to both locations. We formulate the problem via stochastic dynamic programming to minimise the expected total cost. Since the number of actions increases exponentially as the feasible quantities of transshipment and replenishment grow, we develop two-stage dynamic programming to improve the computation efficiency. A near-optimal policy against this two-stage formulation is introduced based on a mixed integer linear programming and receding-horizon approach. numerical experiments are implemented to demonstrate the performance of the two-stage model and the heuristic algorithm

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.

Electric road systems (ERS) are roads that allow compatible vehicles to be powered by grid electricity while in transit, reducing the need for stopping to recharge electric batteries. We investigate how this technology can affect routing and delivery decisions for hybrid heavy good vehicles (HGVs) travelling on a ERS network to support the demand of a single product faced by a set of retailers in the network. We introduce the Electric Roads Routing Problem, which accounts for the costs of electricity and fuel on a ERS network, consumption that are affected by the battery level of the vehicle in each step of the journey, the routing decisions and the variable weight of the vehicle, which depends on vehicle load and delivery decisions. In particular, we study a stochastic demand version of the problem, formulating a mathematical programming heuristic and proving its effectiveness. We use our model on a realistic instance of the problem, showcasing the different strategies that a vehicle may follow depending on fuel costs in relation to the costs of electricity.

The Middle Ages focused obsessively on the old; our era is totally absorbed with the new. In medio stat virtus. In this short note, I advocate a strategy that blends copyright and copyleft for disseminating research results in the sciences. I argue that such a blend may be beneficial in fields such as mathematics and computer science, that it may facilitate the evolution and emergence of improved problem descriptions, whilst at the same time preserving author's rights, and easing researchers' work., Comment: 11 pages, working draft

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.

This paper addresses the single-item single-stocking location non-stationary stochastic lot-sizing problem under a reorder point -- order quantity control strategy. The reorder points and order quantities are chosen at the beginning of the planning horizon. The reorder points are allowed to vary with time and we consider order quantities either to be a series of time-dependent constants or a fixed value; this leads to two variants of the policy: the (st,Qt) and the (st,Q) policies, respectively. For both policies, we present stochastic dynamic programs (SDP) to determine optimal policy parameters and introduce mixed integer non-linear programming (MINLP) heuristics that leverage piecewise-linear approximations of the cost function. Numerical experiments demonstrate that our solution method efficiently computes near-optimal parameters for a broad class of problem instances.

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., Comment: 14 pages

This paper investigates a stochastic inventory management problem in which a cash-constrained small retailer periodically purchases a product from suppliers and sells it to a market while facing non-stationary demands. In each period, the retailer's available cash restricts the maximum quantity that can be ordered. There exists a fixed ordering cost for the retailer when purchasing. We partially characterize the optimal ordering policy by showing it has an $\bf s-C$ structure: for each period, when initial inventory is above the $\bf$s threshold, no product should be ordered no matter how much initial cash it has; when initial inventory is not large enough to be a $\bf s$ threshold, it is also better to not order when initial cash is below the threshold $C$. The values of $C$ may be state-dependent and related to each period's initial inventory. A heuristic policy $(s, C(x), S)$ is proposed: when initial inventory $x$ is less than $s$ and initial cash is greater than $C(x)$, order a quantity that brings inventory as close to $S$ as possible; otherwise, do not order. We first determine the values of the controlling parameters $s$, $C(x)$ and $S$ based on the results of stochastic dynamic programming and test their performance via an extensive computational study. The results show that the $(s, C(x), S)$ policy performs well with a maximum optimality gap of less than 1\% and an average gap of approximately 0.01\%. We then develop a simple and time-efficient heuristic method for computing policy $(s, C(x), S)$ by solving a mixed-integer linear programming problem and approximate newsvendor models: the average gap for this heuristic is approximately 2\% on our test bed.

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 $(\sigma, \vec{S})$ policy. An extensive computational study illustrates the effectiveness of our approach when compared to alternative approaches in the literature.

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.

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., Comment: The modeling framework and the examples used in this work are available at https://gwr3n.github.io/syat-choco/

In this paper, we present the design, fabrication and optical characterization of computer-generated holograms (CGH) encoding information for light beams carrying orbital angular momentum (OAM). Through the use of a numerical code, based on an iterative Fourier transform algorithm, a phase-only diffractive optical element (PH-DOE) specifically designed for OAM illumination has been computed, fabricated and tested. In order to shape the incident beam into a helicoidal phase profile and generate light carrying phase singularities, a method based on transmission through high-order spiral phase plates (SPPs) has been used. The phase pattern of the designed holographic DOEs has been fabricated using high-resolution Electron-Beam Lithography (EBL) over glass substrates coated with a positive photoresist layer (polymethylmethacrylate). To the best of our knowledge, the present study is the first attempt, in a comprehensive work, to design, fabricate and characterize computer-generated holograms encoding information for structured light carrying OAM and phase singularities. These optical devices appear promising as high-security optical elements for anti-counterfeiting applications., Comment: 24 pages, 9 figures. Supplementary info: 8 pages, 3 figures

This paper introduces capital flow to the single item stochastic lot sizing problem. A retailer can leverage business overdraft to deal with unexpected capital shortage, but needs to pay interest if its available balance goes below zero. A stochastic dynamic programming model maximizing expected final capital increment is formulated to solve the problem to optimality. We then investigate the performance of four controlling policies: ($R, Q$), ($R, S$), ($s, S$) and ($s$, $\overline{Q}$, $S$); for these policies, we adopt simulation-genetic algorithm to obtain approximate values of the controlling parameters. Finally, a simulation-optimization heuristic is also employed to solve this problem. Computational comparisons among these approaches show that policy $(s, S)$ and policy $(s, \overline{Q}, S)$ provide performance close to that of optimal solutions obtained by stochastic dynamic programming, while simulation-optimization heuristic offers advantages in terms of computational efficiency. Our numerical tests also show that capital availability as well as business overdraft interest rate can substantially affect the retailer's optimal lot sizing decisions., Comment: 18 pages, 3 figures

We investigate opportunities offered by telematics and analytics to enable better informed, and more integrated, collaborative management decisions on construction sites. We focus on efficient refuelling of assets across construction sites. More specifically, we develop decision support models that, by leveraging data supplied by different assets, schedule refuelling operations by minimising the distance travelled by the bowser truck as well as fuel shortages. Motivated by a practical case study elicited in the context of a project we recently conducted at Crossrail, we introduce the Dynamic Bowser Routing Problem. In this problem the decision maker aims to dynamically refuel, by dispatching a bowser truck, a set of assets which consume fuel and whose location changes over time; the goal is to ensure that assets do not run out of fuel and that the bowser covers the minimum possible distance. We investigate deterministic and stochastic variants of this problem and introduce effective and scalable mathematical programming models to tackle these cases. We demonstrate the effectiveness of our approaches in the context of an extensive computational study designed around data collected on site as well as supplied by our project partners. Keywords: Routing; Dynamic Bowser Routing Problem; Stochastic Bowser Routing Problem; Mixed-Integer Linear Programming; Construction.

Stochastic Constraint Programming (SCP) is an extension of Constraint Programming (CP) used for modelling and solving problems involving constraints and uncertainty. SCP inherits excellent modelling abilities and filtering algorithms from CP, but so far it has not been applied to large problems. Reinforcement Learning (RL) extends Dynamic Programming to large stochastic problems, but is problem-specific and has no generic solvers. We propose a hybrid combining the scalability of RL with the modelling and constraint filtering methods of CP. We implement a prototype in a CP system and demonstrate its usefulness on SCP problems.

The accepted lore is that Operational Research traces its roots back to the First and Second World Wars, when scientific research was used to improve military operations. In this essay we provide a different perspective on the origins of Operational Research by arguing that these are deeply intertwined with the impressive technological advances in Western Europe between the fifteenth and the sixteenth centuries.

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.

We introduce the BIN_COUNTS constraint, which deals with the problem of counting the number of decision variables in a set which are assigned values that lie in given bins. We illustrate a decomposition and a filtering algorithm that achieves generalised arc consistency. We contrast the filtering power of these two approaches and we discuss a number of applications. We show that BIN_COUNTS can be employed to develop a decomposition for the $\chi^2$ test constraint, a new statistical constraint that we introduce in this work. We also show how this new constraint can be employed in the context of the Balanced Academic Curriculum Problem and of the Balanced Nursing Workload Problem. For both these problems we carry out numerical studies involving our reformulations. Finally, we present a further application of the $\chi^2$ test constraint in the context of confidence interval analysis., Comment: 20 pages, working draft

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., Comment: working draft