Dynamic Lot-sizing Problem Research Articles

Forecast Horizon of Dynamic Lot Sizing Problem with Perishable Inventory and Multiple Chain Stores: Shipping and Stockout Cost

Perishable products are very common, but managing inventory of perishable products can be very challenging for firms, especially in distribution systems, including multiple chain stores. In this environment, we consider a dynamic lot-sizing problem faced by a distribution center that dis-patches a single perishable product to multiple chain stores. Demand cannot be backlogged, but it does not have to be satisfied; unsatisfied demand means stockout (lost sale). The first step is to transform the total profit function into a special total cost function. Our next step is to explore the properties of the optimal solution and use them to formulate a dynamic programming algorithm to solve the problem. Furthermore, we establish forecast and decision horizon results, which help the operation manager to decide the precise forecast horizon in a rolling decision-making process. Based on the model setting and the methods of dynamic programming, we obtained two interesting findings: (1) the maximized profit objective function is equivalent to the minimized cost objective function, and (2) the famous zero inventory property conditionally holds in the inventory management of perishable products. On an extensive test bed, useful insights were obtained on the impact of the lifetime of the product and cost parameters on the total cost and length of the forecast horizon. Thus, the contributions of this study are as follows: (1) we explore two structure policies in an optimal solution to devise efficient algorithms to reduce computational complexity; (2) we provide a sufficient condition for forecasting and decision horizons; and (3) we determine that, for a given fixed cost, the median forecast horizon first increases with the lifetime of the product and stockout cost and then remains invariable when it reaches a certain level.

Complexity analysis of integrated dynamic lot sizing and maintenance planning problems

We consider a problem where the planning decisions of producing on and maintaining a single machine are integrated. The age of the machine, and thus when maintenance should be performed, is influenced by the production decisions. Decisions are taken on a planning horizon discretized in periods. The computational complexity of several variants of the single-item problem are studied depending on whether (i) Maintenance can be performed on the machine at any point in time in a period or must be performed at the end of a period, (ii) There is a fixed aging component when starting production, and (iii) There is a minimum age before maintenance can be performed on the machine. The multi-item case is proved to be NP-hard.

A modified variable neighborhood search algorithm for dynamic lot-sizing with supplier selection under varying delivery time quotation

This paper investigates a dynamic lot-sizing problem with supplier selection, where the manufacturer develops an aggregate plan to meet product delivery demand with the objective of minimizing the cost. Specifically, a novel supplier selection scenario with varying delivery time quotation is considered, i.e., the manufacturer places raw material replenishment orders with varying delivery time quotation and dynamically selects suppliers, and the suppliers guarantee delivery within the quoted delivery lead time. In the dynamic lot-sizing problem with supplier selection, a joint decision on the production plan, replenishment order, and supplier selection needs to be made. To tackle this complex problem, a modified Variable Neighborhood Search algorithm with nested two parallel neighborhoods (PN-VNS) is first developed for the decision on the production plan and replenishment order. Then, the decision on supplier selection needs to be made for the given replenishment order, and a pre-pruning optimization rule is proposed to obtain the optimal supplier scheme. Computational experiments are conducted to evaluate the performance of the proposed algorithm, and the experimental results demonstrate that our algorithm performs better both in terms of solution quality and running time.

Efficient approximate algorithms for a class of dynamic lot size problems under product substitution

Efficient approximate algorithms for a class of dynamic lot size problems under product substitution

On inequality constraints in well-known optimisation problems: properties and applications

This work studies transportation, assignment, transshipment, shortest path, and dynamic lot sizing problems. Particularly, it shows how to modify their formulations to handle excess supply and node elimination. The paper proceeds by using the conservation of flow to derive the original constraints of these problems. Thereafter, a general form of the conservation of flow is used to reformulate these problems. The results are problems with inequality constraints. This paper shows that solving each of these new formulations returns the same optimal solution as its original form. It also shows how problems with inequality constraints can handle excess supply and node elimination. Along the way, some additional properties of these alternative formulations are explored.

Reformulations to improve the Lagrangian relaxation approach for the capacitated multi-product dynamic lot sizing problem with batch ordering

In this work, we study the multi-product dynamic lot-sizing problem with capacity constraints and batch ordering. This problem arises in short to medium range production scheduling for several products over a finite number of periods to meet known demand. Each period has a capacity for placing orders, and every order for each product must have a fixed quantity, or batch size, though multiple orders can be placed for each product. We define three mixed-integer linear programming (MILP) models and apply Lagrangian relaxation to formulate the corresponding dual problems by relaxing the capacity constraints. The aim is to identify the dual problem that is the easiest to solve and provides the solution with the smallest duality gap. Subgradient optimisation is applied to solve the preferred Lagrangian dual model, which uses one of two heuristics to find good feasible solutions. We also show that the special case, where the batch sizes for all products are the same, can be modeled as a transportation problem. A set of numerical experiments is designed to compare the performance of the Lagrangian relaxation approach with a commercial MILP solver to identify the version of the subgradient algorithm and the MILP model that provide the best solutions.

A dynamic programming approach for the two-product capacitated lot-sizing problem with concave costs

In this paper, we analyze a two-product multi-period dynamic lot-sizing problem with a fixed capacity constraint in each period. Each product has a known demand in each period that must be satisfied over a finite planning horizon. The aim of this problem is to minimize the overall cost of placing orders and carrying inventory across all periods. The structure of an optimal solution is analyzed with respect to a type of period called regeneration period, which is a period where the inventory of one or both products reach zero. We show that there is an optimal arrangement of placing orders between consecutive regeneration periods. We propose a pseudo-polynomial algorithm to solve the two-product problem. First, we show how the optimal ordering pattern between two consecutive regeneration periods can be solved using a shortest path problem. Then, we explain how the optimal locations for regeneration periods can be found by solving a shortest path problem on a different network, where each arc corresponds to the shortest path in a subproblem network. We then show how this approach can be scaled up to a three-product problem and generalize this technique to any number of products, as long as it is small.

Chance-constrained programming with robustness for lot-sizing and scheduling problems under complex uncertainty

PurposeThe purpose of this paper deals with a capacitated multi-item dynamic lot-sizing problem with the simultaneous sequence-dependent setup scheduling of the parallel resource under complex uncertainty.Design/methodology/approachAn improved chance-constrained method is developed, in which confidence level of uncertain parameters is used to process uncertainty, and based on this, the reliability of the constraints is measured. Then, this study proposes a robust reconstruction method to transform the chance-constrained model into a deterministic model that is easy to solve, in which the robust transformation methods are used to deal with constraints with uncertainty on the right/left. Then, experimental studies using a real-world production data set provided by a gearbox synchronizer factory of an automobile supplier is carried out.FindingsThis study has demonstrated the merits of the proposed approach where the inventory of products tends to increase with the increase of confidence level. Due to a larger confidence level may result in a more strict constraint, which means that the decision-maker becomes more conservative, and thus tends to satisfy more external demands at the cost of an increase of production and stocks.Research limitations/implicationsJoint decisions of production lot-sizing and scheduling widely applied in industries can effectively avert the infeasibility of lot-size decisions, caused by capacity of lot-sing alone decision and complex uncertainty such as product demand and production cost. is also challenging.Originality/valueThis study provides more choices for the decision-makers and can also help production planners find bottleneck resources in the production system, thus developing a more feasible and reasonable production plan in a complex uncertain environment.

Forecast horizons for a two-echelon dynamic lot-sizing problem

This paper examines the forecast horizons under the settings of two-echelon dynamic lot-sizing models with durable and perishable products, where the first echelon also has its own external demands. In the model of durable products, inventory costs are age-independent. In the model of perishable products, inventory deterioration and costs are age-dependent. On the basis of certain significant properties in an optimal solution, forward dynamic programming algorithms are developed for solving the two models of durable and perishable products. Furthermore, we establish monotone properties of production and regeneration points and provide sufficient conditions to obtain forecast horizons under two cases of durable and perishable products. By conducting several numerical experiments, managerial insights are obtained regarding the influence of integrated decision, external demands at the first echelon, perishability and costs’ parameters on forecast horizons. A rolling horizon decision-making is an easy-to-use practice in dynamic lot-sizing problems. We test the performance of the rolling horizon approach in a numerical experiment.

A Class of Two-Level Dynamic Lot Sizing Problems with Complementary Components: The Infinite Horizon Case

A Class of Two-Level Dynamic Lot Sizing Problems with Complementary Components: The Infinite Horizon Case

A Class of Two-Level Dynamic Lot Sizing Problems with Complementary Components: The Infinite Horizon Case

A Class of Two-Level Dynamic Lot Sizing Problems with Complementary Components: The Infinite Horizon Case

Column generation approach for solving uncapacitated dynamic lot-sizing problems with time-varying cost

This study proposes the column generation approach for generating the uncapacitated dynamic lot-sizing problem solution under time-varying cost. The dynamic programming approach is also applied to calculate the cost function value. However, this column generation approach generates only an optimal lot size at period 1. Therefore, using the consistent arc method, search the connecting arc from period 1 to period N. Comparing the lot sizes generated by the proposed and the dynamic programming methods, they are the same value. Therefore, they are an optimal solution. Using GUROBI's simplex method can enhance to reduce the running time on dense matrices to be less than the original simplex method, about 25.4% and 26.1% for 12 and 24 periods. The lowest raw material fluctuation price period is selected obviously to order so that the total inventory cost will still be minimal. Therefore, the raw material price significantly influences procurement decision-making.

Dynamic lot-sizing with short-term financing and external deposits for a capital-constrained manufacturer

The interface between operations and finance has recently been of interest to both practitioners and academics, who recognize that coordinating decisions from both functions provides a great opportunity to improve the performance of firms. However, the link between dynamic lot-sizing and financing decisions remains largely unexplored in the literature. This paper investigates a dynamic lot-sizing problem with short-term financing and external deposits for a capital-constrained manufacturer. In each period, the manufacturer finances its operations using a combination of internal cash, short-term loans, and short-term capital subscriptions from shareholders. Furthermore, the manufacturer decides on an amount to send to an external depository to take into consideration the opportunity cost of carrying cash. Two formulations for this integrated problem are presented, structural properties of the optimal solution are derived, and a dynamic programming algorithm is developed to solve the problem. Subsequently, the integrated problem, analysis, and solution procedure are extended to consider the case of a per-period loan limit. A numerical study is conducted to derive insights on the effect of financial and operational parameters, which are then validated using extensive computational experiments. Our results show that integration leads to significant savings, smoother production, and smaller capital subscriptions and external deposits. In addition, while the discount factor affects the number of setups, inventory levels, capital subscriptions, and external deposits, the short-term interest rate determines the choice between using internal cash or short-term loans from the lender.

Dynamic Mixed Model Lotsizing and Scheduling for Flexible Machining Lines Using a Constructive Heuristic

Dynamic lotsizing and scheduling on multiple lines to meet the customer due dates is significant in multi-line production environments. Therefore, this study investigates dynamic lotsizing and scheduling problems in multiple flexible machining lines considering mixed products. In addition, uncertainty in demand and machine failure is considered. A mathematical model is proposed for the considered problem with an aim to maximize the probability of completion of product models from different customer orders. A constructive heuristic method (CHLP) is proposed to solve the current problem. The proposed heuristic involves the steps to distribute different customer order demands among multiple lines and schedule them considering balancing of makespan between the lines. The performance of CHLP is measured with famous heuristics from the literature, based on the test problem instances. Results indicate that CHLP gives better results in terms of quality of results as compared to other famous literature heuristics.

Integrated disassembly and reprocessing lot-sizing for multi-level structured products in remanufacturing systems

This study addresses a dynamic lot-sizing problem for remanufacturing systems in which end-of-use/life products are disassembled into their parts on a single disassembly workstation and then each part is reprocessed on one of the parallel reprocessing workstations. The problem, which integrates disassembly and reprocessing lot-sizing for multi-level disassembly structures, determines disassembly and reprocessing lot-sizes as well as the workstation on which each part is reprocessed, while satisfying workstation capacities and reprocessed part demands in each period of a planning horizon. An integer programming model is developed that minimizes the sum of set-up, operation and inventory costs. After the NP hardness is shown, two-stage heuristics are proposed in which an initial solution is obtained and then improved by full and partial bi-directional moves. Computational results show that the full move-based heuristic gives near-optimal solutions for small-sized instances and the partial move-based heuristic gives fast solutions with appropriate qualities.

Economic lot-sizing problem with remanufacturing option: complexity and algorithms

In a single item dynamic lot-sizing problem, we are given a time horizon and demand for a single item in every time period. The problem seeks a solution that determines how much to produce and carry at each time period, so that we will incur the least amount of production and inventory cost. When the remanufacturing option is included, the input comprises of number of returned products at each time period that can be potentially remanufactured to satisfy the demands, where remanufacturing and inventory costs are applicable. For this problem, we first show that it cannot have a fully polynomial time approximation scheme. We then provide a polynomial time algorithm, when we make certain realistic assumptions on the cost structure.

A dynamic lot size model with perishable inventory and stockout

This research work deals with the single-item dynamic lot size problem under (i) age-dependent inventory deterioration rates and inventory costs and (ii) stockouts (or lost sales, that is, the demands do not have to be satisfied). Two equivalent forward dynamic programming (DP) algorithms are devised to solve this problem in O(T3) time based on several properties of the optimal solution, where T denotes the length of the problem horizon. Moreover, a special case with no speculative motive cost structures is investigated. Compared to the general time-variant case, two equivalent algorithms for efficient forward DP in O(T2) time are developed. Furthermore, we study the problem under storage capacity constraint and develop a forward DP algorithm to solve this problem. Finally, we offer two useful managerial insights by using a comprehensive test bed of instance: (i) the stockout policy can reduce the total costs when stockout costs are relatively small, and (ii) prolonging the product's lifetime or extending the storage capacity at great financial costs under smaller setup costs and higher holding costs for the firm is not needed.

Column generation approach for solving uncapacitated dynamic lot-sizing problems with time-varying cost

Column generation approach for solving uncapacitated dynamic lot-sizing problems with time-varying cost

An effective heuristic with evolutionary algorithm for the coordinated capacitated dynamic lot-size and delivery problem

This study considers an extensive coordinated capacitated dynamic lot-size problem that contains a delivery scheduling problem. The coordinated capacitated dynamic lot-size and delivery problem (CCLSDP) assumes that a family of products is subject to capacity constraints, and the demand without backorders is dynamic or time-varying but deterministic. The objective is to determine a replenishment and delivery schedule that minimizes the total system costs, including the family and minor setup costs, delivery costs, and inventory holding costs. We develop a six-phase heuristic (SPH) to solve the CCLSDP. The SPH contains six phases that can efficiently reschedule replenishment and delivery planning. Each phase of the SPH evaluates the cost and feasibility of the shifting operations, and determines whether to reschedule the system planning through a series of sequential shifting operations. To further enhance the overall performance, we embed SPH as a lifting procedure into evolutionary algorithms contributing to explore the search space and escapes the local optimum. Experimental results of randomly generated instances that were over a wide range of parameters indicate that the proposed heuristics could efficiently and effectively solve the complicated CCLSDP.

Risk analysis for lot-sizing and maintenance optimization problem under energy constraint with subcontractor solution

PurposeThe main contribution of this manuscript is to suggest new approaches in order to deal with dynamic lot-sizing and maintenance problem under aspect energetic and risk analysis. The authors introduce a new maintenance strategy based on the centroid approach to determine a common preventive maintenance plan for all machines to minimize the total maintenance cost. Thereafter, the authors suggest a risk analysis study further to unforeseen disruption of availability machines with the aim of helping the production stakeholders to achieve the obtained forecasting lot-size plan.Design/methodology/approachThe authors tackle the dynamic lot-sizing problem using an efficient hybrid approach based on random exploration and branch and bound method to generate possible solutions. Indeed, the feasible solutions of random exploration method are used as input for branch and bound to determine the near-optimal solution of lot-size plan. In addition, our contribution to the maintenance part is to determine the optimal common maintenance plan for M machines based on a new algorithm called preventive maintenance (PM) periods means.FindingsFirst, the authors have funded the optimal lot-size plan that should satisfy the random demand under service level requirement and energy constraint while minimizing the costs of production and inventory. Indeed, establishing a best lot-size plan is to determine the appropriate number of available machines and manufactured units per period. Second, for risk analysis study, the solution of subcontracting is proposed by specifying a maximum cost of subcontractor in the context of a calling of tenders.Originality/valueFor maintenance problem, the originality consists in regrouping the maintenance plans of M machines into only one plan. This approach lets us to minimize the total maintenance cost and reduces the frequent breaks of production. As a second part, this paper contributed to the development of a new risk analysis study further to unforeseen disruption of availability machines. This risk analysis developed a decision-making system, for production stakeholders, in order to achieve the forecasting lot-size plan and keeps its profitability, by specifying the unit cost threshold of subcontractor in the context of a calling of tender.