Multi-item Lot Research Articles

A choice-based approach to dynamic capacitated multi-item lot sizing with demand uncertainty

With the purpose of planning and implementing pricing decisions on a tactical level as well as production decisions on an operational level, we consider – in an integrated form – the capacitated multi-item lot sizing problem with uncertain item demands and price-dependent discrete choice demand. The model is embedded into an overarching rolling horizon procedure allowing for adaptations to changes in demand and cost parameters. We first formulate the static problem version as a nonlinear mathematical program with underlying multinomial logit demand and subsequently linearize it to make it viable for mathematical programming solvers. Uncertainty of demands is taken into account by Monte Carlo simulation. More specifically, we generate random demand scenarios and utilize them as input data for the sample average approximation problem version. We further endow the problem setting with possibilities to incorporate pricing policy requirements such as restricting the number of price adaptations or defining periods without price adaptations. Overall, the developed approach yields a powerful tool for balancing item demands via pricing in a way favorable for adhering to available production capacities and thereby striking a balance between revenues and costs. Computational results confirm that adapting prices to time-dependent demand and cost parameters is exploited effectively to maintain a deliberately controlled production environment. Moreover, the integrated pricing and production setting allows to study the effect of pricing policy restrictions and demand uncertainties upon attainable profits.

A Mixed-integer Linear Programming Model for Solving Multi-item Uncapacitated Lot-sizing Problems with Storage Capacities Limitations: A Case Study of an Apparel Manufacturing Company

This study aimed to use a mixed-integer linear programming model to solve the multi-item uncapacitated lot size problem with storage capacity limitations for apparel manufacturing. In real-world operations, sometimes the storage capacity of cloth rolls can be insufficient at any given time. This would be because the sum of the remaining cloth rolls from the previous period and the receiving cloth rolls from the existing period would be a vast number. Furthermore, the extra cloth on the rolls would have to be carried and kept in the production line. Consequently, this caused the late issue of the distribution of the material, the loss of the cloth rolls, and the failure to count the cloth rolls. As a result, we conducted a multi-item cloth roll replenishment plan to meet the production line's demand and keep all the inventory and receiving cloth rolls. Using mixed-integer linear programming, the multi-item lot size of each period and the sum of the cloth, ordering, and holding costs was calculated, including the number of occupied cloth racks that did not exceed the number of available cloth racks in the cloth storage. Then, this result was compared with the calculation of a mixed-integer linear programming model based on the procurement policies of a case study company. This study used OpenSolver and GAMS/CPLEX software to calculate the solutions. The results found that in using the procurement policy, we could provide the maximum number of occupied cloth racks, about 368 racks, or 2.78 times the racks on the available shelves of the cloth storage. The result of the proposed model indicated that the number of occupied cloth racks to be installed did not exceed the number of available racks on the shelves. The total cost of the model of the procurement policy was less than the proposed model by about 0.26 percent because ordering occurred only about 25 times.

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.

Optimization Multi-Item Lot Sizing Model involve Transportation and Capacity Constraint under Stochastic Demand using Aquila Optimizer

Inventory issues are often a major concern as they significantly impact operating costs. Lot sizing is one of the key decisions in managing inventory. However, in a real context, the demand for each item is often uncertain. It is supplied from the same supplier, requiring product orders to be placed in the same period. In addition, limited vehicle capacity and transportation costs are important factors to consider in making multi-item lot sizing decisions. The purpose of this study is to propose a new inventory model multi-item lot sizing model involving transportation cost and capacity constraints under stochastic demand. The decision variables involved in the model are each item's ordering cycle and safety factor with the objective function of minimizing the total inventory cost. To optimize the inventory model, this study also offers the advanced procedure of the Aquila Algorithm. This study also presents sensitivity analysis to the appropriate policy for optimizing the multi-item lot-sizing inventory problem involving transportation cost and capacity constraint under stochastic demand.

Predictive Search for Capacitated Multi-Item Lot Sizing Problems

For capacitated multi-item lot sizing problems, we propose a predictive search method that integrates machine learning/advanced analytics, mathematical programming, and heuristic search into a single framework. Advanced analytics can predict the probability that an event will happen and has been applied to pressing industry issues, such as credit scoring, risk management, and default management. Although little research has applied such technique for lot sizing problems, we observe that advanced analytics can uncover optimal patterns of setup variables given properties associated with the problems, such as problem attributes, and solution values yielded by linear programming relaxation, column generation, and Lagrangian relaxation. We, therefore, build advanced analytics models that yield information about how likely a solution pattern is the same as the optimum, which is insightful information used to partition the solution space into incumbent, superincumbent, and nonincumbent regions where an analytics-driven heuristic search procedure is applied to build restricted subproblems. These subproblems are solved by a combined mathematical programming technique to improve solution quality iteratively. We prove that the predictive search method can converge to the global optimal solution point. The discussion is followed by computational tests, where comparisons with other methods indicate that our approach can obtain better results for the benchmark problems than other state-of-the-art methods. Summary of Contribution: In this study, we propose a predictive search method that integrates machine learning/advanced analytics, mathematical programming, and heuristic search into a single framework for capacitated multi-item lot sizing problems. The advanced analytics models are used to yield information about how likely a solution pattern is the same as the optimum, which is insightful information used to divide the solution space into incumbent, superincumbent, and nonincumbent regions where an analytics-driven heuristic search procedure is applied to build restricted subproblems. These subproblems are solved by a combined mathematical programming technique to improve solution quality iteratively. We prove that the predictive search method can converge to the global optimal solution point. Through computational tests based on benchmark problems, we observe that the proposed approach can obtain better results than other state-of-the-art methods.

Multi-Item Production Lot Sizing with Postponement, External Source for Common Parts, and Adjustable Rate for End Products

This study considers a multi-item production lot-size problem incorporating postponement, an external source for common parts, and an adjustable-rate for end products. Dealing with product variety, timely requirements, and limited in-house capacity has led production managers to seek manufacturing schemes and utilization-reduction strategies that can help them meet customer needs, smoothen fabrication schedules, and lower overall manufacturing expenses. We propose a two-stage manufacturing scheme. The first stage produces common parts for multiproduct incorporating a partial supply from an outside contractor to reduce utilization/uptime. Stage two fabricates all end products using an adjustable-rate to reduce the uptime further. We build a model to characterize the problem’s features and use optimization methods to derive the optimal rotation cycle time in order to help managers make cost-effective lot-size decisions and allow manufacturers to gain competitive advantages. A numerical illustration validates the model’s capability and applicability. This study makes two important contributions: (1) It offers a decision-support model for studying such a particular batch-size problem and deciding the optimal rotation cycle time, and (2) it identifies the individual/collective influence of dual uptime-reduction strategies on the operating policy and various performance indexes to help facilitate managerial decision-making.

Capacitated Multi-item Lot Sizing Problem with Client Prioritization in Tire Industry

Abstract This paper addresses a dedicated lot-sizing problem from off-the-road tire industry. This case study considers a multi-item capacitated lot sizing problem with assignment on unrelated parallel machines. Several specific constraints imposed by the industrial context are considered such as the limitation of number of mold setups, the number of campaign ending per weeks and the demand class prioritization. A novel mixed integer linear program is introduced to solve this problem taking into account several primary and secondary optimization criteria. In order to analyze the sensitivity of the objective function over the different criteria weights, Taguchi method is carried out based on real data issued from the industrial case.

The value of aggregate service levels in stochastic lot sizing problems

Dealing with demand uncertainty in multi-item lot sizing problems poses huge challenges due to the inherent complexity. The resulting stochastic formulations typically determine production plans which minimize the expected total operating cost while ensuring that a predefined service level constraint for each product is satisfied. We extend these stochastic formulations to a more general setting where, in addition to the individual service level constraints, an aggregate service level constraint is also imposed. Such a situation is relevant in practical applications where the service level aggregated from a variety of products must be collectively satisfied. These extended formulations allow the decision maker to flexibly assign different individual service levels to different products while ensuring that the overall aggregate service level is satisfied and these aggregate service level measures can be used in conjunction with the commonly adopted individual service levels. Different mathematical formulations are proposed for this problem with different types of service levels. These formulations are a piece-wise linear approximation for the β, γ, and δ service levels and a quantile-based formulation for the αc service level. We also present a receding horizon implementation of the proposed formulations which can be effectively used in a dynamic environment. Computational experiments are conducted to analyze the impact of aggregate service levels and demonstrate the value of the proposed formulations as opposed to standard service levels imposed on individual items.

Constant Approximation Algorithm for Nonuniform Capacitated Multi-Item Lot Sizing via Strong Covering Inequalities

We study the nonuniform capacitated multi-item lot-sizing problem. In this problem, there is a set of demands over a planning horizon of T discrete time periods, and all demands must be satisfied on time. We can place an order at the beginning of each period s, incurring an ordering cost Ks. In this order, we can order up to Cs units of products. On the other hand, carrying inventory from time to time incurs an inventory holding cost. The goal of the problem is to find a feasible solution that minimizes the sum of ordering and holding costs. Levi et al. [Levi R, Lodi A, Sviridenko M (2008) Approximation algorithms for the capacitated multi-item lot-sizing problem via flow-cover inequalities. Math. Oper. Res. 33(2):461–474.] gave a two-approximation for the problem when the capacities Cs are the same. Extending the result to the case of nonuniform capacities requires new techniques as pointed out in the discussion section of their paper. In this paper, we solve the problem by giving a 10-approximation algorithm for the capacitated multi-item lot-sizing problem with general capacities. The constant approximation is achieved by adding an exponential number of new covering inequalities to the natural facility location–type linear programming (LP) relaxation for the problem. Along the way of our algorithm, we reduce the lot-sizing problem to two generalizations of the classic knapsack-covering problem. We give LP-based constant approximation algorithms for both generalizations via the iterative rounding technique.

The green capacitated multi-item lot sizing problem with parallel machines

Carbon emissions related to energy consumptions from the manufacturing industry have become a substantial part of environmental burdens. To reduce carbon emissions, we introduce carbon emission constraints into the capacitated multi-item lot sizing problem with nonidentical parallel machines. The problem aims to satisfy customer demand for various items over the planning horizon, with an objective to minimize total costs without violating the capacity and carbon emission constraints. We formulate the problem with a mixed integer programming model and propose Lagrangian relaxation and column generation methods to improve lower bounds over the linear programming relaxation. Furthermore, we apply a heuristic named progressive selection to solve the problem and compare the heuristic with other state-of-the-art approaches in the literature. Computational results indicate that the progressive selection heuristic is computationally tractable and can obtain superior results under the same computational resources.

Analytics Branching and Selection for the Capacitated Multi-Item Lot Sizing Problem with Nonidentical Machines

We study a capacitated multi-item lot sizing problem with nonidentical machines. For the problem, we propose several mathematical formulations and their per-item and per-period Dantzig–Wolfe decompositions, followed by exploring their relative efficiency in obtaining lower and upper bounds. Additionally, we observe that the optimum has a correlation with the solution values of the pricing subproblems of Dantzig–Wolfe decompositions, along with the solution values of the uncapacitated problems and linear programming (LP) relaxation. Using these solution values, we build statistical estimation models (i.e., generalized linear models) that give insight on the optimal values, as well as information about how likely a setup variable is to take a value of 1 at an optimal point. We then develop an analytics branching and selection method where the information is utilized for an analytics-based branching and selection procedure to fix setup variables, which is, to our knowledge, the first research using likelihoo...

Analytics Branching and Selection for the Capacitated Multi-Item Lot Sizing Problem with Nonidentical Machines

We study a capacitated multi-item lot sizing problem with nonidentical machines. For the problem, we propose several mathematical formulations and their per-item and per-period Dantzig-Wolfe decomp...

A study of job shop standard setup time quota

For the multi-item and small lot size production mode and single machine job shop scheduling sequence independent setup time, many setup times are difficult to estimate accurately, which influences the ability to achieve accurate production cycles and costs of products. The survey shows that the length of the setup time depends on the level of employee’s knowledge. Therefore, a method for determining the standard setup time quota based on the level of employees’ knowledge is proposed. First, an evaluation index system for the level of employee’s knowledge is built; the level of employee’s knowledge is estimated by the masses and experts fuzzy comprehension evaluation and entropy method. Second, the range and definition of the level of employees’ knowledge index are developed. Third, the relational model of the employee’s knowledge level and the level of employees’ knowledge index is constructed through the least squares method. Finally, an example application is presented to verify the feasibility and effectiveness of the proposed model.

An informative column generation and decomposition method for a production planning and facility location problem

This paper develops an informative column generation and decomposition method for a capacitated multi-item lot sizing and facility location problem with backlogging. The method hybridizes column generation to achieve a relaxed linear solution and a lower bound and a decomposition method (i.e., relax-and-fix) to achieve a feasible solution. The two solutions are used for a neighborhood search procedure that fixes a subset of setup decision variables to iteratively reduce problem sizes. The relax-and-fix method is applied again to solve these smaller-size restricted problems with a purpose of progressively improving solution qualities. To show the effectiveness of the method, a number of computational tests are performed using newly-generated benchmark problems. Computational comparisons with a commercial solver (CPLEX) show that the proposed method provides competitive solution results.

Multi-item uncapacitated lot sizing problem with inventory bounds

We study a multi-item lot sizing problem with inventory bounds, where the production of the items is uncapacitated but a storage capacity is considered, limiting at each period the amount of products that can be held in stock. We prove that the problem is strongly NP-hard even with no holding cost and stationary setup costs. For non-speculative costs, this lot-sizing problem remains NP-hard even when restricted to only 2 periods. However, in this case, we show that it can be polynomially solved for any fixed number of items.

A new approach for solving capacitated lot sizing and scheduling problem with sequence and period-dependent setup costs

Purpose: We aim to examine the capacitated multi-item lot sizing problem which is a typical example of a large bucket model, where many different items can be produced on the same machine in one time period. We propose a new approach to determine the production sequence and lot sizes that minimize the sum of start up and setup costs, inventory and production costs over all periods. Design/methodology/approach: The approach is composed of three steps. First, we compute a lower bound on total cost. Then we propose a three sub-steps iteration procedure. We solve optimally the lot sizing problem without considering products sequencing and their cost. Then, we determine products quantities to produce each period while minimizing the storage and variable production costs. Given the products to manufacture each period, we determine its correspondent optimal products sequencing, by using a Branch and Bound algorithm. Given the sequences of products within each period, we evaluate the total start up and setup cost. We compare then the total cost obtained to the lower bound of the total cost. If this value riches a prefixed value, we stop. Otherwise, we modify the results of lot sizing problem. Findings and Originality/value: We show using an illustrative example, that the difference between the total cost and its lower bound is only 10%. This gap depends on the significance of the inventory and production costs and the machine’s capacity. Comparing the approach we develop with a traditional one, we show that we manage to reduce the total cost by 30%. Research limitations/implications: Our model fits better to real-world situations where production systems run continuously. This model is applied for limited number of part types and periods. Practical implications: Our approach determines the products to manufacture each time period, their economic amounts, and their scheduling within each period. This outcome should help decision makers bearing expensive start up and setup costs, to reduce their Inventory and Production costs and Start up and setup cots. Originality/value: The main idea of the proposed approach is to intelligibly reduce the number of products to manufacture within each period in order to decrease the setup cost; since in case of limited machine’s capacity, we show that setup costs could increase when reducing the total number of parts manufactured over the entire planning horizon. In fact, the triangular inequality in setup costs is proved to be not usually available.

Capacitated Arc Stabbing

In the Capacitated Arc Stabbing problem (CAS) we are given a set of arcs and a set of points on a circle. We say that a point p covers, or stabs, an arc A if p is contained in A. Each point has a weight and a capacity that determines the number of arcs it may cover. The goal is to find a minimum weight set of points that stabs all the arcs. CAS models a periodic multi-item lot sizing problem in which we are given a set of production requests each with its own periodic release time and deadline. Production takes place in batches of bounded capacity: each time unit t is associated with a capacity c(t) and weight w(t), where c(t) bounds the number of requests that can be manufactured at time t, and w(t) is a fixed cost for manufacturing any positive number of requests up to c(t) at time t. The goal is to find a minimum weight periodic schedule. We present a polynomial time algorithm for CAS that is based on a non-trivial reduction to Capacitated Interval Stabbing. Our approach applies to both hard and soft capacities. We also consider two variants of CAS in which some arcs may remain uncovered: in the partial variant there is a covering requirement g, and the goal is to find a minimum weight set of points that covers at least g arcs; and in the prize collecting variant each arc has a penalty that must be paid if this arc is not covered.

An extension of the schedule optimization problem at a public transit terminal to the multiple destinations case

We present a mathematical model to optimize the number of output lines leaving from a transit terminal (in which passengers are supposed to split among different lines, or even change mode of transportation) and their departure times in the aperiodic case. The model is an extension to the multiple destination case of the Schedule Optimization Problem described in (Bruno et al. in OR Spectr. 31(3):465–481, 2009). The proposed model is shown to be NP-hard due to its similarities to the Multi-Item Capacitated Lot Sizing Problem. We provide computational results in order to show that the model can be used to solve instances of significant size.

An HNP-MP Approach for the Capacitated Multi-Item Lot Sizing Problem With Setup Times

In this paper, we consider the capacitated multi-item lot sizing problem with setup times. The problem is to schedule J different items over a horizon of T periods with the objective to minimize the sum of setup cost and inventory holding cost. To achieve feasible high-quality solutions, we propose a new solution approach which hybrids Nested Partitions and Mathematical Programming (HNP-MP). Nested Partitions is a partitioning and sampling based heuristic method with a global perspective on the problem. In the proposed new method the Mathematical Programming method is implemented to calculate the promising index and to provide a good guidance on partitioning in the Nested Partitions framework. A time-oriented decomposition heuristic method, Relax-and-Fix, is also implemented to obtain good promising regions and speed up the computational process. Computational results based on benchmark test problems show that the approach is computationally tractable and is able to obtain good results. The approach outperforms other state-of-the-art approaches found in the literature.

Comparing Dantzig–Wolfe decompositions and branch-and-price algorithms for the multi-item capacitated lot sizing problem

In this article, we consider the multi-item capacitated lot sizing problem with setup times. Starting from an original mixed integer programming model, we apply the standard Dantzig–Wolfe decomposition (DWD) in two different ways: defining the subproblems by items and defining the subproblems by periods. A third decomposition is developed in which the subproblems of both types are integrated in the same model. The linear relaxation of this last approach, which we denote as multiple DWD, provides lower bounds (equal to or) better than the bounds obtained by the other decompositions, which in turn, provide lower bounds (equal to or) better than the ones given by the original model. For solving the three decomposition models, we implemented three branch-and-price algorithms. We describe their main aspects and report on their computational results in instances from the literature.