Solution approaches for the soft drink integrated production lot sizing and scheduling problem

In this paper we present a mixed integer programming model that integrates production lot sizing and scheduling decisions of beverage plants with sequence-dependent setup costs and times. The model considers that the industrial process produces soft drink bottles in different flavours and sizes, and it is carried out in two production stages: liquid preparation (stage I) and bottling (stage II). The model also takes into account that the production bottleneck may alternate between stages I and II, and a synchronisation of the production between these stages is required. A relaxation approach and several strategies of the relax-and-fix heuristic are proposed to solve the model. Computational tests with instances generated based on real data from a Brazilian soft drink plant are also presented. The results show that the solution approaches are capable of producing better solutions than those used by the company.     

Optimization model for a production,inventory, distribution and routing problem in small furniture companies

Production and distribution are two key decisions in supply chain planning. In order to achieve an effective operational performance, it is important for these two decisions to be integrated, especially in supply chains with low inventory levels. In this paper, we propose a mixed integer programming model to integrate production, inventory, distribution and routing decisions in a single framework. The model was inspired by small Brazilian furniture companies and focuses on production and distribution decisions at an operational level. In particular, we consider a scenario in which only one production line and one vehicle, which makes multiple trips over the planning horizon, are available to produce items and deliver final products, respectively. We also take into account some features rarely considered in the literature, but commonly found in real-world applications, such as producing and stocking multiple items, distribution routes extending over one or more periods, multiple time windows and customers’ due dates. Computational tests on a set of randomly generated instances were carried out using a well-known optimization software and six relax-and-fix heuristics, which explore different criteria for partitioning and fixing variables. We also implemented two hybrid heuristics in which an initial solution is first constructed and then fed into the optimization software to improve it. The results showed that one relax-and-fix and the two hybrid heuristics performed better than the solver on the largest instances.     

Joint rolling-horizon scheduling of materials processing and lot-sizing with sequence-dependent setups

A lot sizing and scheduling problem from a foundry is considered in which key materials are produced and then transformed into many products on a single machine. A mixed integer programming (MIP) model is developed, taking into account sequence-dependent setup costs and times, and then adapted for rolling horizon use. A relax-and-fix (RF) solution heuristic is proposed and computationally tested against a high-performance MIP solver. Three variants of local search are also developed to improve the RF method and tested. Finally the solutions are compared with those currently practiced at the foundry.     

Dynamic economic lot size model with perishable inventory and capacity constraints

In this study, we consider a dynamic economic lot sizing problem for a single perishable item under production capacities. We aim to identify the production, inventory and backlogging decisions over the planning horizon, where (i) the parameters of the problem are deterministic but changing over time, and (ii) producer has a constant production capacity that limits the production amount at each period and is allowed to backorder the unmet demand later on. All cost functions are assumed to be concave. A similar problem without production capacities was studied in the literature and a polynomial time algorithm was suggested (Hsu, 2003 [1]). We assume age-dependent holding cost functions and the deterioration rates, which are more realistic for perishable items. Backordering cost functions are period-pair dependent. We prove the NP-hardness of the problem even with zero inventory holding and backlogging costs under our assumptions. We show the structural properties of the optimal solution and suggest a heuristic that finds a good production and distribution plan when the production periods are given. We discuss the performance of the heuristic. We also give a Dynamic Programing-based heuristic for the solution of the overall problem.     

A multi-class multi-level capacitated lot sizing model

When demand loading is higher than available capacity, it takes a great deal of effort for a traditional MRP system to obtain a capacity-feasible production plan. Also, the separation of lot sizing decisions and capacity requirement planning makes the setup decisions more difficult. In a practical application, a production planning system should prioritize demands when allocating manufacturing resources. This study proposes a planning model that integrates all MRP computation modules. The model not only includes multi-level capacitated lot sizing problems but also considers multiple demand classes. Each demand class corresponds to a mixed integer programming (MIP) problem. By sequentially solving the MIP problems according to their demand class priorities, this proposed approach allocates finite manufacturing resources and generates feasible production plans. In this paper we experiment with three heuristic search algorithms: (1) tabu search; (2) simulated annealing, and (3) genetic algorithm, to solve the MIP problems. Experimental designs and statistical methods are used to evaluate and analyse the performance of these three algorithms. The results show that tabu search and simulated annealing perform best in the confirmed order demand class and forecast demand class, respectively.     

Furniture supply chain tactical planning optimization using a time decomposition approach

We study the supply chain tactical planning problem of an integrated furniture company located in the Province of Québec, Canada. The paper presents a mathematical model for tactical planning of a subset of the supply chain. The decisions concern procurement, inventory, outsourcing and demand allocation policies. The goal is to define manufacturing and logistics policies that will allow the furniture company to have a competitive level of service at minimum cost. We consider planning horizon of 1 year and the time periods are based on weeks. We assume that customer’s demand is known and dynamic over the planning horizon. Supply chain planning is formulated as a large mixed integer programming model. We developed a heuristic using a time decomposition approach in order to obtain good solutions within reasonable time limit for large size problems. Computational results of the heuristic are reported. We also present the quantitative and qualitative results of the application of the mathematical model to a real industrial case.     

On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (?, S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures.     

Urban rapid transit network capacity expansion

This paper examines a multi-period capacity expansion problem for rapid transit network design. The capacity expansion is realized through the location of train alignments and stations in an urban traffic context by selecting the time periods. The model maximizes the public transportation demand using a limited budget and designing lines for each period. The location problem incorporates the user decisions about mode and route. The network capacity expansion is a long-term planning problem because the network is built over several periods, in which the data (demand, resource price, etc.) are changing like the real problem changes. This complex problem cannot be solved by branch and bound, and for this reason, a heuristic approach has been defined in order to solve it. Both methods have been experimented in test networks.     

Scheduling batches in flowshop with limited buffers in the shampoo industry

In this paper we address the problem of planning a temporary storage area in a real production system. This temporary storage area is composed of parallel temporary storage units with distinct capacities. The storage operation of a job, also called a batch, has to answer time restrictions such as release dates, due dates, restricted family dependent setup times and time lags, and also a space constraint which is the capacity of the temporary storage unit. The goal is to schedule the batches on the storage units in order to minimize the total setup times and the maximum lateness. First, we model the problem on a single storage unit as a two-machine flowshop problem with a limited buffer capacity and we show that it is NP-hard. We also show that the particular case in which no lateness is allowed is solvable in polynomial time under special conditions on the buffer capacity, both for single or parallel temporary storage units. Next we provide three heuristics: a greedy algorithm, a hybrid heuristic based on Ant Colony Optimization and Simulated Annealing and finally a dedicated heuristic. The latter strongly exploits the structural properties shown in this paper. We provide experimental results which highlight the efficiency of the dedicated heuristic in comparison with the two other heuristics.     

Dynamic lot-sizing with price changes and price-dependent holding costs

This paper proposes a new formulation of the dynamic lot-sizing problem with price changes which considers the unit inventory holding costs in a period as a function of the procurement decisions made in previous periods. In Section 1, the problem is defined and some of its fundamental properties are identified. A dynamic programming approach is developed to solve it when solutions are restricted to sequential extreme flows, and results from location theory are used to derive an O(T²) algorithm which provides a provably optimal solution of an integer linear programming formulation of the general problem. In Section 2, a heuristic is developed for the case where the inventory carrying rates and the order costs are constant, and where the item price can change once during the planning horizon. Permanent price increases, permanent price decreases and temporary price reductions are considered. In Section 3, extensive testing of the various optimal and heuristic algorithms is reported. Our results show that, in this context, the two following intuitive actions usually lead to near optimal solutions: accumulate stock at the lower price just prior to price increase and cut short on orders when a price decrease is imminent.     

A heuristic approach for capacity expansion of packet networks

We address the problem of expanding transmission capacity of an existing packet network over a multiperiod planning horizon, the objective being low total cost of expansion. Discrete capacity choices, interaction with routing decisions, and economy of scale in the cost of capacity make it extremely difficult to decide when, where and how much capacity to add. A fast heuristic solution method is developed based on the well established Flow Deviation routing algorithm. The heuristic begins by making myopic expansion decisions, which are then subsequently adjusted to account for economies of scale in the cost of capacity. Heuristic solutions are compared to a benchmark which approximates the real cost function by its linear lower envelope. Since the number of possible expansion plans is an exponential function of the number of edges, capacity choices, and periods in the planning horizon, a fast heuristic allows one to look beyond small problems at more realistically sized ones.     

Integrating long-term care insurance purchase decisions with saving and investment for retirement

Risk related to long-term care (LTC) is high for the elderly. Planning for LTC is now regarded as the ‘third leg’ of retirement planning. In this paper, planning for LTC is integrated with saving and investment decisions for an integrated approach to retirement planning. Optimal LTC insurance purchase decisions are obtained by developing a trade-off between post-retirement LTC costs and LTC insurance premiums paid and coverage received. Integrating insurance purchase with wealth evolution, consisting of saving and investment decisions, allows addressing affordability issues.Two-way branching models are used for the stochastic health events and asset returns. The problem, formulated as a nonlinearly constrained mixed-integer optimization problem, is solved using a heuristic. Sensitivity analyses are performed for initial health and wealth status. Some important aspects of an individual’s behavioral preferences are also addressed in this framework to provide more robust decision support.     

Heuristics for the flow line problem with setup costs

This paper presents two new heuristics for the flowshop scheduling problem with sequence-dependent setup times (SDSTs) and makespan minimization objective. The first is an extension of a procedure that has been very successful for the general flowshop scheduling problem. The other is a greedy randomized adaptive search procedure (GRASP) which is a technique that has achieved good results on a variety of combinatorial optimization problems. Both heuristics are compared to a previously proposed algorithm based on the traveling salesman problem (TSP). In addition, local search procedures are developed and adapted to each of the heuristics. A two-phase lower bounding scheme is presented as well. The first phase finds a lower bound based on the assignment relaxation for the asymmetric TSP. In phase two, attempts are made to improve the bound by inserting idle time. All procedures are compared for two different classes of randomly generated instances. In the first case where setup times are an order of magnitude smaller than the processing times, the new approaches prove superior to the TSP-based heuristic; for the case where both processing and setup times are identically distributed, the TSP-based heuristic outperforms the proposed procedures.     

A search heuristic for the sequence-dependent economic lot scheduling problem

Almost all of the research on the economic lot scheduling problem (ELSP) has assumed that setup times are sequence-independent even though sequence-dependent problems are common in practice. Furthermore, most of the solution approaches that have been developed solve for a single optimal schedule when in practice it is more important to provide managers with a range of schedules of different length and complexity. In this paper, we develop a heuristic procedure to solve the ELSP problem with sequence-dependent setups. The heuristic provides a range of solutions from which a manager can choose, which should prove useful in an actual stochastic production environment. We show that our heuristic can outperform Dobson's heuristic when the utilization is high and the sequence-dependent setup times and costs are significant.     

Integrating long-term care insurance purchase decisions with saving and investment for retirement

Risk related to long-term care (LTC) is high for the elderly. Planning for LTC is now regarded as the ‘third leg’ of retirement planning. In this paper, planning for LTC is integrated with saving and investment decisions for an integrated approach to retirement planning. Optimal LTC insurance purchase decisions are obtained by developing a trade-off between post-retirement LTC costs and LTC insurance premiums paid and coverage received. Integrating insurance purchase with wealth evolution, consisting of saving and investment decisions, allows addressing affordability issues.Two-way branching models are used for the stochastic health events and asset returns. The problem, formulated as a nonlinearly constrained mixed-integer optimization problem, is solved using a heuristic. Sensitivity analyses are performed for initial health and wealth status. Some important aspects of an individual’s behavioral preferences are also addressed in this framework to provide more robust decision support.     

Simple dynamic location problem with uncertainty: a primal-dual heuristic approach

Abstract

In this paper, the simple dynamic facility location problem is extended to uncertain realizations of the potential locations for facilities and the existence of customers as well as fixed and variable costs. With limited knowledge about the future, a finite and discrete set of scenarios is considered. The decisions to be made are where and when to locate the facilities, and how to assign the existing customers over the whole planning horizon and under each scenario, in order to minimize the expected total costs. Whilst assignment decisions can be scenario dependent, location decisions have to take into account all possible scenarios and cannot be changed according to each scenario in particular. We first propose a mixed linear programming formulation for this problem and then we present a primal-dual heuristic approach to solve it. The heuristic was tested over a set of randomly generated test problems. The computational results are provided.     

A two-phase heuristic approach to the bike repositioning problem

An approach to overcome the bike imbalance problem is to transfer excess bikes to branches with bike shortages. This study develops a constrained mathematical model to deal with a multi-vehicle bike-repositioning problem, and aims to minimize the sum of transportation and unmet demand costs over a planning horizon through bike-transfer strategies under a minimum service requirement. A two-phase heuristic based on linear programming was proposed to solve the problem and produce compromising solutions. In the first phase, the paper developed a linear programming model to quickly develop decisions related to bike inventory, unloading, and loading for all stations for each time slot. In the second phase, this paper proposed an iterative approach through two parameter sensitive mathematical models to sequentially reduce the problem scale to develop decisions related to bike transfers. Computational results show that the proposed approach is superior to a CPLEX optimizer and a hybrid heuristic based on a genetic algorithm. The proposed approach was used to analyze the bicycle system in Taiwan. The impacts of various system parameters on the system were also investigated.     

A column generation approach for the route planning problem in fourth party logistics

In this paper, we address the route planning problem in fourth party logistics (4PL). The problem calls for the selection of the logistics companies by a 4PL provider to optimize the routes of delivering goods through a transportation network. The concept of 4PL emerged in response to the shortfall in services capabilities of traditional third party logistics and has been proven to be capable of integrating logistics resources in order to fulfill complex transportation demands. A mixed-integer programming model is established for the planning problem with setup cost and edge cost discount policies which are commonly seen in practice. We propose a column generation approach combined with graph search heuristic to efficiently solve the problem. The good performance in terms of the solution quality and computational efficiency of our approach is shown through extensive numerical experiments on various scales of test instances. Impacts of cost policies on routing decision are also investigated and managerial insights are drawn.     

Multiple machine continuous setup lotsizing with sequence-dependent setups

We address the short-term production planning and scheduling problem coming from the glass container industry. A furnace melts the glass that is distributed to a set of parallel molding machines. Both furnace and machine idleness are not allowed. The resulting multi-machine multi-item continuous setup lotsizing problem with a common resource has sequence-dependent setup times and costs. Production losses are penalized in the objective function since we deal with a capital intensive industry. We present two mixed integer programming formulations for this problem, which are reduced to a network flow type problem. The two formulations are improved by adding valid inequalities that lead to good lower bounds. We rely on a Lagrangian decomposition based heuristic for generating good feasible solutions. We report computational experiments for randomly generated instances and for real-life data on the aforementioned problem, as well as on a discrete lotsizing and scheduling version.     

Models for capacitated lot-sizing problem with backlogging,setup carryover and crossover

Setup operations are significant in some production environments. It is mandatory that their production plans consider some features, as setup state conservation across periods through setup carryover and crossover. The modelling of setup crossover allows more flexible decisions and is essential for problems with long setup times. This paper proposes two models for the capacitated lot-sizing problem with backlogging and setup carryover and crossover. The first is in line with other models from the literature, whereas the second considers a disaggregated setup variable, which tracks the starting and completion times of the setup operation. This innovative approach permits a more compact formulation. Computational results show that the proposed models have outperformed other state-of-the-art formulation.