Multi-item production routing problem with backordering: a MILP approach

The aim of this paper is to present mixed integer linear programming formulations for the production routing problem with backordering (PRP-B) and a new hybrid heuristic to solve the problem. The PRP-B is considered in the context of a supply chain consisting of a production facility with limited production and storage capacities and geographically dispersed points of sale with limited storage capacities. The PRP-B integrates multiple item lot sizing decisions and vehicle routing decisions to the points of sale, where backordering of end customer demands is allowed at a penalty. Two integrated mixed integer programming models are formulated and a solution procedure consisting of a relax-and-fix heuristic combined with a local search algorithm is proposed. The numerical results show that this hybrid heuristic outperforms a state-of-the-art MIP commercial solver, in terms of solution quality and CPU times.     

Optimal acquisition policy with quantity discounts and uncertain demands

We study the acquisition policy decision problem for a supply network involving one manufacturer and multiple suppliers. The manufacturer produces multiple products under uncertain demands and each supplier provides price discounts. The problem is to determine the manufacturer's acquisition policy and production levels so as to maximise the manufacturer's expected profit, subject to both the manufacturer's and suppliers’ capacities. We present a mixed integer nonlinear programming (MINLP) formulation of the problem, for both single- and multiple-sourcing procurement policies. General algebraic modeling system (GAMS) and its solvers, combining external integration functions, are employed to solve the complex MINLP problem. The preliminary computation results and managerial analysis are reported.     

Multi-objective mixed integer programming and an application in a pharmaceutical supply chain

Multi-objective integer linear and/or mixed integer linear programming (MOILP/MOMILP) are very useful for many areas of application as any model that incorporates discrete phenomena requires the consideration of integer variables. However, the research on the methods for the general multi-objective integer/mixed integer model has been scant when compared to multi-objective linear programming with continuous variables. In this paper, an MOMILP is proposed, which integrates various conflicting objectives. We give importance to the imprecise nature of some of the critical factors used in the modelling that can influence the effectiveness of the model. The uncertainty and the hesitation arising from estimating such imprecise parameters are represented by intuitionistic fuzzy numbers. The MOMILP model with intuitionistic fuzzy parameters is first converted into a crisp MOMILP model, using appropriate defuzzification strategies. Thereafter, the MOMILP is transformed into a single objective problem to yield a compromise solution with an acceptable degree of satisfaction, using suitable scalarisation techniques such as the gamma-connective technique and the minimum bounded sum operator technique. The proposed solution method is applied to several test problems and a multi-objective pharmaceutical supply chain management model with self generated random data.     

Modeling without categorical variables: a mixed-integer nonlinear program for the optimization of thermal insulation systems

Optimal design applications are often modeled by using categorical variables to express discrete design decisions, such as material types. A disadvantage of using categorical variables is the lack of continuous relaxations, which precludes the use of modern integer programming techniques. We show how to express categorical variables with standard integer modeling techniques, and we illustrate this approach on a load-bearing thermal insulation system. The system consists of a number of insulators of different materials and intercepts that minimize the heat flow from a hot surface to a cold surface. Our new model allows us to employ black-box modeling languages and solvers and illustrates the interplay between integer and nonlinear modeling techniques. We present numerical experience that illustrates the advantage of the standard integer model.     

Hybrid (re)manufacturing: manufacturing and operational implications

Remanufacturing presents an option to recover value from used products. However, hybrid (re)manufacturing (i.e. simultaneous manufacturing and remanufacturing) is a challenge, owing to non-uniform availability and heterogeneous quality of returns. In this paper, we examine how heterogeneous quality and non-uniform quantity of returns influence the optimal production rates and inventory levels in a hybrid (re)manufacturing system that incurs costs to readjust manufacturing and remanufacturing capacities on a temporary basis. Specifically, we propose a mixed integer linear programming (MILP) based approach to obtain the optimal production plan for a specified planning horizon. We apply the proposed model to the (representative) operational data of an office equipment manufacturer to evaluate the impacts of quality of returns, quality-based segregation of returns, and capacity readjustment costs. The study provides insights into the effects of trade-offs among different operational costs in a hybrid (re)manufacturing system.     

Two-period vs. multi-period model for supply chain disruption management

A novel two-period modelling approach is developed for supply chain disruption mitigation and recovery and compared with a multi-period approach. For the two-period model, planning horizon is divided into two aggregate periods: before disruption and after disruption. The corresponding mitigation and recovery decisions are: (1) primary supply and demand portfolios and production before a disruption, and (2) recovery supply, transshipment and demand portfolios and production after the disruption. In the multi-period model, a multi-period planning horizon is applied to account for a detailed timing of supplies and production. The primary and recovery portfolios are determined simultaneously and for both approaches the integrated decision-making, stochastic mixed integer programming models are developed. While the simplified two-period setting may overestimate (for best-case capacity constraints) or underestimate (for worst-case capacity constraints) the available production capacity, it can be easily applied in practice for a fast, rough-cut evaluation of disruption mitigation and recovery policy. The findings indicate that for both two- and multi-period setting, the developed multi-portfolio approach leads to computationally efficient mixed integer programming models with an embedded network flow structure resulting in a very strong linear programming relaxation.     

Modelling of multi-period multi-product production planning considering production routes

The multi-period multi-product (MPMP) production planning problems, generally, deal with matching production levels of individual products with fluctuated demands over planning horizon. The conventional MPMP optimisation models suffer from insufficient utilisation of available capacity of machines. This fallacy is due to inappropriate formulation of machine capacity and material handling constraints. In this study, a novel mathematical model is proposed to simultaneously optimise production quantities and provide information about managerial decisions such as subcontracting, carrying inventory/backordering, and also hiring/layoff personnel. The problem is then formulated as a mixed integer linear programming (MILP) model by applying appropriate linearisation of non-linear components. The objective is to minimise production costs comprising of production, storage, shortage, subcontracting costs and costs associated with hiring/dismissing labourers. Superiority of the proposed model over existing ones, has been initially evaluated by solving the case presented by Byrne and Bakir [Byrne, M.D. and Bakir, M.A., 1999. Production planning using a hybrid simulation-analytical approach. International Journal of Production Economics, 59 (1), 305–311], and then evaluated by comparing the results obtained from solving both the proposed and the conventional MPMP production planning models using a 100-randomly-generated-test-problem.     

Comparison of two non-convex mixed-integer nonlinear programming algorithms applied to autoregressive moving average model structure and parameter estimation

In this article, the stochastic modelling approach proposed by Box and Jenkins is treated as a mixed-integer nonlinear programming (MINLP) problem solved with a mesh adaptive direct search and a real-coded genetic class of algorithms. The aim is to estimate the real-valued parameters and non-negative integer, correlated structure of stationary autoregressive moving average (ARMA) processes. The maximum likelihood function of the stationary ARMA process is embedded in Akaike's information criterion and the Bayesian information criterion, whereas the estimation procedure is based on Kalman filter recursions. The constraints imposed on the objective function enforce stability and invertibility. The best ARMA model is regarded as the global minimum of the non-convex MINLP problem. The robustness and computational performance of the MINLP solvers are compared with brute-force enumeration. Numerical experiments are done for existing time series and one new data set.     

Study of a complex water resources system with screening and simulation models

The study describes a sequential iterative modelling process for a complex water resource system. Two types of analytical models are used to find a reasonably small set of possible systems optimal design alternatives for a complex river basin. These models are a linear programming deterministic continuous (lpdc) model and a linear programing deterministic discontinous (lpdd) model. Linear programing has been used with linear approximation of the nonlinear functions. A simulation program has been developed which continues screening on the basis of the information obtained from the linear programing model. The models are developed in the context of analysis of the Narmada river, a large river basin in India, for which in the first instance alternative combinations and capacities of six major dams have to be decided.     

A New Approach to Aggregate Production Planning

This paper proposes a new framework for modelling aggregate production planning problems in which emphasis is placed upon offering the user the flexibility to specify (1) the production options to be employed, (2) the relationships among those options (some of which may be highly situation-specific), and (3) the relevant cost structure. The procedure offered for solving the problem embeds Bowmann's “transportation” approach to aggregate production planning into a large mixed integer programming framework.     

基于变量分离的生产调度空间划分算法

静态生产调度大多形成MILP或MINLP模型，由于调度规模大及混合整数规划的组合优化特性，造成调度求解困难。通过对混合整数规划模型空间的分析，提出依据整数变量和连续变量的分离策略进行空间的自然划分，从而将模型的求解转化为多个较小规模连续子空间的寻优。对典型间歇调度模型的分析表明，将空间划分后进行连续寻优的策略较大地降低了实际运算的规模，降低了求解难度，可以提高问题的求解速度和效率。     

Aggregate and detailed production planning integrating procurement and distribution plans in a multi-site environment

A monolithic mixed integer linear goal programming (MILGP) model that is developed in this paper produces a time and capacity aggregated production plan, a detailed production plan, a detailed procurement plan and a detailed distribution plan simultaneously to overcome the drawbacks of the hierarchical/sequential planning approaches of not yielding a feasible and/or an optimal plan. The model uses different time-grids and planning horizons for aggregate and detailed planning to reduce the computational burden. The limitations of storage space, raw material availability and production capacity at plants and a requirement of maintaining a minimum level of inventory buffer or forward cover have been modelled. Two heuristics that are proposed to solve the MILGP model gave good quality solutions with an average and the worse case optimality gap of 1.17% and 4.4% respectively when applied to one hundred randomly generated industry size problem instances.     

Using the sequential linear integer programming method as a post‐processor for stress‐constrained topology optimization problems

This paper deals with topology optimization of load‐carrying structures defined on discretized continuum design domains. In particular, the minimum compliance problem with stress constraints is considered. The finite element method is used to discretize the design domain into n finite elements and the design of a certain structure is represented by an n‐dimensional binary design variable vector. In order to solve the problems, the binary constraints on the design variables are initially relaxed and the problems are solved with both the method of moving asymptotes and the sparse non‐linear optimizer solvers for continuous optimization in order to compare the two solvers. By solving a sequence of problems with a sequentially lower limit on the amount of grey allowed, designs that are close to ‘black‐and‐white’ are obtained. In order to get locally optimal solutions that are purely {0, 1}ⁿ, a sequential linear integer programming method is applied as a post‐processor. Numerical results are presented for some different test problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimization models for the dynamic facility location and allocation problem

The design of logistic distribution systems is one of the most critical and strategic issues in industrial facility management. The aim of this study is to develop and apply innovative mixed integer programming optimization models to design and manage dynamic (i.e. multi-period) multi-stage and multi-commodity location allocation problems (LAP). LAP belong to the NP-hard complexity class of decision problems, and the generic occurrence requires simultaneous determination of the number of logistic facilities (e.g. production plants, warehousing systems, distribution centres), their locations, and assignment of customer demand to them. The proposed models use a mixed integer linear programming solver to find solutions in complex industrial applications even when several entities are involved (production plants, distribution centres, customers, etc.). Lastly, the application of the proposed models to a significant case study is presented and discussed.     

Aggregate production planning in the automotive industry with special consideration of workforce flexibility

We present a new mixed integer linear programming approach for the problem of aggregate production planning of flowshop production lines in the automotive industry. Our model integrates production capacity planning and workforce flexibility planning. In contrast to traditional approaches, it considers discrete capacity adaptations which originate from technical characteristics of assembly lines as well as from work regulations and shift planning. In particular, our approach takes change costs into account and explicitly represents a working time account via a linear approximation. A solution framework containing different primal heuristics and preprocessing techniques is embedded into a decision support system. Finally, we present an illustrative case study and computational results on problem instances of practically relevant complexity.     

Methods for optimum and near optimum disassembly sequencing

This paper considers disassembly sequencing problems subjected to sequence dependent disassembly costs. In practice, the methods for dealing with such problems rely mainly on metaheuristic and heuristic methods, which intrinsically generate suboptimum solutions. Exact methods are NP-hard and therefore unsuitable to most of the practical problems. Nevertheless, it is useful to have exact methods available that can be applied in order to check, at least medium sized problems, to what extent the heuristically obtained solutions deviate from the optimum solution. The existing exact approaches, which are based on integer linear programming (ILP), become unmanageable, even for the cases of modest product complexity. To alleviate this problem to some extent, the iterative method that has been proposed by Lambert (2006) is applied here. This method is based on repeatedly solving a binary integer linear programming (BILP) problem instead of an ILP problem. The method appears to converge sufficiently quickly to be valuable for dealing with medium sized problems. We then use the iterative method for the validation of a new heuristic method that is also proposed in this paper. Finally, both the heuristic and the iterative BILP methods are implemented on a cellphone from practice consisting of 25 components that are represented, according to a set of precedence relationships, via a disassembly precedence graph.     

Optimal implementation and benefits of rolling inventory

We study a warehouse management problem in which the schedule of incoming supplies and customer orders for a wide variety of products is known over a number of periods. In addition to storage at the warehouse, products can be kept in the shipping trailers (rolling inventory) parked in the warehouse yard, avoiding material handling costs but incurring trailer handling and opportunity costs. Our objective is to determine the amount of product (if any) to leave in each of the incoming trailers, so that it does not have to be stored and then reloaded for an outgoing delivery, in order to minimize overall warehousing costs. We propose three possible implementation policies and show that the search for optimal solutions can be restricted to these three basic policies without loss of generality. Using this result, we formulate the problem as an integer program, in which incoming trailers are assigned to outgoing deliveries. Under one of the proposed policies incoming trailers can only be stored in the yard directly upon arrival, with their original contents. In this case, we show that our formulation possesses the integrality property and thus the optimal solution can be easily obtained. When the three policies are considered jointly, however, this is no longer the case. Nevertheless, computational tests show that the linear programming bound is very strong and commercial integer programming solvers generate an optimal solution very quickly. In most cases, no branch-and-bound nodes are required. Finally, we perform a computational study based on realistic data provided by our industry partner to evaluate the benefits of rolling inventory, the effectiveness of the different implementation policies and the viability of our proposed solution approaches.     

An optimization model for guiding the petrochemical industry development in Saudi Arabia

A mixed integer linear programming model is formulated for determining the optimum plan for the expansion of the Saudi Arabian petrochemical industry. The products selected for consideration fall into four categories: propylene derivatives, ethylene derivatives, synthesis gas derivatives, and aromatic derivatives. The model incorporates new variables and constraints, and realistic estimates of production costs, which are calculated based on local conditions in Saudi Arabia. For each production process, the unit production cost is assumed to be a function of production capacity. The input data for each product includes relevant production technologies, capacities, local production costs, and selling price. The solution of the model gives the recommended products under different scenarios of available capital investment and feedstock. The results are reported and analyzed.     

面向订单的生产计划与调度的综合优化

研究了一个多订单环境下的生产计划与调度集成优化问题,以实现准时生产为目标,综合考虑产品装配结构约束的订单任务计划与订单产品零部件的加工调度,采用直接面向客户订单的工序调度模式建立了计划和调度的综合优化整数规划模型.设计了带精英策略的蚁群算法作为该数学模型的求解方法,并通过对比试验为该算法选取最佳的搜索参数.实例仿真结果表明,所建模型的正确性以及蚁群算法求解该问题的可行性和有效性.     

The equipment maintenance scheduling problem in a coal production system

This paper addresses an equipment maintenance scheduling problem in a coal production system which includes three consecutive stages: the coal mining stage, the coal washing stage and the coal loading stage. Each stage is composed of different equipment that needs maintenance each day. There exists intermediate storage with finite capacities and the finished products are transported by train. Moreover, some equipment has a different preference for (aversion to) the start time of maintenance (STOM). The objective is to minimise the weighted sum of aversion about STOM, changeover times and train waiting time. We first formulate this problem into a mixed integer linear programming (MILP) model, then a hybrid genetic algorithm (HGA) is proposed to solve it. The proposed method has been tested on a practical coal enterprise in China and some randomly generated instances. Computational results indicate that our algorithm can produce near-optimal solutions efficiently.