A flexible MILP model for multiple-shift workforce planning under annualized hours

Flexibility in workforce planning is one of the best ways to respond to fluctuations of the demand. This paper proposes a flexible mixed integer linear programming (MILP) model to solve a multiple-shift workforce planning problem under annualized working hours. The model takes into account laws and collective agreements that impose constraints on overtime and holidays. We consider possible gradual hiring of full time and partial time workers. Several objectives are pursued such as balancing the workload of the employees or minimizing the workforce size. Computational experiments on a real life problem demonstrate the effectiveness of the model.     

MILP-based approaches for medium-term planning of single-stage continuous multiproduct plants with parallel units

In this paper, we address the problem of medium-term planning of single-stage continuous multiproduct plants with multiple processing units in parallel. Sequence-dependent changeover times and costs occur when switching from one type of product to another. A traveling salesman problem (TSP)-based mixed-integer linear programming (MILP) model is proposed based on a hybrid discrete/continuous time representation. We develop additional constraints and variables to ensure that subtours do not occur in the solution. The model is successfully applied to an example of a polymer processing plant to illustrate its applicability. In order to solve larger model instances and planning horizons, a rolling horizon approach is developed to reduce the computational expense. Finally, the proposed model is compared to a recently published approach through literature examples, and the results show that the computational performance of the proposed model is superior.     

A class of manpower scheduling problems

In this paper we formulate a mathematical programming model for solving a class of manpower scheduling problems.The objective is to assign working schedules to a fixed number of employees in order to meet the workforce demand, assumed to be constant over the planning period.Necessary and sufficient conditions of existence of the solution are stated, heuristic methods are presented and the results of computational experiences are reported.     

Production planning with uncertainty in the quality of raw materials: a case in sawmills

Motivated by sawmill production planning, this paper investigates multi-period, multi-product (MPMP) production planning in a manufacturing environment with non-homogeneous raw materials, and consequently random process yields. A two-stage stochastic program with recourse is proposed to address the problem. The random yields are modelled as scenarios with stationary probability distributions during the planning horizon. The solution methodology is based on the sample average approximation (SAA) scheme. The stochastic sawmill production planning model is validated through the Monte Carlo simulation. The computational results for a real medium capacity sawmill highlight the significance of using the stochastic model as a viable tool for production planning instead of the mean-value deterministic model, which is a traditional production planning tool in many sawmills.     

A Dynamic Supplier Selection and Inventory Management Model for a Serial Supply Chain with a Novel Supplier Price Break Scheme and Flexible Time Periods

A new supplier price break and discount scheme taking into account order frequency and lead time is introduced and incorporated into an integrated inventory planning model for a serial supply chain that minimizes the overall incurred cost including procurement, inventory holding, production, and transportation. A mixed-integer linear programming (MILP) formulation is presented addressing this multi-period, multi-supplier, and multi-stage problem with predetermined time-varying demand for the case of a single product. Then, the length of the time period is considered as a variable. A new MILP formulation is derived when each period of the model is split into multiple sub-periods, and under certain conditions, it is proved that the optimal solution and objective value of the original model form a feasible solution and an upper bound for the derived model. In a numerical example, three scenarios of the derived model are solved where the number of sub-period is set to 2, 3, and 4. The results further show the decrease of the optimal objective value as the length of the time period is shortened. Sufficient evidence demonstrates that the length of the time period has a significant influence on supplier selection, lot sizing allocation, and inventory planning decisions. This poses the necessity of the selection of appropriate length of a time period, considering the trade-off between model complexity and cost savings.     

Using MILP to plan annualised working hours

Annualising working hours (AH) is a means to achieve flexibility in the use of human resources in order to face the seasonal nature of the demand. MILP models are proposed to solve the problem of planning the staff's working hours with an annual horizon. The computational experience with the models leads to the conclusion that MILP is an appropriate way to deal with the problem in many real cases.     

Theoretical and algorithmic results for a class of hierarchical fleet mix problems

In this paper a workforce model is studied from both a theoretical and an algorithmic point of view. In the considered hierarchical model workforce units can be substituted by higher qualified ones; external workforce can also be hired to cover low qualified jobs. An exact recursive solution algorithm is proposed to solve the problems and its efficiency is improved by means of cut conditions and discrete convexity properties. Finally, the results of a computational test are provided.     

A robust optimization model for multi-site production planning problem in an uncertain environment

This paper addresses the multi-site production planning problem for a multinational lingerie company in Hong Kong subject to production import/export quotas imposed by regulatory requirements of different nations, the use of manufacturing factories/locations with regard to customers’ preferences, as well as production capacity, workforce level, storage space and resource conditions at the factories. In this paper, a robust optimization model is developed to solve multi-site production planning problem with uncertainty data, in which the total costs consisting of production cost, labor cost, inventory cost, and workforce changing cost are minimized. By adjusting penalty parameters, production management can determine an optimal medium-term production strategy including the production loading plan and workforce level while considering different economic growth scenarios. The robustness and effectiveness of the developed model are demonstrated by numerical results. The trade-off between solution robustness and model robustness is also analyzed.     

A mathematical model and algorithms for the aircraft hangar maintenance scheduling problem

An aircraft hangar maintenance scheduling problem is studied, motivated by the aircraft heavy maintenance conducted in a hangar operated by an independent maintenance service company. The aircraft hangar maintenance scheduling problem in such context consists of determining a maintenance schedule with minimum penalty costs in fulfilling maintenance requests, and a series of hangar parking plans aligned with the maintenance schedule through the planning period. A mixed-integer linear programming (MILP) mathematical model, integrating the interrelations between the maintenance schedule and aircraft parking layout plans, is presented at first. In the model, the variation of parking capacity of the maintenance hangar and the blocking of the aircraft rolling in and out path are considered. Secondly, the model is enhanced by narrowing down the domain of the time-related decision variables to the possible rolling in and out operations time of each maintenance request. Thirdly, to obtain good quality feasible solutions for large scale instances, a rolling horizon approach incorporating the enhanced mathematical model is presented. The results of computational experiments are reported, showing: (i) the effectiveness of the event-based discrete time MILP model and (ii) the scalability of the rolling horizon approach that is able to provide good feasible solutions for large size instances covering a long planning period.     

Design and planning of supply chains with integration of reverse logistics activities under demand uncertainty

In this paper, a mixed integer linear programming (MILP) formulation is developed for the design and planning of supply chains with reverse flows while considering simultaneously production, distribution and reverse logistics activities. It is also considered products’ demand uncertainty using a scenario tree approach. As main goal the model defines the maximization of the expected net present value and the results provide details on sizing and location of plants, warehouses and retailers, definition of processes to install, establishment of forward and reverse flows and inventory levels to attain. The model is applied to a representative European supply chain case study and its applicability is demonstrated.     

The Development of a Mathematical Model of the Operations of a National Glass Manufacturer

In this paper, a mathematical model of the entire operations of a national glass manufacturer is developed. This includes the float glass manufacture, distribution, storage operations and the technical considerations dictated by the plant as well as the operating procedures. The model is initially for a planning year, and is generated from a ‘monthly model’. This monthly model interconnects with other monthly models primarily via stock flows. The mathematical model is formulated in a unique way that allows certain production aspects to be modelled using a ‘pseudo-continuous’ time frame, rather than a discrete one. The generation of the overall model (as a mixed integer linear programming problem) and its solution is also discussed.     

On solving a large-scale problem on facility location and customer assignment with interaction costs along a time horizon

A 0-1 quadratic programming model is presented for solving the strategic problem of timing the location of facilities and the assignment of customers to facilities in a multi-period setting. It is assumed that all parameters are known and, on the other hand, the quadratic character of the objective function is due to considering the interaction cost incurred by the joint assignment of customers belonging to different categories to a facility at a period. The plain use of a state-of-the-art MILP engine with capabilities for dealing with quadratic terms does not give any advantage over the matheuristic algorithm proposed in this work. In fact, the MILP engine was frequently running out of memory before reaching optimality for the equivalent mixed 0-1 linear formulation, being its best lower bound at that time instant too far from the incumbent solution for the large-sized instances which we have worked with. As an alternative, a fix-and-relax algorithm is presented. A deep computational comparison between MILP alternatives is performed, such that fix-and-relax provides a solution value very close to (and, frequently, a better than) the one provided by the MILP engine. The time required by fix-and-relax is very affordable, being frequently two times smaller than the time required by the MILP engine.     

Planning annualised hours with a finite set of weekly working hours and cross-trained workers

Annualising working hours (i.e., the possibility of irregularly distributing the total number of working hours over a year) permits companies to adapt capacity to fluctuations in demand, thus reducing overtime, temporary workers and inventory costs. Since annual hours can lead to a worsening of the staff’s working conditions, many laws and collective bargaining agreements contain constraints that affect the distribution of working time. This paper proposes a MILP model to solve an annualised working hours planning problem in which workers are considered to be cross-trained, and in which the number of weekly working hours must belong to a previously agreed finite set. A computational experiment demonstrates the effectiveness of the model.     

An adaptive large neighborhood search algorithm for a selective and periodic inventory routing problem

We study a selective and periodic inventory routing problem (SPIRP) and develop an Adaptive Large Neighborhood Search (ALNS) algorithm for its solution. The problem concerns a biodiesel production facility collecting used vegetable oil from sources, such as restaurants, catering companies and hotels that produce waste vegetable oil in considerable amounts. The facility reuses the collected waste oil as raw material to produce biodiesel. It has to meet certain raw material requirements either from daily collection, or from its inventory, or by purchasing virgin oil. SPIRP involves decisions about which of the present source nodes to include in the collection program, and which periodic (weekly) routing schedule to repeat over an infinite planning horizon. The objective is to minimize the total collection, inventory and purchasing costs while meeting the raw material requirements and operational constraints. A single-commodity flow-based mixed integer linear programming (MILP) model was proposed for this problem in an earlier study. The model was solved with 25 source nodes on a 7-day cyclic planning horizon. In order to tackle larger instances, we develop an ALNS algorithm that is based on a rich neighborhood structure with 11 distinct moves tailored to this problem. We demonstrate the performance of the ALNS, and compare it with the MILP model on test instances containing up to 100 source nodes.     

Mixed Integer Linear Programming in Process Scheduling: Modeling,Algorithms, and Applications

This paper reviews the advances of mixed-integer linear programming (MILP) based approaches for the scheduling of chemical processing systems. We focus on the short-term scheduling of general network represented processes. First, the various mathematical models that have been proposed in the literature are classified mainly based on the time representation. Discrete-time and continuous-time models are presented along with their strengths and limitations. Several classes of approaches for improving the computational efficiency in the solution of MILP problems are discussed. Furthermore, a summary of computational experiences and applications is provided. The paper concludes with perspectives on future research directions for MILP based process scheduling technologies.     

Chance constrained programming models for refinery short-term crude oil scheduling problem

The main objective of this work is to put forward chance constrained mixed-integer nonlinear stochastic and fuzzy programming models for refinery short-term crude oil scheduling problem under demands uncertainty of distillation units. The scheduling problem studied has characteristics of discrete events and continuous events coexistence, multistage, multiproduct, nonlinear, uncertainty and large scale. At first, the two models are transformed into their equivalent stochastic and fuzzy mixed-integer linear programming (MILP) models by using the method of Quesada and Grossmann [I. Quesada, I E. Grossmann, Global optimization of bilinear process networks with multicomponent flows, Comput. Chem. Eng. 19 (12) (1995) 1219–1242], respectively. After that, the stochastic equivalent model is converted into its deterministic MILP model through probabilistic theory. The fuzzy equivalent model is transformed into its crisp MILP model relies on the fuzzy theory presented by Liu and Iwamura [B.D. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets Syst. 94 (2) (1998) 227–237] for the first time in this area. Finally, the two crisp MILP models are solved in LINGO 8.0 based on scheduling time discretization. A case study which has 267 continuous variables, 68 binary variables and 320 constraints is effectively solved with the solution approaches proposed.     

An integrated staff-sizing approach considering feasibility of scheduling decision

This paper presents an integrated staff-sizing system for analyzing and determining workforce management policies with consideration of staff flexibility in service organizations, which addresses and captures the integrated requirements between long-term manpower planning and short-term staff scheduling in the service sector. Multiple Objective Linear Programming (MOLP) is applied to optimize several diversified goals. Solution methods to the MOLP models for the staff planning and staff scheduling are developed respectively, then a solution approach is proposed to iteratively revise the unacceptable staff-sizing plan or scheduling plan. Finally, an example of nurse sizing is analyzed and computational studies are carried out to investigate managerial insights.     

A robust optimization model for multi-product two-stage capacitated production planning under uncertainty

Production planning (PP) is one of the most important issues carried out in manufacturing environments which seeks efficient planning, scheduling and coordination of all production activities that optimizes the company’s objectives. In this paper, we studied a two-stage real world capacitated production system with lead time and setup decisions in which some parameters such as production costs and customer demand are uncertain. A robust optimization model is developed to formulate the problem in which minimization of the total costs including the setup costs, production costs, labor costs, inventory costs, and workforce changing costs is considered as performance measure. The robust approach is used to reduce the effects of fluctuations of the uncertain parameters with regards to all the possible future scenarios. A mixed-integer programming (MIP) model is developed to formulate the related robust production planning problem. In fact the robust proposed model is presented to generate an initial robust schedule. The performance of this schedule could be improved against of any possible occurrences of uncertain parameters. A case from an Iran refrigerator factory is studied and the characteristics of factory and its products are discussed. The computational results display the robustness and effectiveness of the model and highlight the importance of using robust optimization approach in generating more robust production plans in the uncertain environments. The tradeoff between solution robustness and model robustness is also analyzed.     

A stochastic multi-period industrial hazardous waste location-routing problem: Integrating NSGA-II and Monte Carlo simulation

The present study extends a multi-objective mathematical model in the context of industrial hazardous waste management, which covers the integrated decisions of three levels with locating, vehicle routing, and inventory control. Analyzing these decisions simultaneously not only may lead to the most effective structure in the waste management network, but also may reduce the potential risk of managing the hazardous waste. Furthermore, because of the inherent complexity of the waste management system, uncertainty is inevitable and should be acknowledged to guarantee reliability in the decision-making process. From this perspective, the proposed model is novel in the following three aspects: (1) shifting from a deterministic to stochastic environment; (2) considering a multi-period planning horizon; and (3) incorporating the inventory decisions into the problem. The problem is formulated as a multi-objective stochastic Mixed-Integer Nonlinear Programming (MINLP) model, which can be easily converted into a MILP one. In terms of methodological contribution, a new simheuristic approach that is an integration of Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) and Monte Carlo simulation is developed to overcome the stochastic combinatorial optimization problem of this study. Our findings verify the efficiency of the proposed approach as it is able to find a high-quality solution within a relatively reasonable computational time.     

An integrated approach to worker assignment,workforce flexibility acquisition,and task rotation

In a manufacturing environment, workforce flexibility can be achieved by cross-training and improved via job rotation. In firms with a flexible workforce, employees perform different tasks and functions in response to fluctuations in both product demands and labour resources. This paper presents a mathematical programming model that assigns workers to tasks, rotates workers between the tasks, and determines the training schedule. The objective is to minimize the total costs including training cost, flexibility cost, and productivity loss cost. A constructive-search heuristic is also developed to solve the proposed model. The algorithm provides good solutions in two phases: construction and improvement. At the construction phase, a solution is built using some problem-specific information. The quality of the solution is then enhanced by changing worker assignments at a particular time point during a planning horizon. Our computational results for a number of randomly generated test problems confirms the efficiently of the proposed method.