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.     

Planning production and working time within an annualised hours scheme framework

Production flexibility is essential for industrial companies that have to deal with seasonal demand. Human resources are one of the main sources of flexibility. Annualising working hours (i.e., the possibility of irregularly distributing the total number of working hours over the course of a year) is a tool that provides organisations with flexibility; it enables a firm to adapt production capacity to fluctuations in demand. However, it can involve a worsening of the staff working conditions. To take this into account, the planning and scheduling of working time should comply with constraints derived from the law or from a collective bargaining agreement. Thus, new and more difficult working-time and production planning and scheduling problems are arising. This paper proposes two mixed-integer linear program models for solving the problem of planning the production, the working hours and the holiday weeks of the members of a human team operating in a multi-product process in which products are perishable, demand can be deferred and temporary workers are hired to stand in for employees. The results of a computational experiment are presented. Supported by the Spanish MCyT projects DPI2001-2176 and DPI2004-05797, co-financed by FEDER.     

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.     

Using a MILP model to establish a framework for an annualised hours agreement

Production flexibility is essential for industrial companies that have to deal with seasonal demand. Human resources are one of the main sources of flexibility. Annualising working hours (i.e., the possibility of irregularly distributing the total number of working hours over the course of a year) is a tool that provides flexibility to organizations; it enables a firm to adapt production capacity to fluctuations in demand. However, it can imply a worsening of the staff’s working conditions. To take the human aspect into account, the planning and scheduling of working time should comply with constraints derived from the law or from a collective bargaining agreement. Furthermore, new and more difficult working-time planning and scheduling problems are arising. This paper proposes a mixed-integer linear program model to solve the problem of planning the production and the working hours of a human team that operates in a multi-product process. Solving the model for different settings provides the essential quantitative information to negotiate the best conditions of the annualised hours system (the elements to establish the trade-off between weekly flexibility and economic or working-time reduction compensation can be obtained). The results achieved in a computational experiment were very satisfactory.     

Single-site strategic capacity planning considering renewal,maintenance, inventory,taxes and cash flow management

This paper deals with strategic capacity planning of a single-site manufacturing system. We propose a MILP model that includes relevant business aspects and possibilities, some of which are only partially or not at all found in the literature. Specifically, we consider decisions on expansion, reduction and renewal of production capacity, and acquisition of storage capacity. In addition, we model aspects such as (a) maintenance costs and unit variable costs depending, respectively, on age and characteristics of facilities, (b) seasonality of the demand and (c) cash flow management, including taxes and, therefore, depreciation of assets. The model maximises the after-tax cash balance at the end of the planning horizon. We also present a computational experiment with 54 instances to show that the model can be solved for a wide range of sizes in a reasonable computing time using comercial software.     

Heuristics for workforce planning with worker differences

This study considers decisions in workforce management assuming individual workers are inherently different as measured by general cognitive ability (GCA). A mixed integer programming (MIP) model that determines different staffing decisions (i.e., hire, cross-train, and fire) in order to minimize workforce related costs over multiple periods is described. Solving the MIP for a large problem instance size is computationally burdensome. In this paper, two linear programming (LP) based heuristics and a solution space partition approach are presented to reduce the computational time. A genetic algorithm was also implemented as an alternative method to obtain better solutions and for comparison to the heuristics proposed. The heuristics were applied to realistic manufacturing systems with a large number of machine groups. Experimental results shows that performance of the LP based heuristics performance are surprisingly good and indicate that the heuristics can solve large problem instances effectively with reasonable computational effort.     

Column generation based heuristic for tactical planning in multi-period vehicle routing

The periodic vehicle routing problem (PVRP) consists in establishing a planning of visits to clients over a given time horizon so as to satisfy some service level while optimizing the routes used in each time period. The tactical planning model considered here restricts its attention to scheduling visits and assigning them to vehicles while leaving sequencing decisions for an underlying operational model. The objective is twofold: to optimize regional compactness of the routes in a desire to specialize routes to restricted geographical area and to balance the workload evenly between vehicles. Approximate solutions are constructed using a truncated column generation procedure followed by a rounding heuristic. This mathematical programming based procedure can deal with problems with 50–80 customers over five working days which is the range of size of most PVRP instances treated in the literature with meta-heuristics. The paper highlights the importance of alternative optimization criteria not accounted for in standard operational models and provides insights on the implementation of a column generation based rounding heuristic.     

Improved program planning with formal models? The case of high risk crop farming in Northeast Germany

In this paper we explore whether the incorporation of systematic time series analyses and mathematical optimization procedures in the practical planning process has the potential to improve production program decisions. The cases of four German cash crop farms are investigated over six planning periods. In order to avoid solutions that simply exceed the farmer’s risk tolerance, the apparently accepted variance of the observed program’s total gross margin is used as an upper bound in the optimization. For each of the 24 planning occasions, the formal model is used to generate optimized alternative programs. The total gross margins that could have been realized if the formally optimized programs had been implemented are then compared to those that were actually realized. We find that the farmers could have increased their total gross margins significantly if—instead of using simple routines and rules of thumb—they had used adequate methods of statistical analysis combined with the formal optimization model. Norbert Hirschauer thanks the German Research Foundation (DFG) for the opportunity to work on this paper during a research leave.     

Development of a new approach for deterministic supply chain network design

This paper proposes a mixed integer linear programming model and solution algorithm for solving supply chain network design problems in deterministic, multi-commodity, single-period contexts. The strategic level of supply chain planning and tactical level planning of supply chain are aggregated to propose an integrated model. The model integrates location and capacity choices for suppliers, plants and warehouses selection, product range assignment and production flows. The open-or-close decisions for the facilities are binary decision variables and the production and transportation flow decisions are continuous decision variables. Consequently, this problem is a binary mixed integer linear programming problem. In this paper, a modified version of Benders’ decomposition is proposed to solve the model. The most difficulty associated with the Benders’ decomposition is the solution of master problem, as in many real-life problems the model will be NP-hard and very time consuming. In the proposed procedure, the master problem will be developed using the surrogate constraints. We show that the main constraints of the master problem can be replaced by the strongest surrogate constraint. The generated problem with the strongest surrogate constraint is a valid relaxation of the main problem. Furthermore, a near-optimal initial solution is generated for a reduction in the number of iterations.     

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.     

Formation of a strategic manufacturing and distribution network with transfer prices

We propose a profit maximization model for the decision support system of a firm that wishes to establish or rationalize a multinational manufacturing and distribution network to produce and deliver finished goods from sources to consumers. The model simultaneously evaluates all traditional location factors in a manufacturing and distribution network design problem and sets intra-firm transfer prices that take account of tax and exchange rate differentials between countries. Utilizing the generalized Benders decomposition approach, we exploit the partition between the product flow and the cash allocation (i.e., the pricing and revenue assignment) decisions in the supply chain to find near optimal model solutions. Our proposed profit maximizing strategic planning model produces intuitive results. We offer computational experiments to illustrate the potential valuable guidance the model can provide to a firm's supply chain design strategic planning process.     

Multi-horizon stochastic programming

Infrastructure-planning models are challenging because of their combination of different time scales: while planning and building the infrastructure involves strategic decisions with time horizons of many years, one needs an operational time scale to get a proper picture of the infrastructure’s performance and profitability. In addition, both the strategic and operational levels are typically subject to significant uncertainty, which has to be taken into account. This combination of uncertainties on two different time scales creates problems for the traditional multistage stochastic-programming formulation of the problem due to the exponential growth in model size. In this paper, we present an alternative formulation of the problem that combines the two time scales, using what we call a multi-horizon approach, and illustrate it on a stylized optimization model. We show that the new approach drastically reduces the model size compared to the traditional formulation and present two real-life applications from energy planning.     

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.     

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.     

Optimizing the supply chain configuration for make-to-order manufacturing

We consider a make-to-order (MTO) manufacturer who has won multiple contracts with specified quantities to be delivered by certain due dates. Before production starts, the company must configure its supply chain and make sourcing decisions. It also needs to plan the starting time for each production task under limited availability of resources such as machines and workforce. We develop a model for simultaneously optimizing such sourcing and planning decisions while exploiting their tradeoffs. The resulting multi-mode resource-constrained project scheduling problem (MMRCPSP) with a nonlinear objective function is NP-complete. To efficiently solve it, a hybrid Benders decomposition (HBD) algorithm combining the strengths of both mathematical programming and constraint programming is developed. The HBD exploits the structure of the model formulation and decomposes it into a relaxed master problem handled by mixed-integer nonlinear programming (MINLP), and a scheduling feasibility sub-problem handled by constraint programming (CP). Cuts are iteratively generated by solving the feasibility sub-problem and added back to the relaxed master problem, until an optimal solution is found or infeasibility is proved. Computational experiments are conducted to examine performance of the model and algorithm. Insights about optimal configuration of MTO supply chains are drawn and discussed.     

Speciality oils supply chain optimization: From a decoupled to an integrated planning approach

We study a problem of tactical planning in a divergent supply chain. It involves decisions regarding production, inventory, internal transportation, sales and distribution to customers. The problem is motivated by the context of a company in the speciality oils industry. The overall objective at tactical level is to maximize contribution and, in order to achieve this, the planning has been divided into two separate problems. The first problem concerns sales where the final sales and distribution planning is decentralized to individual sellers. The second problem concerns production, transportation and inventory planning through refineries, hubs and depots and is managed centrally with the aim of minimizing costs. Due to this decoupling, the solution of the two problems needs to be coordinated in order to achieve the overall objective. In the company, this is pursued through an internal price system aiming at giving the sellers the incentives needed to align their decisions with the overall objective. We propose and discuss linear programming models for the decoupled and integrated planning problems. We present numerical examples to illustrate potential effects of integration and coordination and discuss the advantages and disadvantages of the integrated over the decoupled approach. While the total contribution is higher in the integrated approach, it has also been found that the sellers’ contribution can be considerably lower. Therefore, we also suggest contribution sharing rules to achieve a solution where both the company and the sellers attain a better outcome under the integrated planning.     

MODELING HARVEST SCHEDULING IN MULTIFUNCTIONAL PLANNING OF FORESTS FOR LONGTERM WATER YIELD OPTIMIZATION

In this study, wood production and hydrologic functions of forests were accommodated within a planning procedure for separate working circles (areas dedicated to certain forest functions) that were delineated according to an Ecosystem‐Based Functional Planning approach. Mixed integer goal programming was used as the optimization technique. The timing and scheduling of a maintenance cutting (partial harvest) was the decision variable in the modeling effort, and an original formulation was developed as a multiobjective planning procedure. Four sample planning strategies were developed and model outputs were evaluated according to these strategies. Spatial characteristics of stands were considered, and used to prohibit the regeneration of adjacent stands during the same time period. Because of the positive relationship between qualified water production and standing timber volume in the forest, the model attempts to maximize qualified water production levels by increasing standing volume stocks in the forest through the delay of regeneration activities.     

A stochastic programming approach to cash management in banking

The treasurer of a bank is responsible for the cash management of several banking activities. In this work, we focus on two of them: cash management in automatic teller machines (ATMs), and in the compensation of credit card transactions. In both cases a decision must be taken according to a future customers demand, which is uncertain. From historical data we can obtain a discrete probability distribution of this demand, which allows the application of stochastic programming techniques. We present stochastic programming models for each problem. Two short-term and one mid-term models are presented for ATMs. The short-term model with fixed costs results in an integer problem which is solved by a fast (i.e. linear running time) algorithm. The short-term model with fixed and staircase costs is solved through its MILP equivalent deterministic formulation. The mid-term model with fixed and staircase costs gives rise to a multi-stage stochastic problem, which is also solved by its MILP deterministic equivalent. The model for compensation of credit card transactions results in a closed form solution. The optimal solutions of those models are the best decisions to be taken by the bank, and provide the basis for a decision support system.     

Multi-product production planning: A goal programming approach

Cost minimization multi-product production problems with static production resource usage and internal product flow requirements have been solved by linear programming (LP) with input/output analysis. If the problem is complicated by interval resource estimates, interval linear programming (ILP) can be used. The solution of realistic problems by the above method is cumbersome. This paper suggests that linear goal programming (LGP) can be used to model a multi-product production system. LGP's unique modeling capabilities are used to solve a production planning problem with variable resource parameters. Input/output analysis is used to determine the technological coefficients for the goal constraints and is also used to derive an information sub-model that is used to reduce the number of variable resource goal constraints. Preliminary findings suggest that the LGP approach is more cost-efficient (in terms of CPU time) and in addition provides valuable information for aggregate planning.     

Fuzzy Multi-fractional Programming for Land Use Planning in Agricultural Production System

In this paper, we present a multi-objective linear fractional programming (MOLFP) approach for multi-objective linear fuzzy goal programming (MOLFGP) problem. Here, we consider a problem in which a set of pair of goals are optimized in ratio rather than optimizing them individually. In particular, we consider the optimization of profit to cash expenditure and crop production in various seasons to land utilization as a fractional objectives and used remaining goals in its original form. Further, the goals set in agricultural production planning are conflicting in nature; thus we use the concept of conflict and nonconflict between goals for computation of appropriate aspiration level. The method is illustrated on a problem of agricultural production system for comparison with Biswas and Pal [1] method to show its suitability.