A framework for the probabilistic analysis of hierarchical planning systems

As we have argued in previous papers, multi-level decision problems can often be modeled as multi-stage stochastic programs, and hierarchical planning systems designed for their solution, when viewed as stochastic programming heuristics, can be subjected to analytical performance evaluation. The present paper gives a general formulation of such stochastic programs and provides a framework for the design and analysis of heuristics for their solution. The various ways to measure the performance of such heuristics are reviewed, and some relations between these measures are derived. Our concepts are illustrated on a simple two-level planning problem of a general nature and on a more complicated two-level scheduling problem.     

Geometric and algorithmic developments for a hierarchical planning problem

This paper presents a new model for multiobjective planning in hierarchical systems that explicitly takes into consideration the order in which decisions are made. Interactions and conflicts that normally exist among the levels are introduced by specifying jointly controlled feasible regions and interdependent objective functions. At each level in the system, planners attempt to maximize net benefits in light of all higher-level decisions, and thus may influence but not control the behavior of others. The resultant formulation leads to the multilevel programming problem. The geometry of an all linear case is first examined wherein it is shown that the optimal solution must lie at a vertex of the original polyhedral constraint region. Next, a set of first order optimality conditions is derived for the general case and used as the basis of an algorithm for the linear problem. A number of examples are given to highlight the results.     

Distribution planning in a turbulent environment

This paper looks at planning within the present distribution environment, paying particular attention to recent trends and developments. The idea of a depot as the traditional building block of a distribution network is questioned and the various functions of a regional depot are explored.The concept of a transit point as a simple alternative to a depot is developed and examples are given of its use in practice. Several refinements are available, particularly in the area of trailer technology, and these are discussed. The necessary support system for transit point operation is identified and finally the implications for distribution planning are considered.     

Process planning in a fuzzy environment

Mixed-integer optimization models for chemical process planning typically assume that model parameters can be accurately predicted. As precise forecasts are difficult to obtain, process planning usually involves uncertainty and ambiguity in the data. This paper presents an application of fuzzy programming to process planning. The forecast parameters are assumed to be fuzzy with a linear or triangular membership function. The process planning problem is then formulated in terms of decision making in a fuzzy environment with fuzzy constraints and fuzzy net present value goals. The model is transformed to a deterministic mixed-integer linear program or mixed-integer nonlinear program depending on the type of uncertainty involved in the problem. For the nonlinear case, a global optimization algorithm is developed for its solution. This algorithm is applicable to general possibilistic programs and can be used as an alternative to the commonly used bisection method. Illustrative examples and computational results for a petrochemical complex with 38 processes and 24 products illustrate the applicability of the developed models and algorithms.     

Linking strategic and tactical forestry planning decisions

In this paper a two-level model and optimization algorithms areintroduced to assist forestry companies in simultaneously considering strategicinvestment and tactical planning decisions. A procedure to reduce thediscrepancy produced in the aggregation and disaggregation process used to linkthese two-level decisions is also presented. This procedure is based onboth cluster analysis over economic parameters defined on the standmacro-units to be harvested and on the information transmittedbottom-up and top-down between the strategic and tactical models. Wealso show a new approach for solving the tactical mixed integer model.     

Robust hierarchical production planning under uncertainty

We consider aggregation of products with similar characteristics in a two-level hierarchical production planning model. A robust aggregate plan at the upper level is such that, at the lower level, a disaggregation policy exists even when detailed demands may vary within some given bounds. We provide necessary and sufficient conditions for the robustness of an aggregate plan and obtain a closed form expression of these conditions. A set of more manageable sufficient conditions is also presented.Institut National des Sciences Appliquées de Toulouse.     

Introducing parametric uncertainty into a nonlinear friction model

We introduce parametric uncertainty into a nonlinear model for static and kinetic friction of solids. The uncertain parameters are modeled by possibility distributions and an analytical approach is used to propagate the uncertainties through the computations. As a result, we obtain a symbolic expression for the whole spectrum of possible calculation results. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Disaggregation under uncertainty in hierarchical production planning

This study deals with a hierarchical planning system that is used for planning of a single-stage manufacturing system. We consider two decision levels, aggregate and detailed planning, and formulate a model for evaluation of aggregate plans and optimal disaggregation in case of independent stochastic demand. It is shown how the optimal solution can be obtained with the aid of a dynamic programming algorithm. Furthermore, we give conditions that will guarantee optimality of a simple intuitive disaggregation rule.     

Manpower planning procedures for hierarchical manpower systems

In this paper, a model which describes manpower planning procedures for a hierarchical system having certain desired properties to be achieved is presented. These properties will act as constraints on the system. It will be assumed that eventual, desired properties are achieved gradually over a period of time. Promotion policies will be wastagelinked and designed to satisfy vacancy-expansion demands. The model will be applied to data from a group of English County Police Constabularies.     

Aggregation and disaggregation in hierarchical production planning

This paper reports on a simulation study of hierarchical planning procedures, which can support a material requirements planning system. Data for this study have been obtained from a Swedish manufacturing company. The three final products considered in the simulations represent a major part of this company.An aggregate plan in terms of product groups and machine groups is derived with the aid of an aggregate model. This plan is then disaggregated by changing order release times obtained from material requirements planning, and by distributing extra capacity among individual machines.The results indicate that our methods in general perform significantly better than a comparable reference case without the supporting hierarchical planning process. In our simulation experiments we evaluate different design features, like disaggregation procedures and methods for aggregating items and machines.     

Handling fuzzy temporal constraints in a planning environment

An interleaved integration of the planning and scheduling process is presented with the idea of including soft temporal constraints in a partial order planner that is being used as the core module of an intelligent decision support system for the design forest fire fighting plans. These soft temporal constraints have been defined through fuzzy sets. This representation allows us a flexible representation and handling of temporal information. The scheduler model consists of a fuzzy temporal constraints network whose main goal is the consistency checking of the network associated to each partial order plan. Moreover, we present a model of estimating this consistency, and show the monitoring and rescheduling capabilities of the system. The resulting approach is able to tackle problems with ill defined knowledge, to obtain plans that are approximately consistent and to adapt the execution of plans to unexpected delays. This work has been partially supported under the project MCyT TIC2002-04146-C05-2 and the contract NET033957 with the Andalusian Regional Government.     

A hierarchical location model for public facility planning

In this article, we present a discrete hierarchical location model for public facility planning. The main features of the model are: an accessibility maximization objective; several levels of demand and of facilities; a nested hierarchy of facilities (i.e. a facility of a given level can serve demand of equal and lower levels); maximum and minimum capacity constraints; and user-to-facility assignment constraints. The latter include single-assignment and closest-assignment constraints, as well as a new type of constraints called path-assignment constraints. Their purpose is to enforce some desirable properties for the spatial pattern of assignments. If they are not included, model solutions are difficult to interpret and to explain in a public facility planning context, therefore being less likely to be accepted by the users. The usefulness of the model is illustrated through a real-world application to school network planning.     

Fuzzy hierarchical analysis

This paper extends hierarchical analysis to the case where the participants are allowed to employ fuzzy ratios in place of exact ratios. If a person considers alternative A more important than alternative B, then the ratio used might be approximately 3 to 1, or between 2 to 1, and 4 to 1, or at most 5 to 1. The pairwise comparison of the issues and the criteria in the hierarchy produce fuzzy positive reciprocal matrices. The geometric mean method is employed to calculate the fuzzy weights for each fuzzy matrix, and these are combined in the usual manner to determine the final fuzzy weights for the alternatives. The final fuzzy weights are used to rank the alternatives from highest to lowest. The highest ranking contains all the undominated issues. The procedure easily extends to the situation where many experts are utilized in the ranking process, or to the case of missing data. Two examples are presented showing the final fuzzy weights and the final ranking.     

Design of insurance contracts using stochastic programming in forestry planning

This work addresses a tactical planning problem faced by a forestry firm, deciding which timber units to harvest and what roads to build to obtain the greatest possible benefits. We include uncertainty in prices by means of utility theory. This enables solutions to be found that the firm finds preferable to those obtained when risk aversion is ignored and makes it possible to design insurance contracts that benefit the firm while also being attractive to an insurer. Two types of contract are designed; one dependent on the firm’s operating result and the other independent of it. Metrics are then developed to quantify the benefits conferred by a contract, demonstrating that the latter contract type dominates the former. These results are then illustrated by applying them to a simplified planning problem of a forest owned by the Chilean forestry operator Millalemu.     

Production planning in fuzzy environment

This paper describes a fuzzy system designed to support production planning in an industrial unit producing cardboard boxes. In this industrial unit, orders forn boxes of widthw, lengthl, heighth, made ofq layers of typek paper for delivery int units of time are produced. In the production of such orders apart from meeting the orders specifications, it is usually tried to minimize the margin trim loss, the number of machine setups and the holding cost of the finished orders. Considering the dynamism of production systems that are influenced by such factors as market demand fluctuations, changes in commercial priorities, raw material availability and production capabilities, we solve this multi-objective problem by fuzzy set theory.     

A stochastic linear programming approach to hierarchical production planning

This paper examines production planning decisions. The process is formulated as a hierarchical production planning (HPP) model under uncertain demand. A review of HPP articles indicates that while current models do consider uncertainty as a part of their solution methods, a deficiency persists since these models fail to incorporate the uncertain demand explicitly in the formulation of the problem. A stochastic linear programming model (SLP) is proposed to better reflect reality and to provide a superior solution. The model remains computationally tractable despite the precise incorporation of uncertainty and the imposition of penalties when constraints are violated. A problem is introduced which illustrates the superiority of the proposed model over those currently being applied.     

Fuzzy hierarchical analysis revisited

We present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by Saaty in the analytic hierarchy process. We test our new procedure in two cases where there are formulas for the crisp weights. An example is presented where there are five criteria and three alternatives.     

Introducing oblique norms into multiple criteria programming

We propose to use block norms to generate nondominated solutions of multiple criteria programs and introduce the new concept of the oblique norm that is specially tailored to handle general problems. We prove the equivalence of finding the properly nondominated solutions of a multiple criteria program and solving its scalarization by means of oblique norms.     

Packing two disks into a polygonal environment

We consider the following problem. Given a polygon P, possibly with holes, and having n vertices, compute a pair of equal radius disks that do not intersect each other, are contained in P, and whose radius is maximized. Our main result is a simple randomized algorithm whose expected running time, on any input, is O(nlogn). This is optimal in the algebraic decision tree model of computation.     

Airport ground workforce planning with hierarchical skills: a new formulation and branch-and-price approach

Annals of Operations Research - We address the problem of strategic workforce planning for airport ground staff. The goal is to find a minimum workforce mix that satisfies a target coverage rate...