On the approximability of adjustable robust convex optimization under uncertainty

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.     

Collaborative production planning of supply chain under price and demand uncertainty

This research is motivated by an automobile manufacturing supply chain network. It involves a multi-echelon production system with material supply, component fabrication, manufacturing, and final product distribution activities. We address the production planning issue by considering bill of materials and the trade-offs between inventories, production costs and customer service level. Due to its complexity, an integrated solution framework which combines scatter evolutionary algorithm, fuzzy programming and stochastic chance-constrained programming are combined to jointly take up the issue. We conduct a computational study to evaluate the model. Numerical results using the proposed algorithm confirm the advantage of the integrated planning approach. Compared with other solution methodologies, the supply chain profits from the proposed approach consistently outperform, in some cases up to 13% better. The impacts of uncertainty in demand, material price, and other parameters on the performance of the supply chain are studied through sensitivity analysis. We found the proposed model is effective in developing robust production plans under various market conditions.     

Time-dependent and independent control rules for coordinated production and pricing under demand uncertainty and finite planning horizons

We address the effect of uncertainty on a manufacturer’s dynamic production and pricing decisions over a finite planning horizon. The demand for products, which depends on their price, is characterized by two stochastic processes: potential demand and customer price sensitivity. An optimal policy for coordinating production and pricing is a time-dependent feedback rule with respect to the state of the manufacturer’s inventories. We show that when the volatility of customer sensitivity to the product price is negligible, the optimal policy can be obtained analytically. Moreover, our simulations demonstrate that the volatility of stochastic customer price sensitivity does not have a strong effect on the manufacturer’s expected profit. Therefore, the solution derived for the case of customer price sensitivity with zero volatility can serve as a good approximation heuristic for the optimal policy if the true volatility of customer price sensitivity is within 40 % of its mean and the volatility of potential demand is within 25 % of its mean. Moreover, under these conditions, a simplified, time-independent control rule deteriorates expected profits by only 1.5 %.     

Control and system-theoretic identification of the supply chain dynamics domain for planning,analysis and adaptation of performance under uncertainty

The analysis of how to achieve planned economic performance in a real-time, uncertain and perturbed execution environment is a vital and up-to-date issue in many supply chains. Although it is intuitive that uncertainty is likely to have impacts on performance, the research on systematic terminology and quantitative analysis in this domain is rather limited as compared with the well-established domain of supply chain optimal planning. This study is among the first to address the operative perspective of the supply chain dynamics domain. The methodology of this conceptual paper is based on the business and technical literature analysis and fundamentals of control and systems theory. In contributing to the existing studies in this domain, the paper proposes a possible systemization and classification of related terminology from different theoretical perspectives, and important practical problems. For the supply chain dynamics domain, the paper identifies and groups possible problem classes of research, corresponding quantitative methods, and describes the general mathematical formulations. The results of this study may be of interest to both academics and practitioners.     

Evaluation of three production planning procedures for the use of recipe flexibility

Process industries often obtain their raw materials from mining or agricultural industries. These raw materials usually have variations in quality, which often lead to variations in the recipes used for manufacturing a product. Another reason for varying the recipe is to minimize production costs by using the cheapest materials that still lead to a satisfactory quality in the product. A third reason for using recipe flexibility is that it may occur that at the time of production not all materials for the standard recipe are available. In earlier research we showed under what conditions the use of this type of recipe flexibility should be preferred to the use of high materials stock to avoid materials shortages. We also showed that the use of recipe flexibility to account for material shortages can be justified if the material replenishment leadtime is long, the demand uncertainty is high and the required service level is high. In this paper we assume that these conditions are satisfied and we investigate three different production planning procedures that make use of recipe flexibility to cope with the uncertainty in demand and supply. We assume that the customer order leadtime is much smaller than the material replenishment leadtime, and therefore demand uncertainty is high. The optimal procedure optimizes material use over a planning horizon equal to the material replenishment leadtime, taking into account the customers orders and knowledge of the distribution function of future demand. The deterministic procedure also optimizes the material use over the material replenishment leadtime, but it assumes a deterministic demand level for unknown orders. The simplest, myopic procedure optimizes material use over only the accepted customer orders. These three procedures are investigated via an experimental design of computer simulations of an elementary small scale model of the production planning situation. The results show that the optimal procedure outperforms the other two procedures. Furthermore, for a realistic cost structure in feed industry under certain circumstances the use of the optimal procedure may lead to a 4% increase in profit. However, this improvement must be weighted against the cost incurred by the operational use of this complex procedure. Based on these considerations and the numerical results in this paper, we may expect that for some situations in practice the use of the simplest myopic procedure, optimizing material use only over the available customer orders, will be justified from an overall cost point of view.     

Planning and scheduling of the make-and-pack dairy production under lifetime uncertainty

In the dairy processing, the rapid quality decay of milk-based intermediate mixture to make and pack restricts productivity and, forces organizations to carefully plan and schedule their production. Hereby, in this study, we consider a planning and scheduling problem encountered in the dairy industry and propose a chance-constrained programming model accounting for uncertainty in quality decay of intermediate mixture. The aim of the model is to find the optimal lot sizes and production schedule with minimum makespan (total time needed to finish the daily production). The proposed schedule allows the storage duration of intermediate mixture to be within a stochastic lifetime. A case study is presented to illustrate the typical structure of the two-stage semi-continuous make-and-pack production process. The numerical study reflects real settings from a set (Balkan type) yoghurt production process. Accordingly, a simulation of the production process is introduced to evaluate the proposed production plan and schedule in terms of product waste. The model is examined with 32 scenarios consisting of different distribution parameters, confidence levels and demand patterns. Overall in the scenarios, the proposed plan and schedule result in decreasing 2693 l of product waste with 3.24 h increase of makespan in total.     

Accessibility as the objective of public transportation planning: an integrated transportation and land use model

The subject of this paper is planning of public transportation in sparsely populated areas.The paper consists of four main parts. In part I an analysis of the system to be planned is carried out. A comparison with a survey on the traffic and transportation models available in the literature, though very voluminous, reveals a lack of a general model-framework for planning public transportation in sparsely populated areas. It is the aim of the present paper to fill this gap.Part II discusses how to evaluate optional plans for the physical and socio-economic structure of a region and concludes that this must be done by means of accessibility. Acessibility is defined in this paragraph.In part III a model is presented which describes the interdependance between on one hand the demand of accessibility of the population and on the other hand the transportation system, both public and private, and the location of facilities and residential areas. The applicability of some standard transportation and traffic models is discussed briefly in this section.Finally part IV describes a case study, in which the models set out in part III are applied. Some results are presented with conclusions.     

Efficient algorithms for combined heat and power production planning under the deregulated electricity market

Combined heat and power (CHP) production is an important energy production technology that can yield much higher total energy efficiency than separate heat and power generation. In CHP production, the heat and power production follows a joint characteristic, which means that the production planning must be done in coordination. Cost-efficient operation of a CHP system can be planned by using an optimization model. A long-term planning model decomposes into thousands of hourly models. Earlier, in the regulated electric power market, the planning problem was symmetrically driven by heat and power demand. The liberalization of the power market has created an asymmetrical planning problem, where heat production responds to the demand and power production to the volatile market price. In this paper, we utilize this asymmetry to develop novel envelope-based dual algorithms for solving the hourly CHP models efficiently. The basic idea is to transform the three-dimensional characteristic operating region for heat and power production of each CHP plant into a two-dimensional envelope by taking the power price as a parameter. Then the envelopes of each plant are used for looking up the optimal solution rapidly. We propose two versions of the algorithm: the on-line envelope construction algorithm (ECON) where the envelopes are constructed for each hour based on the power price and the off-line envelope construction algorithm (ECOFF) where envelopes are pre-computed for all different power price ranges. We derive the theoretical time complexity of the two algorithms and compare their performance empirically with realistic test models against the ILOG CPLEX solver and the Power Simplex (PS) algorithm. PS is an extremely efficient specialized primal algorithm developed for the symmetrical CHP planning problem under the regulated market. On average, when reusing previous basic solutions, ECON is 603 times faster than CPLEX and 1.3 times faster than PS. ECOFF is 1860 times faster than CPLEX and four times faster than PS.     

Design of mathematical models for the integration of purchase and production lot-sizing and scheduling problems under demand uncertainty

This study addresses the multi-level lot-sizing and scheduling problem with complex setups and considers supplier selection with quantity discounts and multiple modes of transportation. The present research proposes a mixed-integer linear programming (MILP) model in which the purchase lot-sizing from multiple suppliers, production lot-sizing with multiple machines and scheduling of various products of different families are accomplished at the same time. However, these decisions are not integrated in traditional environments and are taken separately. In this study, two different types of lot-sizing models called aggregated and disaggregated are developed for the problem to evaluate and compare the computational efficiency of them under deterministic and stochastic demands and provide some managerial insights. To deal with the stochastic demands, Chance-Constrained Programming (CCP) approach is applied. Based on the results of this study, the average profit of the separated (purchase from production) lot-sizing model under demand choice flexibility and stochastic demand is 24% and 22% less than the integrated model, respectively. Moreover, the results also confirm the effect of discount structure on the amount of purchases, productions, revenues and costs.     

A production planning model for an unreliable production facility: Case of finite horizon and single demand

We study a two-level inventory system that is subject to failures and repairs. The objective is to minimize the expected total cost so as to determine the production plan for a single quantity demand. The expected total cost consists of the inventory carrying costs for finished and unfinished items, the backlog cost for not meeting the demand due-date, and the planning costs associated with the ordering schedule of unfinished items. The production plan consists of the optimal number of lot sizes, the optimal size for each lot, the optimal ordering schedule for unfinished items, and the optimal due-date to be assigned to the demand. To gain insight, we solve special cases and use their results to device an efficient solution approach for the main model. The models are solved to optimality and the solution is either obtained in closed form or through very efficient algorithms.     

On the problem of output feedback control synthesis under uncertainty and finite-time observers

We consider the problem of controlling a linear system of ordinary differential equations with a linear observable output. The system contains uncertain items (disturbances), for which we know only “hard” pointwise constraints. The problem of synthesizing a control that brings the trajectories of the system into a given target set in finite time is solved under weakened conditions without assuming that the control and the disturbance are of the same type. To this end, we suggest an approach that amounts to constructing an information set and a weakly invariant set with subsequent “aiming” of the first set at the second. Both stages are carried out in a finite-dimensional space, which permits one to use an efficient algorithm for solving the synthesis problem approximately on the basis of the ellipsoidal calculus technique. The results are illustrated by an example in which the control of a linear oscillation system is constructed.     

Supply planning for industrial ecology and remanufacturing under uncertainty: a numerical study of leaded-waste recovery from television disposal

The use of waste as a raw material for manufacturing is hampered by the uncertainty associated with the availability of supply. Technological change and obsolescence further complicates the ability of decision makers to consider discarded durable products as a potential source of raw materials. This uncertainty complicates remanufacturing and industrial ecology. A problem since remanufacturing and industrial ecology need to be (and can be) profitable as well as environmentally desirable if they are to be encouraged. To address this problem the modelling of the waste flow of durable goods is considered. The disposal of televisions in the United States is used to illustrate the challenges and requirements for forecasting in an environment with supply uncertainty. This example is timely since the diposal of cathode ray tubes (CRTs) in municipal landfills is being banned and an alternate technology trajectory for televisions exists—the flat panel display and phase-out of analogue broadcasting in the US. This paper estimates the waste stream resulting from three different scenarios of CRT leaded-waste disposal patterns. The reuse of lead-containing CRT glass is found to offer potential. The elimination of this controversial waste stream, as a result of replacement by the adoption of flat panel television technology, is still decades away. The findings in this study indicate the range of the quantity of waste that will require an alternative infrastructure as it is displaced from municipal landfills. This study provides important information for both developing a collection infrastructure and processing alternatives to extract the residual value of the disposed of televisions.     

Aggregation and the design of models for medium-term planning of production

A major issue in the design of an aggregate multi-period model for medium-term planning of production is the determination of the aggregation level of the variables. In the process of model design, one has to examine if and how the outcomes of the model are effected by applying different modes and levels of aggregation. This is especially important with regard to those outcomes which will be used for medium-term decisions and for laying out short-term and detailed plans. Also dependent on the aggregation level is the operational convenience of the model for decision support. In this paper it will be shown that simulation experiments with a preliminary model for medium-term planning can yield valuable information to aid this design problem.     

Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis

ZDM – Mathematics Education - Mathematical working space (MWS) is a model that is used in research in mathematics education, particularly in the field of geometry. Some MWS elements are...     

Error linearization as an effective tool for experimental analysis of the numerical stability of algorithms

After introducing briefly the principles of theerror linearization method, which is able to determine the coefficients of the first order error approximation, a collection of examples is presented to demonstrate its efficiency as a test bench for analyzing numerical algorithms. These examples illustrate the propagation of initial errors, the effect of cancellation, the easy location of the most unstable parts of an algorithm, calculation of condition numbers, approximating the statistical behavior of accumulated errors and the convergence of iterative methods.     

Optimal production time and number of maintenance actions for an imperfect production system under equal-interval maintenance policy

This paper deals with the optimal production/maintenance (PM) policy for a deteriorating production system which may shift from the in-control state to the out-of-control state while producing items. The process is assumed to have a general shift distribution. Under the commonly used maintenance policy, equal-interval maintenance, the joint optimizations of the PM policy are derived such that the expected total cost per unit time is minimized. Different conditions for optimality, lower and upper bounds and uniqueness properties on the optimal PM policy are provided. The implications of another commonly used policy, to perform a maintenance action only at the end of the production run, are also discussed. Structural properties for the optimal policy are established so that an efficient solution procedure is obtained. In the exponential case, some extensions of the results obtained previously in the literature are presented. A numerical example is provided to illustrate the solution procedure for the optimal production and maintenance policy.