Sensitivity Analysis in Linear Programming Models with Common Inputs

The importance of sensitivity analysis information in linear programming has been stressed in the management science literature for some time. Indeed, Gal [3] has devoted an entire text to just this issue. Linear programs with common inputs (cost coefficients or right-hand-side values) present a problem in that classical sensitivity analysis does not allow for the simultaneous changes required to determine the sensitivity of these models to common inputs. We first survey the approaches previously developed for simultaneous-change sensitivity analysis and cast them in the framework of the special common input case. These general techniques are compared to a simple aggregate variable technique that has not received attention in the literature.     

A Generalization of Wendell's Tolerance Approach to Sensitivity Analysis in Linear Programming

A criticism of linear programming has been that the data which are available in practice are too inexact and unreliable for linear programming to properly work. Managers are therefore concerned with how much actual values may differ from the estimates that were used in the model before the results become irrelevant. Sensitivity analysis emerged to help deal with the uncertainties inherent in the linear programming model. However, the ranges calculated are generally valid only when a single coefficient is varied. An extension of sensitivity analysis, the 100 Percent Rule, allows the simultaneous variation of more than one element in a vector, but does not permit the independent variation of the elements. A tolerance approach to sensitivity analysis enables the consideration of simultaneous and independent change of more than one coefficient. However, the ranges developed are unnecessarily restricted and may be reduced in width to zero when primal or dual degeneracy exists. This paper presents an extension of the tolerance approach which reduces the limitations of both the traditional and tolerance approaches to sensitivity analysis.     

Sensitivity Analysis Revisted and Extended

The tolerance approach to sensitivity analysis in linear programming provides a framework for dealing with simultaneous and independent variations of the coefficients and terms. Specifically, the approach yields a maximum tolerance percentage which in turn characterizes a region of variability for the coefficients or terms. This paper shows how to expand this region of variability yielding ordinary sensitivity analysis as a special case of the tolerance approach.     

EVOLVING INTERNATIONAL INFLATION DYNAMICS: WORLD AND COUNTRY‐SPECIFIC FACTORS

The decline in the level and persistence of inflation over the 1980s is a common feature of the most industrialized economies in the world. The rise in inflation volatility of the late 1970s and the subsequent fall of the 1980s is country specific for the UK, Canada, and, to a lesser extent, the United States, Italy, and Japan. Since the late 1980s, inflation predictability has declined significantly across the industrialized world. We link the empirical results to recent theories of international inflation.     

Refining Two-Group Multivariable Classification Models Using Univariate Optimal Discriminant Analysis*

Fisher's discriminant analysis (FDA) is often used to obtain a prediction model for dichotomous classifications on the basis of two or more independent variables. FDA provides an equation whereby values on independent variables are combined into a single predicted value (Y*) that is compared against a cutpoint and direction in order to make classifications. Theoretically, univariate optimal discriminant analysis employed on these Y* will maximize training classification accuracy. This methodology is illustrated using three examples.     

Concepts,Theory, and Techniques PRINCIPLES OF MULTIOBJECTIVE OPTIMIZATION*

This paper attempts to isolate and analyze the principal ideas of multiobjective optimization. We do this without casting aspersions on single-objective optimization or championing any one multiobjective technique. We examine each fundamental idea for strengths and weaknesses and subject two—efficiency and utility—to extended consideration. Some general recommendations are made in light of this analysis. Besides the simple advice to retain single-objective optimization as a possible approach, we suggest that three broad classes of multiobjective techniques are very promising in terms of reliably, and believably, achieving a most preferred solution. These are: (1) partial generation of the efficient set, a rubric we use to unify a wide spectrum of both interactive and analytic methods; (2) explicit utility maximization, a much-overlooked approach combining multiattribute decision theory and mathematical programming; and (3) interactive implicit utility maximization, the popular class of methods introduced by Geoffrion, Dyer, and Feinberg [24] and extended significantly by others.     

Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: A geometric programming approach

A multi-item inventory model with constant demand and infinite replenishment is developed under the restrictions on storage area, total average shortage cost and total average inventory investment cost. These restrictions may be precise or imprecise. Here, it is assumed that inventory costs are directly proportional to the respective quantities, and unit purchase/production cost is inversely related to the demand. Restricted shortages are allowed but fully backlogged. First, the problem is formulated in crisp environment taking the deterministic and precise inventory parameters. It is solved by both geometric programming (GP) and gradient-based non-linear programming (NLP) methods. Later, the problem is formulated with fuzzy goals on constraints and objectives where impreciseness is introduced through linear membership functions. It is solved using the fuzzy geometric programming (FGP) method. The inventory models are illustrated with numerical values and compared with the crisp results. A sensitivity analysis on the optimum order quantity and average cost is also presented due to the variation in the tolerance of total average inventory investment cost and total average shortage cost following Dutta et al., 1993, Fuzzy Sets and Systems, 55, 133-142.     

SENSITIVITY ANALYSIS IN DECISION THEORY

This paper explores linear programming-like sensitivity analysis in decision theory. In particular, the paper considers the sensitivity of an optimal decision to changes in probabilities of the states of nature and the development of “confidence spheres” to bound arbitrary parametric changes in the probability vector. Such information can be used to assess the accuracy required in assigning probabilities and the confidence in the maximumutility decision.     

Climate Change Beliefs and Perceptions of Weather‐Related Changes in the United Kingdom

Public perception research in different countries has suggested that real and perceived periods of high temperature strengthen people's climate change beliefs. Such findings raise questions about the climate change beliefs of people in regions with moderate climates. Relatively little is known about whether public concerns about climate change may also be associated with perceived changes in other weather‐related events, such as precipitation or flooding. We examine the relationship between perceived changes in weather‐related events and climate change beliefs among U.K. residents at a time of below‐average winter temperatures and recent flooding. National survey data (n = 1,848) revealed that heat waves and hot summers were perceived to have become less common during respondents’ lifetimes, while flooding, periods of heavy rainfall, coastal erosions, and mild winters were perceived to have increased in frequency and cold winters were perceived to be unchanged. Although perceived changes in hot‐weather‐related events were positively associated with climate change beliefs, perceived changes in wet‐weather‐related events were found to be an even stronger predictor. Self‐reported experience of “flooding in own area” and “heat‐wave discomfort” also significantly contributed to climate change beliefs. These findings highlight the importance of salient weather‐related events and experiences in the formation of beliefs about climate change. We link our findings to research in judgment and decision making, and propose that those wishing to engage with the public on the issue of climate change should not limit their focus to heat.     

Notes and Communications A NON-DUAL APPROACH TO SENSITIVITY ANALYSIS—THE RIGHT-HAND-SIDE CASE

Linear programming and sensitivity analysis in linear programming have long been integral parts of business school curricula, both at the undergraduate and graduate levels. Most business school students are required to take an introductory course in management science emphasizing linear programming and sensitivity analysis. This note is educational and presents a non-dual approach to restoring the optimal solution to a linear program when changes in the right-hand-side vector render the current solution infeasible and thus nonoptimal.     

Theoretical Distributions of Optima for Univariate Discrimination of Random Data*

Optimal linear discriminant models maximize percentage accuracy for dichotomous classifications, but are rarely used because a theoretical framework that allows one to make valid statements about the statistical significance of the outcomes of such analyses does not exist. This paper describes an analytic solution for the theoretical distribution of optimal values for univariate optimal linear discriminant analysis, under the assumption that the data are random and continuous. We also present the theoretical distribution for sample sizes up to N= 30. The discovery of a statistical framework for evaluating the performance of optimal discriminant models should greatly increase their use by scientists in all disciplines.     

Effect of forecast errors on rolling horizon master production schedule cost performance for various replanning intervals

Master production schedules are usually updated by the use of a rolling schedule. Previous studies on rolling schedules seem to form the consensus that frequent replanning of a master production schedule (MPS) can increase costs and schedule instability. Building on previous research on rolling schedules, this study addresses the impact of overestimation or underestimation of demand on the rolling horizon MPS cost performance for various replanning frequencies. The MPS model developed in this paper is based on actual data collected from a paint company. Results indicate that under both the forecast errors conditions investigated in this study, a two-replanning interval provided the best MPS cost performance for this company environment. However, results from the sensitivity analysis performed on the MPS model indicate that when the setup and inventory carrying costs are high, a 1-month replanning frequency (frequent replanning) seems more appropriate for both of the above forecast error scenarios.     

Decision Support System of Truck Routing and Refueling: A Dual‐Objective Approach

The variable‐route vehicle‐refueling problem (VRVRP) is a variant of the network‐flow problem which seeks, for a vehicle traveling from origin s to destination d, both the route and the refueling policy (sequence of fuel stations to use between s and d) that jointly minimize the fuel cost of operating the vehicle. Commercial‐grade decision support systems that solve the VRVRP are widely used by motor carriers, but they provide heuristic solutions only. Exact methods are available from the academic side, but because they focus on minimizing costs, they tend to cut fuel costs in exchange for increased vehicle miles (which can increase fuel consumptions and pollutants emission). We propose a new approach to the VRVRP that allows carriers to jointly seek the two possibly conflicting goals; minimizing fuel cost and vehicle miles. Computational testing shows that our approach (i) outperforms the commercial software products in both goals, and (ii) finds solutions that require significantly less vehicle miles than those given by the exact method proposed in the academic literature, without incurring unacceptable increases in fuel cost.     

Vendor Selection with Bundling: A Comment*

In a recent article by Rosenthal, Zydiak, and Chaudhry (1995), a mixed integer linear programming model was introduced to solve the vendor selection problem for the case in which the vendor can sell items individually or as part of a bundle. Each vendor offered only one type of bundle, and the buyer could purchase at most one bundle per vendor. The model employed n(m+ 1) binary variables, where n is the number of vendors and m is the number of products they sell. The existing model can lead to a purchasing paradox: it may force the buyer to pay more to receive less. We suggest a reformulation of the same problem that (i) eliminates this paradox and reveals a more cost-effective purchasing strategy; (ii) uses only n integer variables and significantly reduces the computational workload; and (iii) permits the buyer to purchase more than one bundle per vendor.     

ON THE (RE)DISCOVERY OF FUZZY GOAL PROGRAMMING

An earlier paper and subsequent commentaries in Decision Sciences described purportedly new methods for formulating and solving goal programming problems with fuzzy goals. This note suggests that much of what these articles propose is but a rediscovery of well-known results.     

Separation Failure in Linear Programming Discriminant Models

Linear programming discriminant analysis (LPDA) models are designed around a variety of objective functions, each representing a different measure of separation of the training samples by the resulting discriminant function. A separation failure is defined to be the selection of an “optimal” discriminant function which incompletely separates a pair of completely separable training samples. Occurrence of a separation failure suggests that the chosen discriminant function may have an unnecessarily low classification accuracy on the actual populations involved. In this paper, a number of the LPDA models proposed for the two-group case are examined to learn which are subject to separation failure. It appears that separation failure in any model can be avoided by applying the model twice, reversing group designations.     

A Note on “Discrete Sequential Search with Group Activities”

This note comments on a paper published by Wagner and Davis [Decision Sciences (2001), 32(4), 557–573]. These authors present an integer‐programming model for the single‐item discrete sequential search problem with group activities. Based on their experiments, they conjecture that the problem can be solved as a linear program. In this note, we provide a counterexample for which the optimal value of the linear program they propose is different from the optimal value of the integer‐programming model, hence contradicting their conjecture for the specific linear program that they specify. To the best of our knowledge, the conjecture settled in this note was still an open question.     

Modeling Take-or-Pay Contract Decisions

Long-term purchase contracts for natural resources, such as natural gas, often contain take-or-pay provisions that penalize the buyer for not purchasing (taking) a minimum quantity of output over some period of time. In some contracts for a limited time interval, known as the make-up period, the penalty payments can be credited against future “takes” in excess of the take-or-pay level. Additionally, options to “buy down” or “buy out” existing contracts, or to initiate new contracts, may exist. The purchaser, faced with projected requirements over some planning horizon, must determine purchase levels from a selected set of take-or-pay contracts so as to minimize purchase, inventory holding, penalty costs, contract initiation, and buy-out or buy-down costs. This paper presents a mixed-integer programming model of take-or-pay decisions with and without make-up provisions.     

SCHEDULING PARALLEL PROCESSORS WITH SETUP COSTS AND RESOURCE LIMITATIONS

An integer linear programming model is presented for the scheduling of n products on m identical machines. The particular problem studied is one that occurs frequently in the fiberglass and textile industries. The model incorporates setup costs, lost production costs, and overtime costs. Due to the structure of the model, integer solutions can be obtained by explicitly restricting only a small number of the integer variables. This allows those responsible for scheduling to solve realistically sized problems in an efficient manner. Computational results are provided for a set of generated test problems.     

Branch-and-Bound Strategies for the Log Bucking Problem

Tree bucking is the initial production process in converting felled trees into useable wood products. This process has been previously modelled as a dynamic programming problem. Unlike other production problems that have been modelled as dynamic programming problems, there have been no serious attempts to formulate this problem as a branch-and-bound model and then examine the model's performance. This research develops the tree bucking problem as a branch-and-bound model to be tested by varying several parameters. The testing is performed in two phases: (1) a sensitivity analysis is performed to test two key parameters used by the model, and (2) branching strategies are tested on various problem scenarios. The size of the solution sets searched by the technique vary from as low as 40 to as many as 41,000 possible combinations.