A spreadsheet modelling approach to the assortment problem

A list of similar items of different sizes are required on a regular basis, but it is impractical to stock each of the different sizes. Demands for any size that is not stocked must be met by supplying the nearest acceptable size that is stocked, resulting in increased cost or trim wastage for example. The problem of determining which sizes should be stocked to minimise or maximise an appropriate objective function is formulated as a shortest or longest path problem on a directed acyclic network. A spreadsheet model is used to solve the problem, in such a way that showing precedents on the spreadsheet results in the basis tree for the shortest or longest path solution being drawn without the need for special software. The basis tree produced by this method is shown to be planar for practical applications, enabling improved efficiency of the algorithm used. Examples are given to illustrate the application of this approach.     

Consolidating Product Sizes to Minimize Inventory Levels for a Multi-Stage Production and Distribution System

The optimal design of a supply chain was approached in two phases by using: (1) a mathematical programming formulation and heuristic solution approach to minimize the distinct number of product types held at various points in the supply chain; and (2) a spreadsheet inventory model to estimate the safety stock needed to absorb random fluctuations in both demand and lead time throughout the system. This two-phased approach allowed management to quantify the effects of inventory required for locating parts of the supply chain in different geographic areas. The quantification of projected inventory requirements was a critical input used by senior management to clarify their final decision-making process.     

Investment timing and optimal capacity choice for small hydropower projects

This paper presents a method for assessing small hydropower projects that are subject to uncertain electricity prices. We present a real options-based method with continuous scaling, and we find that there is a unique price limit for initiating the project. If the current electricity price is below this limit it is never optimal to invest, but above this limit investment is made according to the function for optimal size. The connection between the real option and the physical properties of a small hydropower plant is dealt with using a spreadsheet model that performs a technical simulation of the production in a plant, based on all the important choices for such a plant. The main results of the spreadsheet are simulated production size and the investment costs, which are in turn used for finding the value of the real option and the price limit. The method is illustrated on three different Norwegian small hydropower projects.     

An optimal homotopy-analysis approach for strongly nonlinear differential equations

In this paper, an optimal homotopy-analysis approach is described by means of the nonlinear Blasius equation as an example. This optimal approach contains at most three convergence-control parameters and is computationally rather efficient. A new kind of averaged residual error is defined, which can be used to find the optimal convergence-control parameters much more efficiently. It is found that all optimal homotopy-analysis approaches greatly accelerate the convergence of series solution. And the optimal approaches with one or two unknown convergence-control parameters are strongly suggested. This optimal approach has general meanings and can be used to get fast convergent series solutions of different types of equations with strong nonlinearity.     

Identifying the efficiency status in network DEA

Data envelopment analysis (DEA) is commonly employed to evaluate the efficiency performance of a decision making unit (DMU) that transforms exogenous inputs into final outputs. In such a black-box DEA approach, details of an internal production process of the DMU are typically ignored and hence the locations of inefficiency are not adequately provided. In view of this, DEA researchers have recently developed various network approaches by looking into the black box, where the inputs that enter the box and the outputs that come out of it are only considered. However, most of these network approaches evaluate divisional efficiency by using an optimal solution of their respective optimization problem. If such an optimal solution is used in the case when there are multiple optima, then managerial guidance based on this solution alone may be inappropriate because more appropriate targets from the viewpoint of management may be ignored. Taking this fact into account, therefore, we propose a network approach for identifying the efficiency status of each DMU and its divisions. This approach provides a practical computational procedure.     

Instantaneous rate of change: a numerical approach

The calculus reform movement has encouraged numerical and graphical approaches to functions in addition to the more traditional analytical approach. While valiant efforts have been made to use these other approaches in newer calculus curricula, more numerical approaches should be introduced. Research on student learning in calculus indicates that particular numerical approaches hold promise for students' learning of instantaneous rate of change. Numerical approaches involving the average rate of change over successively smaller intervals can be used to obtain the instantaneous rate of change for a given function at a given value of x. These approaches can help students appreciate the fundamental relationship between average and instantaneous rates of change. They can also be used to obtain general expressions for the derivative of most elementary functions. Standard computer spreadsheet programs facilitate this process and make numerical approaches a more viable option for calculus instruction. These are underutilized resources for instruction in calculus, even in reform or other new calculus curricula.     

Error estimates for generalized barycentric interpolation

We prove the optimal convergence estimate for first-order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.     

Simple expressions for finding recovery system inventory control parameter values

In recent years considerable effort has been devoted to the development of inventory control models for joint manufacturing and remanufacturing. Optimality of control policies is analyzed and algorithms for the determination of parameter values have been developed. However, there is still a lack of formulae or algorithms that allow for an easy computation of optimal or near optimal policy parameter values. This paper addresses the problem of computing the produce-up-to level S and the remanufacture-up-to level M in a periodic review inventory control model. We provide simple formulae for the policy parameter values, which can easily be implemented within spreadsheet applications. The approach is to derive news-vendor-type formulae that are based on underage and overage cost considerations. We propose different formulae depending on whether lead times for production and remanufacturing are identical or not. A numerical study shows that the obtained solutions provide relatively small cost deviations compared to the optimal solution within the investigated class of inventory control policies.     

Allocation of Control Points in Stochastic Dynamic-Programming Models

The question of allocation of control points in stochastic dynamic-programming models from different areas is considered. A widely used approach is to choose equal time intervals between successive control points. Conditions under which this choice may cease to be optimal are discussed. Two different approaches to statement and solution of a production-control problem are presented.     

On Applied Nonlinear and Bilevel Programming or Pursuit-Evasion Games

Motivated by the benefits of discretization in optimal control problems, we consider the possibility of discretizing pursuit-evasion games. Two approaches are introduced. In the first approach, the solution of the necessary conditions of the continuous-time game is decomposed into ordinary optimal control problems that can be solved using discretization and nonlinear programming techniques. In the second approach, the game is discretized and transformed into a bilevel programming problem, which is solved using a first-order feasible direction method. Although the starting points of the approaches are different, they lead in practice to the same solution algorithm. We demonstrate the usability of the discretization by solving some open-loop representations of feedback solutions for a complex pursuit-evasion game between a realistically modeled aircraft and a missile, with terminal time as the payoff. The solutions are compared with those obtained via an indirect method.     

A surrogate and Lagrangian approach to constrained network problems

This paper presents the use of surrogate constraints and Lagrange multipliers to generate advanced starting solutions to constrained network problems. The surrogate constraint approach is used to generate a singly constrained network problem which is solved using the algorithm of Glover, Karney, Klingman and Russell [13]. In addition, we test the use of the Lagrangian function to generate advanced starting solutions. In the Lagrangian approach, the subproblems are capacitated network problems which can be solved using very efficient algorithms.The surrogate constraint approach is implemented using the multiplier update procedure of Held, Wolfe and Crowder [16]. The procedure is modified to include a search in a single direction to prevent periodic regression of the solution. We also introduce a reoptimization procedure which allows the solution from thekth subproblem to be used as the starting point for the next surrogate problem for which it is infeasible once the new surrogate constraint is adjoined.The algorithms are tested under a variety of conditions including: large-scale problems, number and structure of the non-network constraints, and the density of the non-network constraint coefficients.The testing clearly demonstrates that both the surrogate constraint and Langrange multipliers generate advanced starting solutions which greatly improve the computational effort required to generate an optimal solution to the constrained network problem. The testing demonstrates that the extra effort required to solve the singly constrained network subproblems of the surrogate constraints approach yields an improved advanced starting point as compared to the Lagrangian approach. It is further demonstrated that both of the relaxation approaches are much more computationally efficient than solving the problem from the beginning with a linear programming algorithm.     

Junction times in singular optimal control

Finding junction times on partially singular optimal control trajectories is in general difficult analytically and especially so when the optimal control is explicitly dependent on both the state and adjoint variables. This paper presents a new approach for linear problems whereby variations in the final state, measured from a reference optimal solution, can be used to find corresponding junction times. The approach is shown to be effective when applied to a problem which has an optimal solution involving a single bang/singular junction for certain choices of end state.     

Optimal Planning of a Multi-Station System with Sojourn Time Constraints

This paper studies a dynamic production system where multiple products must visit stations where inventories are constrained by maximum and minimum sojourn times with neither negative flow nor backlog being allowed. A resource availability constraint limits the aggregate throughput of the stations. The objective is to minimize the sum of flow and inventory cost. The problem is broken down into several single-product serial systems that serve as subroutines of a Lagrangian relaxation routine. This model is implemented in a spreadsheet so that it can be used by the officials of a Chilean institution for planning the operations and defining the optimal allocation of resources.     

A new MIP model for mine equipment scheduling by minimizing maintenance cost

Mining investment has been recognized as capital intensive due mainly to the cost of large equipment. Equipment capital costs for a given operation are usually within the order of hundreds of million dollars but may reach to billion dollars for large companies operating multiple mines. Such large investments require the optimum usage of equipment in a manner that the operating costs are minimized and the utilization of equipment is maximized through optimal scheduling. This optimum usage is required to ensure that the business remains sustainable and financially stable. Most mining operations utilize trucks to haul the mined material. Maintenance is one of the major operating cost items for these fleets as it can reach approximately one hundred million dollars yearly. There is no method or application in the literature that optimizes the utilization for truck fleet over the life of mine. A new approach based on mixed integer programming (MIP) techniques is used for annually scheduling a fixed fleet of mining trucks in a given operation, over a multi-year time horizon to minimize maintenance cost. The model uses the truck age (total hours of usage), maintenance cost and required operating hours to achieve annual production targets to produce an optimum truck schedule. While this paper focuses on scheduling trucks for mining operation, concept can be used in most businesses using equipment with significant maintenance costs. A case study for a large scale gold mine showed an annual discounted (10% rate) maintenance cost saving of over $2M and more than 16% ($21M) of overall maintenance cost reduction over 10 years of mine life, compared with the spreadsheet based approach used currently at the operation.     

Integrated revenue management approaches for capacity control with planned upgrades

In many service industries, firms offer a portfolio of similar products based on different types of resources. Mismatches between demand and capacity can therefore often be managed by using product upgrades. Clearly, it is desirable to consider this possibility in the revenue management systems that are used to decide on the acceptance of requests. To incorporate upgrades, we build upon different dynamic programming formulations from the literature and gain several new structural insights that facilitate the control process under certain conditions. We then propose two dynamic programming decomposition approaches that extend the traditional decomposition for capacity control by simultaneously considering upgrades as well as capacity control decisions. While the first approach is specifically suited for the multi-day capacity control problem faced, for example, by hotels and car rental companies, the second one is more general and can be applied in arbitrary network revenue management settings that allow upgrading. Both approaches are formally derived and analytically related to each other. It is shown that they give tighter upper bounds on the optimal solution of the original dynamic program than the well-known deterministic linear program. Using data from a major car rental company, we perform computational experiments that show that the proposed approaches are tractable for real-world problem sizes and outperform those disaggregated, successive planning approaches that are used in revenue management practice today.     

The crane scheduling problem: models and solution approaches

In this paper, we study the crane scheduling problem for a vessel after the vessel is moored on a terminal and develop both exact and heuristic solution approaches for the problem. For small-sized instances, we develop a time-space network flow formulation with non-crossing constraints for the problem and apply an exact solution approach to obtain an optimal solution. For medium-sized instances, we develop a Lagrangian relaxation approach that allows us to obtain tight lower bounds and near-optimal solutions. For large-sized instances, we develop two heuristics and show that the error bounds of our heuristics are no more than 100%. Finally, we perform computational studies to show the effectiveness of our proposed solution approaches.     

A library for continuous convex separable quadratic knapsack problems

The Continuous Convex Separable Quadratic Knapsack problem (CQKnP) is an easy but useful model that has very many different applications. Although the problem can be solved quickly, it must typically be solved very many times within approaches to (much) more difficult models; hence an efficient solution approach is required. We present and discuss a small open-source library for its solution that we have recently developed and distributed.     

Optimal cutting directions and rectangle orientation algorithm

The first stage in hierarchical approaches to Floorplan Design defines topological relations between components that intend to optimize a given objective in a circuit board. These relations determine a placement that is subsequently optimized in order to minimize a cost measurement (that will probably be one between chip area or perimeter). The board optimization gives rise to multiple subproblems that need to be answered in order to obtain a good solution. Among the most relevant ones we find the problem of defining the optimal orientation of cells and the definition of the optimal cutting sequence that minimize the placement board area. We will present a generalization of an algorithm due to Stockmeyer so that it obtains a solution that not only defines the optimal cell orientation but also the slicing cuts sequence that will lead to this optimal orientation and overall area minimization.     

Game theoretic modelling of service agent warranty fraud

When a warranty provider outsources warranty servicing to an external service agent this agent may act in a fraudulent manner. In this paper we consider a special case of service agent fraud—with the service agent overbilling the warranty provider for some of the warranty claims. A detailed inspection of a claim may be made to identify whether or not the service agent has committed fraud, but this inspection involves an additional cost to the warranty provider. This cost may be recovered by imposing a penalty on the service agent whenever a fraud is committed and it is detected. This penalty is specified in the maintenance service contract. A game theoretic approach is used to find the optimal overbilling strategy for the service agent and the optimal inspection strategy for the warranty provider. The optimal solution is the mixed strategy Nash equilibrium of a static game between the two parties.     

Fuzzy goal programming for multiobjective transportation problems

Several fuzzy approaches can be considered for solving multiobjective transportation problem. This paper presents a fuzzy goal programming approach to determine an optimal compromise solution for the multiobjective transportation problem. We assume that each objective function has a fuzzy goal. Also we assign a special type of nonlinear (hyperbolic) membership function to each objective function to describe each fuzzy goal. The approach focuses on minimizing the negative deviation variables from 1 to obtain a compromise solution of the multiobjective transportation problem. We show that the proposed method and the fuzzy programming method are equivalent. In addition, the proposed approach can be applied to solve other multiobjective mathematical programming problems. A numerical example is given to illustrate the efficiency of the proposed approach.