Total Cost Function Research Articles

Analysis of screening decisions in inventory models with imperfect quality items

We consider an Economic Order Quantity (EOQ) model with imperfect quality items. All the items undergo a quality screening process for which both the cost and the defect detection rate depend on the time devoted to this operation. Contrary to most previous research in the field, the speed of screening is taken as a decision variable in the presented model jointly with the order quantity. Additionally, we introduce various functional forms for the screening cost and defect detection functions. In particular, nonlinear forms lead to nonconvexities in the total cost function consisting of multiple cost components, e.g. inspection cost, penalty costs, etc. We solve the model analytically for a selected set of assumptions and provide numerical examples and sensitivity analysis. The results provide interesting insights. While lower screening costs motivate longer screening times, the effectiveness of this operation is only obtained in combination with the reliability of the defect detection method. It is also observed that reducing return rates and improving customer satisfaction can be achieved by adjusting the operational parameters, without a significant increase in system cost or new investments.

Nonconvex Haar-TV denoising

The anisotropic total variation (TV) denoising model suppresses noise for two-dimensional signals that are vertically and horizontally piecewise constant. However, two-dimensional signals may have sparse derivatives in other directions. We propose a modification of the classical anisotropic two-dimensional TV regularizer from a spectral point of view. In the frequency domain, the TV regularizer can be considered as penalizing the high-frequency component of original signals and promoting only low-frequency components. The classical anisotropic TV, which applies l1-norm on vertical and horizontal differences, suppresses high-frequency components of the signals. The proposed operator, named Haar total variation (Haar-TV), penalizes two-dimensional signals that have more varied high-frequency regions. Furthermore, we propose non-convex penalties based on the Haar-TV operator since non-convex penalties can preserve edges and thus enhance the quality of the estimation. We derive a condition that preserves the strong convexity of the total cost function so the global minimizer can be reached.

Minimization of Total Pumping Cost from an Aquifer to a Water Tank, Via a Pipe Network

In this paper, we have investigated minimization of total cost to pump a given flow rate from any number (n) of wells up to a water tank, under steady-state flow conditions. Regarding groundwater flow, we have considered infinite or semi-infinite aquifers, to which the method of images applies. Additional regional groundwater flow can be taken into account, too. The pipe network connecting the wells to the tank can include junctions at the locations of the wells only. Moreover, all pumps have equal efficiency. We have derived a new analytical formula, which holds at the critical points of the total cost function. Based on this formula, we derived a system of n equations and n unknowns, to calculate the well flow rate combinations which correspond to the critical points of the total cost function. The n-1 equations are 2nd degree polynomials, while the remaining one is linear, expressing the constraint that the sum of well flow rates must be equal the required total flow rate. The solution of the system can be achieved using commercial solvers. Moreover, we have concluded that there is one feasible solution that minimizes the total cost. Finally, we present a tabulation process to facilitate the use of solvers and we provide and discuss two illustrative examples.

E2E QoS Guarantee for the Tactile Internet via Joint NFV and Radio Resource Allocation

The Tactile Internet (TI) is one of the next generation wireless network services with end to end (E2E) delay as low as 1 ms. Since this ultra low E2E delay cannot be met in the current 4G network architecture, it is necessary to investigate this service in the next generation wireless network by considering new technologies such as networks function virtualization (NFV). On the other hand, given the importance of E2E delay in the TI service, it is crucial to consider the delay of all parts of the network, including the radio access part and the NFV core part. In this paper, for the first time, we investigate the joint radio resource allocation (R-RA) and NFV resource allocation (NFV-RA) in a heterogeneous network where queuing delays, transmission delays, and delays resulting from virtual network function (VNF) execution are jointly considered. For this setup, we formulate a new resource allocation (RA) problem to minimize the total cost function subject to guaranteeing E2E delay of each connection. Since the proposed optimization problem is highly non-convex, we exploit alternative search method (ASM), successive convex approximation (SCA), and heuristic algorithms to solve it. Besides, for the NFV-RA, we propose an online heuristic algorithm, and analyze its performance for the TI service. Simulation results reveal that the proposed scheme can significantly reduce the network costs compared to the case where the two problems are optimized separately. Moreover, we compare the online algorithm with its offline counterpart as well as a baseline approach and it is shown that the online algorithm outperforms both of them.

A simple derivation of the optimal solution for the EOQ model for deteriorating items with planned backorders

We consider the Economic Order Quantity (EOQ) model for exponentially deteriorating items in which the items deteriorate at a rate proportional to the inventory level per unit time, which leads to an exponentially decreasing inventory level function over time. The coexistence of the exponential and polynomial terms in the total cost function makes closed-form exact solutions impossible, so approximations are used. Methods that optimize management science models without derivatives have recently been used to provide insights to managers who may not have a background in calculus. We propose such an approach for the deteriorating items inventory model with and without planned backorders and compare it with an existing one. Our derivation is much simpler and succinct; our closed-form solution is more intuitive than the existing one. We conduct numerical experiments to analyze the accuracy of the closed-form solution.

State-of-the-Art of Research on Optimization of Shell and Tube Heat Exchangers by Methods of Evolutionary Computation

This paper presents a comprehensive state-of-the-art review of optimization by evolutionary computation methods of shell and tubes heat exchangers (STHE) of single segmental baffles. It is seen that the heat transfer coefficient to the shell side is calculated by the Kern method or by the Bell Delaware method, and that the pressure drop to the shell side is calculated by the Kern method or by the Bell Delaware method or by the Peters and Timmershaus method. It is verified the use of evolutionary computation algorithms in single-objective and multiobjective optimization of STHE, and that most related work used the total cost function of STHE. And among the contributions presented, there were implementations and applications of different evolutionary computation algorithms, or study of the STHE optimization problem by different objective functions, such as ecological, entransy dissipation, field synergy, and cost of the life cycle. Also, it was possible to verify some gaps in the work related to STHE optimization, among them, can be cited the need for greater application of decision-making methods that relate technical, economic and managerial aspects applied to the solution of the STHE. And it is suggested for future work the development of the STHE model subject to delimited uncertainties; and the formalization of a standard STHE optimization problem, so that it can serve as a reference for comparing the different optimization algorithms, and applying metrics to evaluate the advantages that are obtained when using a new algorithm for the same problem.

QuateRA: The Quaternion Regression Algorithm

This work proposes a batch solution to the problem of estimating fixed angular velocity using orientation measurements. Provided that the angular velocity remains constant with time, it is shown that the orientation quaternion belongs to a constant plane of rotation as time evolves. Motivated by this fundamental property, the angular velocity’s direction can be determined by estimating the quaternion plane of rotation. Under the small angle assumption on the attitude measurement noise, the plane of rotation is estimated by minimizing a constrained total least-squares cost function, and the proposed algorithm produces a unique optimizing solution through a batch approach (no need for iterations). The angular velocity magnitude is estimated by projecting the measured quaternions onto the estimated plane of rotation, and then computing the least-squares evolution of the quaternion angle in the plane. A Monte Carlo analysis of the proposed algorithm is performed, validating the method and comparing it with a multiplicative extended Kalman Filter, which is a traditional method in the literature.

Reinforcement Learning-Based Optimal Tracking Control of an Unknown Unmanned Surface Vehicle.

In this article, a novel reinforcement learning-based optimal tracking control (RLOTC) scheme is established for an unmanned surface vehicle (USV) in the presence of complex unknowns, including dead-zone input nonlinearities, system dynamics, and disturbances. To be specific, dead-zone nonlinearities are decoupled to be input-dependent sloped controls and unknown biases that are encapsulated into lumped unknowns within tracking error dynamics. Neural network (NN) approximators are further deployed to adaptively identify complex unknowns and facilitate a Hamilton-Jacobi-Bellman (HJB) equation that formulates optimal tracking. In order to derive a practically optimal solution, an actor-critic reinforcement learning framework is built by employing adaptive NN identifiers to recursively approximate the total optimal policy and cost function. Eventually, theoretical analysis shows that the entire RLOTC scheme can render tracking errors that converge to an arbitrarily small neighborhood of the origin, subject to optimal cost. Simulation results and comprehensive comparisons on a prototype USV demonstrate remarkable effectiveness and superiority.

The residual time approach for (Q, r) model under perishability, general lead times, and lost sales

We consider a (Q, r) perishable inventory system with state-dependent compound Poisson demands with a random batch size, general lead times, exponential shelf times, and lost sales. We assume $$r<Q$$ and analyze the system using an embedded Markov process at the replenishment points. Using the queueing and Markov chain decomposition approach, we characterize the distribution of the residual lead time and derive the stationary distribution of the inventory level. We construct closed-form expressions for the expected total long-run average cost function. The closed form allows us to efficiently obtain, numerically, the optimal Q and r parameters. Numerical study provides several guidelines for the optimal control. For example, we show that approximating the lead time distribution by an exponential one only works when the optimal reorder point of the approximation is very small; in other cases the usage of the exact distribution can lead to substantial cost savings (up to 14%). We further provide intuition insight on the optimal controls and how they depend on different factors, e.g., the lead time variability, and the demand features (arrival rate, size and variability).

A note about ‘on replenishment schedule for deteriorating items with time-proportional demand’

In the Economic Order Quantity (EOQ) model for exponentially deteriorating items, the inventory deterioration rate is proportional to the inventory level, which leads to an exponentially decreasing inventory level over time. Due to polynomial and exponential terms in the total cost function, an exact closed form solution is not possible. Previously, an approximation for the total cost function and a closed form optimal solution for the order interval have been proposed for this problem. However, the proposed closed form formula is complicated and it is not intuitive. It is also assumed without justification that the average inventory level is equal to half of the order quantity. We justify this assumption and show that the same approximation yields a much more intuitive formula when it is used to determine the optimal order quantity instead of the order interval. Our intuitive optimal order quantity formula is more likely to be adopted in practice.

Multiproduct Economic Lot Scheduling Problem with Returns and Sorting Line

This work studies a hybrid manufacturing–remanufacturing system with a sorting line and disposal. In particular, it models a company that collects used product, remanufactures returned products that have been evaluated as suitable to be recovered, and manufactures new products to satisfy customer demand. Specifically, the system is modeled as a multilevel inventory system, with three types of stock (used products inventory, recoverable inventory, and serviceable inventory), each characterized by an inventory holding cost, and three limited capacity resources: a sorting line, which enables the company to distinguish those returns that are remanufacturable from those that are not; a remanufacturing line to carry out operations on sorted remanufacturable returns; and a manufacturing line to produce new products in order to satisfy customer demand. Each resource is characterized by a setup cost, as well as a constant production rate, while each type of stock is associated with an inventory holding cost. The aim of the paper is to develop a model for the considered production system in order to minimize the setup and inventory holding costs. In particular, the objective is to evaluate the behavior of a controllable disposal rate with the minimization of the total cost function, by considering the effect on the remanufacturing and manufacturing lines.

Optimization of Pressure-Swing Distillation for iC5-Methanol Azeotropic Mixture Purification

Typical industrial technologies to produce tert-amyl methyl ether (TAME), a gasoline oxygenated additive, require some advanced purification process for methanol recovery from the reactor effluent, due to the presence of the iC5-methanol minimum boiling point azeotrope. Pressure-swing distillation (PSD) is widely used as an efficient method for separating pressure-sensitive azeotropic mixtures in industrial processes. However, it is well known that distillation units are responsible for the highest energy consumption in chemical separation facilities, reaching in some cases to 95% of the total energy cost. Thus, the objective of this work was to perform an optimization methodology of the main design and operational variables of the PSD process applied to the separation of the azeotropic iC5-methanol mixture, considering minimizing the objective function of total annual cost (TAC). For this purpose, Honeywell’s UniSim Design® Suite software was used to simulate the process flowchart and Matlab® was used to perform TAC optimization. The optimized PSD design revealed potential for saving of up to 12% per year in terms of TAC when compared to the non-optimized process, thus proving the feasibility of the optimization methodology for this specific process. Furthermore, it reinforces the main objective of the optimization technique that is to make chemical processes more economically competitive in the industrial sector.

Optimal Strategies for CLSC considering Supply Disruption and Carbon Tax

Under the background of economic globalization, supply chain is becoming more and more complex, which is manifested in the instability of external environment. On the one hand, with the improvement of global environmental protection awareness, the government's policy tools for environmental impact (carbon tax) on the whole supply chain have become one of the major external problems faced by the supply chain enterprises; on the other hand, the intensification of competition between upstream and downstream in supply chains makes supply disruption an important proposition to be solved urgently. In this paper, the two propositions of green and supply disruption are reduced to two factors affecting the cost. The average total cost function of the manufacturer as a recycler is established. The practicability of the algorithm and the effectiveness of the model are verified by Lingo, Particle Swarm Optimization, and Genetic Algorithm, with the purpose of obtaining the optimal strategies for manufacturers who play the role of the recycler in the closed-loop supply chain.

Dynamic Path Planning of the UAV Avoiding Static and Moving Obstacles

This paper introduces a dynamic path planning method for the UAV that can avoid both static and moving obstacles. The condition with sudden threats can better reflect the real situation of the UAV in the real environment. First of all, the A* algorithm is adopted to generate an optimal path in a known environment in this method. Then, in the situation of static sudden threats, a series of candidate paths are generated by the principle of cubic spline second-order continuity. In order to make the static sudden threat at the center of a cluster of candidate paths, they need to be adjusted. After that, this path cluster completely surrounds the sudden threat and has symmetry about the sudden threat. When encountering a sudden threat of movement, factors such as the speed, acceleration and certain parameters of the movement obstacle or the UAV are considered, and a correlation model of the dynamic sudden threat is established. Finally, the total cost function is established to select the optimal obstacle avoidance path, and the total cost function contains four sub-cost functions, they are static security cost function, smoothness cost function, consistency cost function and dynamic security cost function. The simulation results demonstrate the effectiveness of the proposed method.

Event-Triggering Interaction Scheme for Discrete-Time Decentralized Optimization With Nonuniform Step Sizes.

In this article, we study the discrete-time decentralized optimization problems of multiagent systems by an event-triggering interaction scheme, in which each agent privately knows its local convex cost function, and collectively minimizes the total cost functions. The underlying interaction and the corresponding weight matrix are required to be undirected connected and doubly stochastic, respectively. To resolve this optimization problem collaboratively, we propose a decentralized event-triggering algorithm (DETA) that is based on the consensus theory and inexact gradient tracking technique. DETA involves each agent interacting with its neighboring agents only at some independent event-triggering sampling time instants. Under the assumptions that the global convex cost function is coercive and has Lipschitz continuous gradient, we prove that DETA steers all agents' states to an optimal solution even with nonuniform constant step sizes. Moreover, our analysis also shows that DETA converges at a rate of O(1/√t) if the step sizes are uniform and do not exceed some upper bounds. We illustrate the effectiveness of DETA on a canonical simple decentralized parameter estimation problem.

Canadian Regional Agriculture Model: PMP Applied to the Beef Sector in Canada

This paper describes the Positive Mathematical Programming (PMP), the method for calibrating models of agricultural livestock production and resource use by a nonlinear total cost function. The PMP method is applied to agricultural sectoral models to study changes in policy and market signals. The Canadian Regional Agriculture Model (CRAM) is a regional, multi-sectoral, comparative static, partial equilibrium, mathematical programming model developed and maintained by Agriculture and Agri-Food Canada (AAFC) since mid-eighties. The PMP process converts a linear model using flexibility constraints into a nonlinear model in the absence of the flexibility constraints. A component of CRAM is the beef sector. The elements of the set of total cost curves are defined as quadratic function in terms of the number of cows and calves in the beef production activities. The marginal cost curves were then approximated using the shadow values from linear programming solution with linear curves. Once the flexibility constraints were removed, the model automatically calibrates to the base year production levels. The results from four scenarios indicated the beef sector of CRAM could predict the impact of the scenarios on the size of beef herd. In Scenario 1 where cash costs were increased by 10 percent, the breeding herd size decreased from 3.73 percent in New Brunswick to 0.0 percent in Ontario and Quebec. In Scenario 2 where barley costs were decreased by 10 percent, the breeding herd size increased from 0 percent for British Columbia, Alberta, Ontario, Quebec, Prince Edward Island and Nova Scotia to 1.93 percent for New Brunswick. In Scenario 3 where carcass weight per beef cow could be increased by 10 percent, the increase in beef herd size ranged from 0 percent for Ontario and Quebec to 2.56 percent for New Brunswick. In Scenario 4 where world beef prices were increased by 10 percent increase in beef herd size ranged from 4.48 percent for Manitoba to 25.78 percent for New Brunswick.

Imperfect Production Inventory Model with Uncertain Elapsed Time

Most of the classical inventory control model assumes that all items received conform to quality characteristics. However, in practice, items may be damaged due to production conditions, transportation and environmental conditions. Modelling such real world problems involve various indeterminate phenomena which can be estimated through human beliefs. The uncertainty theory proposed by Liu (2015) is extensively regarded as an appropriate tool to deal with such uncertainty. This paper investigates the optimum production run time and optimum cost in an imperfect production process, where the rate of imperfect items are different in different states of the process. The process may be shifting from ‘in-control’ state to the ‘out-of-control’ state is an uncertain variable with certain uncertainty distribution. Some propositions are derived for the optimal production run time and optimized the expected total cost function per unit time. Finally, numerical examples have been illustrated to study the practical feasibility of the model.

Canadian Regional Agriculture Model: Positive Mathematical Programming Applied to the Beef Sector in Canada

This paper describes the Positive Mathematical Programming (PMP), the method for calibrating models of agricultural livestock production and resource use using a nonlinear total cost function. The PMP method is applied to agricultural sectoral models to study changes in policy and market signals. The Canadian Regional Agriculture Model (CRAM) is a regional, multi-sectoral, comparative static, partial equilibrium, mathematical programming model developed and maintained by Agriculture and Agri-Food Canada (AAFC) since mid-eighties. The PMP process converts a linear model using flexibility constraints into a nonlinear model in the absence of the flexibility constraints. A component of CRAM is the beef sector. The elements of the set of total cost curves are defined as quadratic function in terms of the number of cows and calves in the beef production activities. The marginal cost curves are then approximated using the shadow values from linear programming solution with linear curves. Once the flexibility constraints were removed, the model automatically calibrates to the base year production levels. The results from four scenarios indicated the beef sector of CRAM could predict the impact of the scenarios on the size of beef herd. In Scenario 1 where cash costs were increased by 10 percent, the breeding herd size decreased from 3.73 percent in New Brunswick to 0.0 percent in Ontario and Quebec. In Scenario 2 where barley costs were decreased by 10 percent, the breeding herd size increased from 0 percent for British Columbia, Alberta, Ontario, Quebec, Prince Edward Island and Nova Scotia to 1.93 percent for New Brunswick. In Scenario 3 where carcass weight per beef cow could be increased by 10 percent, the increase in beef herd size ranged from 0 percent for Ontario and Quebec to 2.56 percent for New Brunswick. In Scenario 4 where world beef prices were increased by 10 percent increase in beef herd size ranged from 4.48 percent for Manitoba to 25.78 percent for New Brunswick.

Canadian Regional Agriculture Model: Positive Mathematical Programming Applied to the Beef Sector in Canada

This paper describes the Positive Mathematical Programming (PMP), the method for calibrating models of agricultural livestock production and resource use using a nonlinear total cost function. The PMP method is applied to agricultural sectoral models to study changes in policy and market signals. The Canadian Regional Agriculture Model (CRAM) is a regional, multi-sectoral, comparative static, partial equilibrium, mathematical programming model developed and maintained by Agriculture and Agri-Food Canada (AAFC) since mid-eighties. The PMP process converts a linear model using flexibility constraints into a nonlinear model in the absence of the flexibility constraints. A component of CRAM is the beef sector. The elements of the set of total cost curves are defined as quadratic function in terms of the number of cows and calves in the beef production activities. The marginal cost curves are then approximated using the shadow values from linear programming solution with linear curves. Once the flexibility constraints were removed, the model automatically calibrates to the base year production levels. The results from four scenarios indicated the beef sector of CRAM could predict the impact of the scenarios on the size of beef herd. In Scenario 1 where cash costs were increased by 10 percent, the breeding herd size decreased from 3.73 percent in New Brunswick to 0.0 percent in Ontario and Quebec. In Scenario 2 where barley costs were decreased by 10 percent, the breeding herd size increased from 0 percent for British Columbia, Alberta, Ontario, Quebec, Prince Edward Island and Nova Scotia to 1.93 percent for New Brunswick. In Scenario 3 where carcass weight per beef cow could be increased by 10 percent, the increase in beef herd size ranged from 0 percent for Ontario and Quebec to 2.56 percent for New Brunswick. In Scenario 4 where world beef prices were increased by 10 percent increase in beef herd size ranged from 4.48 percent for Manitoba to 25.78 percent for New Brunswick.

Fast evaluation of a periodic review inventory policy

We consider a single-echelon inventory system facing stochastic demand controlled by the standard (r,nQ,T) batch ordering policy. While an exact algorithm for optimizing all three policy variables exists, excessive computational requirements hinder industrial applications, particularly in multi-product environments. Aiming at reducing computation time, we revisit the exact total average cost function. After establishing a new analytical property, we show that the exact ordering cost can be very accurately approximated by two convex functions. Using these, the original problem is decomposed in two separate sub-problems easier to solve. For each sub-problem we establish properties linking each solution with the classical EOQ, which lead to a fast heuristic search algorithm for the policy variables evaluation. Numerical comparisons with both the exact algorithm and two popular meta-heuristics demonstrate the excellent performance of the heuristic in terms of solution quality and run-time.