Unique Optimal Solution Research Articles

Closed-loop supply chain models with product remanufacturing under random demand

This study addresses the problem of choosing the appropriate closed-loop supply chain structure for the collection of used products from customers under random demand. We consider a two-echelon supply chain with three decision-making structures: (1) the manufacturer who coordinates with the retailer to obtain optimal channel profit, (2) the manufacturer who collects used products directly from customers, and (3) the retailer who collects used products from customers and the manufacturer who pays a transfer price to reacquire the products. We show that a unique optimal solution exists when the collection cost function satisfies the specified conditions under different models. We also identify differences among the three models and provide managerial guidelines for various marketing environments and collection cost functions. Surprisingly, contrary to previous studies, we find that the collection rate is not always largest under a centrally coordinated system compared with other making structures. As a special case, we investigate the effect of the coefficient of collection effort cost when cost structure is given.

Applying Social Choice Theory to Solve Engineering Multi-objective Optimization Problems

Multi-objective optimization problems usually do not have a single unique optimal solution, for either discrete or continuous domains. Furthermore, there are usually many possible available algorithms for solving these problems, and one typically does not know in advance which of these will be the most effective for solving a particular problem instance. Hyper-heuristics (HHs) are often used as a means to make this choice. In particular, the underlying idea of HHs is to run several algorithms or heuristics and dynamically decide, based on different criteria, which problem or part of the problem should be solved by which algorithm or heuristic. On the other hand, the domain of social choice theory studies how to design collective decision processes by aggregating individual inputs into collective ones. In this paper, we explore the use of social choice theory in creating HHs. By using HHs based on different voting methods, like Borda, Copeland and Kemeny–Young, we show how we can solve both continuous and discrete engineering multi-objective optimization problems and discuss the results obtained by each of these methods. Our obtained results show that our strategy has found solutions that are at least equals to the ones generated by the best algorithm among the studied ones, and sometimes even overcomes these results.

Numerical Optimization of a Fully Cross-Coupled Rectifier Circuit for Wireless Passive Ultra Low Power Sensor Nodes.

In the context of the Internet of Things, billions of devices—especially sensors—will be linked together in the next few years. A core component of wireless passive sensor nodes is the rectifier, which has to provide the circuit with sufficient operating voltage. In these devices, the rectifier has to be as energy efficient as possible in order to guarantee an optimal operation. Therefore, a numerical optimization scheme is proposed in this paper, which is able to find a unique optimal solution for an integrated Complementary Metal-Oxide-Semiconductor (CMOS) rectifier circuit with Self-Vth-Cancellation (SVC). An exploration of the parameter space is carried out in order to generate a meaningful target function for enhancing the rectified power for a fixed communication distance. In this paper, a mean conversion efficiency is introduced, which is a more valid target function for optimization than the Voltage Conversion Efficiency (VCE) and the commonly used Power Conversion Efficiency (PCE) and is defined as the arithmetic mean between PCE and VCE. Various trade-offs between output voltage, PCE, VCE and MCE are shown, which provide valuable information for low power rectifier designs. With the proposed method, a rectifier in a low power 55 nm process from Globalfoundries (GF55LPe) is optimized and simulated at −30 dBm input power. A mean PCE of 63.33% and a mean VCE of 63.40% is achieved.

Effects of a hybrid-price-stock dependent demand on the optimal solutions of a deteriorating inventory system and trade credit policy on re-manufactured product

Abstract This study contributes to formulate a deteriorating inventory model with a hybrid-price-stock dependent demand with credit for units in stock period in the presence of trade credit policy by using the discounted cash flow approach on re-manufactured products. In real life, the demand function is not always constant or linear or nonlinear. The demand rate of any items goes up and down in the real market due to the sum of two different types of demand. This study observes that the demand is depending on price and stock of the product, time value of money and trade credit policy. The objective of this study is to find optimum cycle time, optimum selling price and present profit of all future cash flows. The model starts with mathematical formulating and thereafter studies the necessary and sufficient conditions for obtaining the unique optimal solution. The theoretical solution procedure and case examples have studied for validation of this model. Moreover, the sensitivity analysis is expressed out with respect to main parameters and managerial application is obtained. This study uses the different demand functions of a simple pricing-inventory system to find out the maximum profit on re-manufactured products.

Single-machine lot scheduling problem for deteriorating items with negative exponential deterioration rate

Determining production-inventory level is one of the most important and challenging problems in a manufacturing system.Economic Batch Quantity(EBQ) is the simplest and the most employed inventory model in this field. Here, a single-source production lot-sizing model for deteriorating products with negative exponential deterioration rate is developed. In this manufacturing system, we assume that all items are manufactured using a single source. In this paper the optimal values for cycle length and lot size are derived in a way that the total cost (consists of machine setup cost, manufacturing cost, carrying and disposal costs of the deteriorating items) is minimized. Consequently, the developed model for the proposed problem is formulated. According to the derived mathematical model, we could not provide a closed form optimal solution, but we show that there is a unique optimal solution for the cycle length. At first, the upper and lower bounds of the optimal solution are determined, then using a Bi-section method (root-finding method) the problem is solved. To demonstrate the applicability of this method, we solve a simple example for a manufacturing system with three products. Finally, results of ten experiment that obtain from this method compare to answers from Newton-Raphson method, then it has been shown that this method has great effectiveness and less calculation efforts for the single-machine problem with deteriorating items.

Weak solution of the Novikov equation and optimal control

The aim of this paper is to discuss optimal distributed control problem governed by the Novikov equation. We first investigate the existence and uniqueness of weak solution for the controlled system. In contrast to the existing research result, the proof of solution mapping is local Lipschitz continuous that is one major novelty. Moreover, using the well-posedness result, we obtain a unique optimal control and optimal solution for the controlled system. At last, we also demonstrate the sufficient and necessary optimality condition of an optimal control by means of the optimal control theory, not limited to the necessary condition, which is another significant improvement of this paper.

Multi-Objective Model Updating Optimization Considering Orthogonality

The coordinate orthogonality check (CORTHOG) and multi-objective optimization considering pseudo-orthogonality as an objective function are introduced to overcome several limitations present in current model updating methods. It was observed that the use of the CORTHOG to remove four inaccurate degrees-of-freedom (DOF) was able to increase the orthogonality between mode shape vectors. The multi-objective model updating process generated a Pareto front with 38 unique optimal solutions. Four critical points were identified along the Pareto front, of which decreased the natural frequency error by greater than 2.84% and further increased the orthogonality between mode shape vectors. Therefore, it has been demonstrated that both steps of the methodology are critical to significantly reduce the overall errors of the system and to generate a finite element (FE) model that best describes physical reality. Additionally, the methodology introduced in this work generated a feasible computational runtime allowing for it to be easily adapted to widespread applications.

Payoff Optimization Through Wireless Network Virtualization for IoT Applications: A Three Layer Game Approach

Performance of wireless networks for Internet of Things (IoT) is suffering from exponential growth of lightweight hand-held and smart IoT devices as well as the data flow in the network. Static assignment of wireless resources is creating bottleneck to the further development of emerging applications, including IoT and cyber-physical systems. Wireless network virtualization through slicing of primary wireless resources is regarded as an emerging approach for enhancing overall network performance to solve the bottleneck problem for those emerging applications. In this paper, we study wireless virtualization where a three-layer game is formulated for wireless infrastructure providers (WIPs), mobile virtual network operators (MVNOs), and IoT devices (or end users). In the proposed three-layer game, players play sequentially to find out their optimal strategies. With the help of a controller, WIPs divide and reconfigure their RF bands to sublease frequency slices to MVNOs based on their service-level-agreement. MVNOs let end users use their subleased frequency slices. For the game, we present a unique optimal solution that facilitates a tradeoff between quality of service of end users, payoffs of MVNOs and payoff of WIPs. We also present a necessary condition and closed form for the existence of an equilibrium in the game. Furthermore, performance of the proposed approach is evaluated using numerical results which show that the payoffs of WIPs and MVNOs are maximized while offering competitive price for the subleased RF spectrum for their users.

Continuous-Time Distributed Subgradient Algorithm for Convex Optimization With General Constraints

The distributed convex optimization problem is studied in this paper for any fixed and connected network with general constraints. To solve such an optimization problem, a new type of continuous-time distributed subgradient optimization algorithm is proposed based on the Karuch–Kuhn–Tucker condition. By using tools from nonsmooth analysis and set-valued function theory, it is proved that the distributed convex optimization problem is solved on a network of agents equipped with the designed algorithm. For the case that the objective function is convex but not strictly convex, it is proved that the states of the agents associated with optimal variables could converge to an optimal solution of the optimization problem. For the case that the objective function is strictly convex, it is further shown that the states of agents associated with optimal variables could converge to the unique optimal solution. Finally, some simulations are performed to illustrate the theoretical analysis.

Feedback Capacity of Stationary Gaussian Channels Further Examined

It is well known that the problem of computing the feedback capacity of a stationary Gaussian channel can be recast as an infinite-dimensional optimization problem; moreover, necessary and sufficient conditions for the optimality of a solution to this optimization problem have been characterized, and based on this characterization, an explicit formula for the feedback capacity has been given for the case that the noise is a first-order autoregressive moving-average Gaussian process. In this paper, we further examine the above-mentioned infinite-dimensional optimization problem. We prove that unless the Gaussian noise is white, its optimal solution is unique, and we propose an algorithm to recursively compute the unique optimal solution, which is guaranteed to converge in theory and features an efficient implementation for a suboptimal solution in practice. Furthermore, for the case that the noise is a k-th order autoregressive moving-average Gaussian process, we give a relatively more explicit formula for the feedback capacity; more specifically, the feedback capacity is expressed as a simple function evaluated at a solution to a system of polynomial equations, which is amenable to numerical computation for the cases k=1, 2 and possibly beyond.

The Optimal Emission Decisions of Sustainable Production with Innovative Baseline Credit Regulations

In the era of the fourth industrial revolution, the international community is striving to establish a coordinated system to prevent fatal climate change in a global sense. As a result of such changes in business environments, a new issue, sustainability, has recently presented a paradigm shift and new research opportunity in which the theories and practices in traditional production and operations management are being reinterpreted and reapplied in relation to this emerging issue. Under this research background, we consider an optimal emission-trading problem under a cap-and-trade (CAT) emission regulation when the customers’ demand is given as an arbitrary probability distribution. Such a CAT approach to reduce the amount of emissions is a normative system for the sustainable production of manufacturing firms, which is also closely related to a well-known open innovation in literature of inventory management. Then, we formulate two stochastic inventory optimization models, which can be applied immediately for two famous CAT policies that exist in reality. In particular, our objective is to draw theoretical and practical implications for baseline credit emission regulations, which are innovative and government-led emission regulation policies, with a well-known newsvendor analysis. For our analytical results, we first show that our objective functions are piecewise linear and (quasi)-concave. Thus, it is found that there exists a unique optimal solution to the problem. Second, we successfully obtain the closed-form optimal solutions for the two models considered. Finally, we conduct a sensitivity analysis through a comparative static analysis to examine how the model parameters can affect the optimal solution in each model. All these analytical results and implications are consistent with previous studies in the literature, as well as with our insights for the models.

Minimum variance allocation among constrained intervals

We propose a weighted minimum variance allocation model, denoted by WMVA, which distributes an amount of a divisible resource as fairly as possible while satisfying all demand intervals. We show that the problem WMVA has a unique optimal solution and it can be characterized by the uniform distribution property (UDP in short). Based on the UDP property, we develop an efficient algorithm. Theoretically, our algorithm has a worst-case $$O(n^2)$$ complexity, but we prove that, subject to slight conditions, the worst case cannot happen on a 64-bit computer when the problem dimension is greater than 129. We provide extensive simulation results to support the argument and it explains why, in practice, our algorithm runs significantly faster than most existing algorithms, including many O(n) algorithms.

Convergence and correctness of belief propagation for the Chinese postman problem

Belief Propagation (BP), a distributed, message-passing algorithm, has been widely used in different disciplines including information theory, artificial intelligence, statistics and optimization problems in graphical models such as Bayesian networks and Markov random fields. Despite BP algorithm has a great success in many application fields and many progress about BP algorithm has been made, the rigorous analysis about the correctness and convergence of BP algorithm are known in only a few cases for arbitrary graph. In this paper, we will investigate the correctness and convergence of BP algorithm for determining the optimal solutions of the Chinese postman problem in both undirected and directed graphs. As a main result, we prove that BP algorithm converges to the optimal solution of the undirected Chinese postman problem within O(n) iterations where n represents the number of vertices, provided that the optimal solution is unique. For the directed case, we consider the directed Chinese postman problem with capacity and show that BP algorithm also converges to its optimal solution after O(n) iterations, as long as its corresponding linear programming relaxation has the unique optimal solution.

Optimal Risk-Sharing Across a Network of Insurance Companies

Risk transfer is a key risk and capital management tool for insurance companies. Transferring risk between insurers is used to mitigate risk and manage capital re- quirements. We investigate risk transfer in the context of a network environment of insurers and consider capital costs and capital constraints at the level of individual insurance companies. We demonstrate that the optimisation of profitability across the network can be achieved through risk transfer. Considering only individual in- surance companies, there is no unique optimal solution and, a priori, it is not clear which solutions are fair. However, from a network perspective, we derive a unique fair solution in the sense of cooperative game theory. Implications for systemic risk are briefly discussed.

Distributed Frequency Control With Operational Constraints, Part I: Per-Node Power Balance

This paper addresses the distributed optimal frequency control of multi-area power system with operational constraints, including the regulation capacity of individual control area and the power limits on tie-lines. Both generators and controllable loads are utilized to recover nominal frequencies while minimizing regulation cost. We study two control modes:the per-node balance mode and the network balance mode. In Part I of the paper, we only consider the per-node balance case, where we derive a completely decentralized strategy without the need for communication between control areas. It can adapt to unknown load disturbance. The tie-line powers are restored after load disturbance, while the regulation capacity constraints are satisfied both at equilibrium and during transient. We show that the closed-loop systems with the proposed control strategies carry out primal-dual updates for solving the associated centralized frequency optimization problems. We further prove the closed-loop systems are asymptotically stable and converge to the unique optimal solution of the centralized frequency optimization problems and their duals. Finally, we present simulation results to demonstrate the effectiveness of our design. In Part II of the paper, we address the network power balance case, where transmission congestions are managed continuously.

Optimal pricing and inventory strategies for introducing a new product based on demand substitution effects

This paper studies a single-period inventory-pricing problem with two substitutable products, which is very important in the area of Operations Management but has received little attention. The proposed problem focuses on determining the optimal price of the existing product and the inventory level of the new product. Inspired by practice, the problem considers various pricing strategies for the existing product as well as the cross elasticity of demand between existing and new products. A mathematical model has been developed for different pricing strategies to maximize the expected profit. It has been proven that the objective function is concave and there exists the unique optimal solution. Different sets of computational examples are conducted to show that the optimal pricing and inventory strategy generated by the model can increase profits.

A Common Set of Weights for Ranking Decision-Making Units with Undesirable Outputs: A Double Frontiers Data Envelopment Analysis Approach

Major drawbacks of the traditional data envelopment analysis (DEA) method include selecting optimal weights in a flexible manner, lacking adequate discrimination power for efficient decision-making units, and considering only desirable outputs. By introducing the concept of global efficiency optimization, this study proposed a double frontiers DEA approach with undesirable outputs to generate a common set of weights for evaluating all decision-making units from both the optimistic and pessimistic perspectives. For a unique optimal solution, compromise models for individual efficiency optimization were developed as a secondary goal. Finally, as an illustration, the models were applied to evaluate the energy efficiency of the Chinese regional economy. The results showed that the proposed approach could improve discrimination power and obtain a fair result in a case where both desirable and undesirable outputs exist.

Finitely forcible graph limits are universal

The theory of graph limits represents large graphs by analytic objects called graphons. Graph limits determined by finitely many graph densities, which are represented by finitely forcible graphons, arise in various scenarios, particularly within extremal combinatorics. Lovász and Szegedy conjectured that all such graphons possess a simple structure, e.g., the space of their typical vertices is always finite dimensional; this was disproved by several ad hoc constructions of complex finitely forcible graphons. We prove that any graphon is a subgraphon of a finitely forcible graphon. This dismisses any hope for a result showing that finitely forcible graphons possess a simple structure, and is surprising when contrasted with the fact that finitely forcible graphons form a meager set in the space of all graphons. In addition, since any finitely forcible graphon represents the unique minimizer of some linear combination of densities of subgraphs, our result also shows that such minimization problems, which conceptually are among the simplest kind within extremal graph theory, may in fact have unique optimal solutions with arbitrarily complex structure.

Quadratic convergence to the optimal solution of second-order conic optimization without strict complementarity

ABSTRACTUnder primal and dual nondegeneracy conditions, we establish the quadratic convergence of Newton's method to the unique optimal solution of second-order conic optimization. Only very few approaches have been proposed to remedy the failure of strict complementarity, mostly based on nonsmooth analysis of the optimality conditions. Our local convergence result depends on the optimal partition of the problem, which can be identified from a bounded sequence of interior solutions. We provide a theoretical complexity bound for identifying the quadratic convergence region of Newton's method from the trajectory of central solutions. By way of experimentation, we illustrate quadratic convergence of Newton's method on some SOCO problems which fail strict complementarity condition.

Unique (optimal) solutions: Complexity results for identifying and locating–dominating codes

We investigate the complexity of four decision problems dealing with the uniqueness of a solution in a graph: “Uniqueness of an r-Locating–Dominating Code with bounded size” (U-LDCr), “Uniqueness of an Optimal r-Locating–Dominating Code” (U-OLDCr), “Uniqueness of an r-Identifying Code with bounded size” (U-IdCr), “Uniqueness of an Optimal r-Identifying Code” (U-OIdCr), for any fixed integer r≥1.In particular, we describe a polynomial reduction from “Unique Satisfiability of a Boolean formula” (U-SAT) to U-OLDCr, and from U-SAT to U-OIdCr; for U-LDCr and U-IdCr, we can do even better and prove that their complexity is the same as that of U-SAT, up to polynomials. Consequently, all these problems are NP-hard, and U-LDCr and U-IdCr belong to the class DP.