Piecewise Linear Cost Functions Research Articles

Discrete intermodal freight transportation network design with route choice behavior of intermodal operators

We consider a discrete intermodal network design problem for freight transportation, in which the network planner needs to determine whether or not to build up or expand a link to minimize the total operating cost of carriers and hub operators under a general route choice model of intermodal operators. We formulate the problem as a mixed-integer nonlinear and non-convex program that involves congestion effects, piecewise linear cost functions, and a fixed-point constraint. We develop a series of relaxed and equivalent models to reduce the hardness of the problem and provide theoretical results to show the equivalences. We present two solution methods to solve the problem with one returning heuristic solutions and the other generating a globally optimal solution. We offer two numerical experiments to test the two solution algorithms and also shed light on their performance comparisons.

Accelerating NCUC Via Binary Variable-Based Locally Ideal Formulation and Dynamic Global Cuts

This paper presents tighter formulations and dynamic global cutting plane solution approaches for accelerating network-constrained unit commitment (NCUC) problems, as the computational bottleneck of NCUC problems is always a concern to independent system operators (ISOs)/regional transmission owners (RTOs). Specifically, this paper explores tighter mixed-integer linear programming (MILP)-based NCUC formulations, via locally ideal reformulation for piece-wise linear cost functions of generators and valid cover inequalities by linking system load balance/power flow constraints with generation capacity limits. The paper also discusses an enhanced solution approach by dynamically generating global cutting planes at infeasible/suboptimal nodes of the branch-and-cut (BAC) tree, which would prevent similar infeasible/suboptimal nodes from being explored repeatedly. Since the major computational complexity of NCUC is caused by the large number of binary UC variables, the proposed strategies, being directly related to binary UC variables, could quickly identify infeasible/suboptimal binary solutions, reduce the search space in the BAC tree, and in turn improve the computational performance. Numerical case studies show that the proposed approach could significantly reduce the overall computational time as compared to the traditional NCUC approach.

Piecewise Linear 비용함수의 최소화를 위한 가상 네트워크 매핑 알고리즘

Piecewise Linear 비용함수의 최소화를 위한 가상 네트워크 매핑 알고리즘

A Note on Appointment Scheduling with Piecewise Linear Cost Functions

We consider the problem of determining the optimal schedules for a given sequence of jobs on a single processor. The objective is to minimize the expected total cost incurred by job waiting and processor idling, where the job processing times are random variables. It is known in the prior literature that if the processing times are integers and the costs are linear functions satisfying a mild condition, then the problem can be solved in a polynomial number of expected cost evaluations. In this work, we extend the result to piecewise linear cost functions, which include many useful objective functions in practice. Our analysis explores the (hidden) dual network flow structure of the appointment scheduling problem and thus greatly simplifies that of prior work. We also find the number of samples needed to compute a near optimal solution when only some independent samples of processing times are known.

The two-dimensional vector packing problem with piecewise linear cost function

The two-dimensional vector packing problem with piecewise linear cost function (2DVPP-PLC) is a practical problem faced by a manufacturer of children׳s apparel that ships products using courier service. The manufacturer must ship a number of items using standard-sized cartons, where the cost of a carton quoted by the courier is determined by a piecewise linear function of its weight. The cost function is not necessarily convex or concave. The objective is to pack all given items into a set of cartons such that the total delivery cost is minimized while observing both the weight limit and volume capacity constraints. This problem is commonly faced by many manufacturers that ship products using courier service. We formulate the problem as an integer programming model. Since the 2DVPP-PLC generalizes the classical bin packing problem, it is more complex and challenging. Solving it directly using CPLEX is successful only for small instances. We propose a simple heuristic that is extremely fast and produces high-quality solutions for instances of practical size. We develop an iterative local search algorithm to improve the solution quality further. We generate two categories of test data that can serve as benchmark for future research.

Optimal and Compact Control Policies for Energy Storage Units With Single and Multiple Batteries

In this paper we investigate the use of energy storage units (ESUs) to reduce the average cost of power supply. Using a dynamic programming (DP) argument, we first show that the optimal control policy for a single-battery ESU, under piecewise linear cost functions, has a threshold structure that can be expressed and stored in a very compact form. We then extend the analysis to an ESU with multiple (heterogeneous) batteries, where DP is generally impractical because of the explosion of the computational complexity with the number of batteries. We hence define a simple method to map the optimal solution of a suitable single-battery ESU into an admissible policy for the multiple-battery ESU, whose complexity grows only linearly with the number of batteries. Extensive simulation results show that the proposed approach can yield significant cost reduction in comparison with other policies, under realistic users' power requests, and with different costs functions. Furthermore, the admissible policy for the multiple-battery ESU is shown to achieve almost optimal performance under realistic assumptions on the battery parameters.

A Minimax Distribution Free Procedure for a Multi-constraint Inventory Model with Deterministic Variable Lead Time

Nowadays, considering real environment limitations, the effect of some constraints such as storage space and budget in determining optimal policies of inventory/production systems is a momentous concern of inventory managers. Hence, here, a budget constraint on total inventory investment and a maximum permissible storage space constraint are added simultaneously to a stochastic continuous review inventory system. The piece-wise linear crashing cost function which is widely used in project management is considered in this paper. The study relaxes the assumption about the form of distribution function of lead time demand by only assuming the mean and standard deviation of lead time demand are known. A minimax distribution free procedure is developed for the constrained problem. This study also shows that the respective budget and storage space constrained inventory model to be minimized is jointly convex in the decision variables. A numerical example is given to illustrate the discussed model.

Markov Decision Processes with Threshold Based Piecewise Linear Optimal Policies

This letter investigates the structure of the optimal policy for a class of Markov decision processes (MDPs), having convex and piecewise linear cost function. The optimal policy is proved to have a piecewise linear structure that alternates flat and constant-slope pieces, resembling a staircase with tilted rises and as many steps (thresholds) as the breakpoints of the cost function. This particular structure makes it possible to express the policy in a very compact manner, particularly suitable to be stored in low-end devices. More importantly, the threshold-based form of the optimal policy can be exploited to reduce the computational complexity of the iterative dynamic programming (DP) algorithm used to solve the problem. These results apply to a rather wide set of optimization problems, typically involving the management of a resource buffer such as the energy stored in a battery, or the packets queued in a wireless node.

Designing a stochastic distribution network model under risk

In this paper, we present a stochastic supply chain network model under risk, in which three tiers of decision makers of suppliers, distribution centers (DCs), and customers seek to determine their optimal plans. Unlike other studies in the extant literature, we use financial risk measure to model and for the first time, a new methodology based on finance literature is used to optimize the conditional value-at-risk (CVaR) measure for the problem. In fact the aim of this paper is to show the benefits of considering CVaR to control the risks in the supply chain networks. Also in order to obtain the economics of scale, a piecewise linear transportation cost function is applied between suppliers and DCs. We formulate the problem as a convex mixed integer program. A two-phase (constructive and improvement) heuristic method is developed to solve the problem. In the constructive phase, an initial solution is built randomly, and in improvement phase, this solution is improved iteratively by using a hybrid algorithm combining Tabu search and simulated annealing methods. Numerical experiments demonstrate the practicability of considering CVaR in the model which is promising in the supply chain network area, and also the significant efficiency of the heuristic method in terms of CPU times.

The value of travel time variability with trip chains, flexible scheduling and correlated travel times

This paper extends the analysis of the value of mean travel time (VMTT) and day-to-day travel time variability (VTTV) from single, isolated trips to daily trip chains, considering the effects of flexibility in activity scheduling and within-day correlation of travel times. Using a multi-stage stochastic programming approach, we show that the VMTT and VTTV on a trip is conditional on the realized travel times on preceding trips, first through the arrival time to the preceding activity and second through the information provided about subsequent travel times. Analytical formulas for the VMTT and VTTV are obtained for two special cases with piecewise constant and linear marginal cost functions, respectively. With flexible scheduling, there is typically a cost associated with a positive correlation of travel times, arising from persistent deviations from typical travel demand or supply on a given day. However, there is also a strict benefit in the dependence since it allows for a more efficient scheduling of later trips.

Performance bounds for min-max uncertain constrained systems

We present a method to bound the performance of causal controllers for uncertain linear systems with mixed state and input constraints. The performance is measured by the worst-case value (over all possible disturbance inputs) of a convex piecewise linear cost function over a finite horizon. Our method computes a causal affine control policy that satisfies the constraints robustly, while simultaneously providing a lower bound on the achievable performance of this, or any other causal control policy constructed by any other method. The controller and performance guarantee can be found by solving a number of convex conic optimization problems which are closely connected.

MATPOWER: Steady-State Operations, Planning, and Analysis Tools for Power Systems Research and Education

MATPOWER is an open-source Matlab-based power system simulation package that provides a high-level set of power flow, optimal power flow (OPF), and other tools targeted toward researchers, educators, and students. The OPF architecture is designed to be extensible, making it easy to add user-defined variables, costs, and constraints to the standard OPF problem. This paper presents the details of the network modeling and problem formulations used by MATPOWER, including its extensible OPF architecture. This structure is used internally to implement several extensions to the standard OPF problem, including piece-wise linear cost functions, dispatchable loads, generator capability curves, and branch angle difference limits. Simulation results are presented for a number of test cases comparing the performance of several available OPF solvers and demonstrating MATPOWER's ability to solve large-scale AC and DC OPF problems.

Optimization Based Production Planning With Hybrid Dynamics and Constraints

Optimizing production planning has tremendous economic impact for many industrial production systems. In this paper, the planning problem of a class of production systems with hybrid dynamics and constraints is considered with practical background of power generation planning and other applications. The problem is solved within the Lagrangian relaxation framework, with the system wide demand and resource limit constraints relaxed by Lagrange multipliers. A new method is developed in this paper to obtain the exact optimal solutions to the subproblems with hybrid dynamics and constraints efficiently without discretizing the continuous production levels or introducing intermediate levels of relaxation. A novel definition of the discrete state associated with a consecutive time span is introduced so that solving each subproblem is converted into solving a number of continuous optimization problems and a discrete optimization problem separately. An efficient double dynamic programming (DP) method is developed to solve these subproblems and the principle of optimality is guaranteed for both the continuous and discrete problem. The production levels in a consecutive running span with non-convex piecewise linear cost functions are determined in a DP forward sweep without discretization. The DP method is then applied to determine the optimal discrete operating states across time efficiently. Numerical testing results demonstrate that the new method is efficient and effective for optimization based production planning with the complex hybrid dynamics and constraints.

Low-power and high-performance techniques in global interconnect signaling

In this article, two accurate and efficient approaches are proposed to optimize the power and delay of global interconnects in VLSI ICs. We modify the conventional buffer insertion and low swing methods for delay and power optimization of various lengths of the global interconnects. As such, we address non-equidistance buffer insertion (NEBI) and current-mode driver and receiver (CMDR) techniques along with our smart optimization procedure. It is shown that the optimized low swing CMDR technique is efficient for global interconnects of the length equal or longer than 5mm, and the improved buffer insertion technique, NEBI, is a perfect choice for the short global interconnects. Additionally, a random search algorithm known as simulated annealing (SA), improved by an intelligent method using a piecewise linear and exponential cost function, is employed for optimization of the power and delay. To this end, we have implemented a smart CAD tool that works interactively with HSPICE to achieve accurate and reliable design results. For verification purposes, several circuits are designed and simulated in 0.25, 0.18, and 0.13μm CMOS technologies. The simulation results verify a significant reduction in the power and delay of global interconnects compared to other methods in the literature.

Benders-based winner determination algorithm for volume discount procurement auctions

This paper considers an industrial procurement of large volume of a single good, such as a raw material or a subcomponent. The procurement is implemented using volume discount auctions, in which the bids are nonconvex piecewise linear cost functions. The winner determination problem faced by the buyer is an NP-hard nonlinear knapsack problem. We propose a mixed integer linear formulation for the problem which has the following primal structure: if the integer variables are fixed at some feasible values, then solving the remaining linear variables is a continuous knapsack problem. Exploiting this feature, Benders' decomposition algorithm is developed and various techniques are proposed to accelerate the convergence of the algorithm. Experimental results for the random test data show the efficacy of the accelerating techniques.

Optimal requirement allocation among quantity-discount quoting suppliers

We consider a problem motivated by a central purchasing organization for a major office products distributor. This purchasing organization must source a quantity of a particular resale item from a set of capacitated suppliers. In our case each supplier offers an incremental quantity discount purchase price structure. The purchaser’s objective is to obtain a quantity of a required item at minimum cost. The resulting problem is one of allocating order quantities among an approved supply base and involves minimizing the sum of separable piecewise linear concave cost functions. We develop a branch and bound algorithm that arrives at an optimal solution by generating linear knapsack subproblems with feasible solutions to the original problem.

Nonconvex, lower semicontinuous piecewise linear optimization

A branch-and-cut algorithm for solving linear problems with continuous separable piecewise linear cost functions was developed in 2005 by Keha et al. This algorithm is based on valid inequalities for an SOS2 based formulation of the problem. In this paper we study the extension of the algorithm to the case where the cost function is only lower semicontinuous. We extend the SOS2 based formulation to the lower semicontinuous case and show how the inequalities introduced by Keha et al. can also be used for this new formulation. We also introduce a simple generalization of one of the inequalities introduced by Keha et al. Furthermore, we study the discontinuities caused by fixed charge jumps and introduce two new valid inequalities by extending classical results for fixed charge linear problems. Finally, we report computational results showing how the addition of the developed inequalities can significantly improve the performance of CPLEX when solving these kinds of problems.

The price of anarchy of serial, average and incremental cost sharing

We compute the price of anarchy (PoA) of three familiar demand games, i.e., the smallest ratio of the equilibrium to efficient surplus, over all convex preferences quasi-linear in money. For any convex cost, the PoA is at least \(\frac{1}{n}\) in the average and serial games, where n is the number of users. It is zero in the incremental game for piecewise linear cost functions. With quadratic costs, the PoA of the serial game is \(\theta (\frac{1}{\log n})\) , and \(\theta (\frac{1}{n})\) for the average and incremental games. This generalizes if the marginal cost is convex or concave, and its elasticity is bounded.

A Case Study of Joint Online Truck Scheduling and Inventory Management for Multiple Warehouses

For a real-world problem—transporting pallets between warehouses to guarantee sufficient supply for known and additional stochastic demand—we propose a solution approach via convex relaxation of an integer programming formulation, suitable for online optimization. The essential new element linking routing and inventory management is a convex piecewise-linear cost function that is based on minimizing the expected number of pallets that still need transportation. For speed, the convex relaxation is solved approximately by a bundle approach yielding an online schedule in five to 12 minutes for up to three warehouses and 40,000 articles; in contrast, computation times of state-of-the-art LP solvers are prohibitive for online application. In extensive numerical experiments on a real-world data stream, the approximate solutions exhibit negligible loss in quality; in long-term simulations the proposed method reduces the average number of pallets needing transportation due to short-term demand to less than half the number observed in the data stream.

Comparative studies on dynamic programming and integer programming approaches for concave cost production/inventory control problems

This paper is concerned with classical concave cost multi-echelon production/inventory control problems studied by W. Zangwill and others. It is well known that the problem with m production steps and n time periods can be solved by a dynamic programming algorithm in O(n 4 m) steps, which is considered as the fastest algorithm for solving this class of problems. In this paper, we will show that an alternative 0–1 integer programming approach can solve the same problem much faster particularly when n is large and the number of 0–1 integer variables is relatively few. This class of problems include, among others problem with set-up cost function and piecewise linear cost function with fewer linear pieces. The new approach can solve problems with mixed concave/convex cost functions, which cannot be solved by dynamic programming algorithms.