A generalized recursive algorithm for a class of non-stationary regeneration (scheduling) problems

Recent efforts in the field of dynamic programming have explored the feasibility of solving certain classes of integer programming problems by recursive algorithms. Special recursive algorithms have been shown to be particularly effective for problems possessing a 0–1 attribute matrix displaying the “nesting property” studied by, Ignall and Veinott in inventory theory and by Glover in network flows. This paper extends the class of problem structures that has been shown amenable to recursive exploitation by providing an efficient dynamic programming approach for a general transportation scheduling problem. In particular, we provide alternative formulations lor the scheduling problem and show how the most general of these formulations can be readily solved vis a vis recursive techniques.     

Minimizing logistic costs in multistage supply chains

We consider the problem in which a set of products has to be shipped from a common origin to a common destination through one or several intermediate nodes with the objective of minimizing the sum of inventory and transportation costs when a set of possible shipping frequencies is given on each link. From the theoretical point of view, the main issue is the computation of the inventory cost in the intermediate nodes. From the computational point of view, given that the simpler single link problem is known to be NP-hard, we present four classes of heuristic algorithms. The first two classes are based on the decomposition of the sequence in links, the third class on the adaptation of the EOQ-type solution known for the continuous case, and the fourth on the optimal solution of a simpler problem through dynamic programming techniques. Finally, we compare them on a set of randomly generated problem instances. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 399–417, 1999     

Parallel machine scheduling with batch deliveries to minimize total flow time and delivery cost

Motivated by the flow of products in the iron and steel industry, we study an identical and parallel machine scheduling problem with batch deliveries, where jobs finished on the parallel machines are delivered to customers in batches. Each delivery batch has a capacity and incurs a cost. The objective is to find a coordinated production and delivery schedule that minimizes the total flow time of jobs plus the total delivery cost. This problem is an extension of the problem considered by Hall and Potts, Ann Oper Res 135 (2005) 41–64, who studied a two‐machine problem with an unbounded number of transporters and unbounded delivery capacity. We first provide a dynamic programming algorithm to solve a special case with a given job assignment to the machines. A heuristic algorithm is then presented for the general problem, and its worst‐case performance ratio is analyzed. The computational results show that the heuristic algorithm can generate near‐optimal solutions. Finally, we offer a fully polynomial‐time approximation scheme for a fixed number of machines. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 492–502, 2016     

Mathematical control theory solution of an interactive accounting flows model

In this paper we have applied the mathematical control theory to the accounting network flows, where the flow rates are constrained by linear inequalities. The optimal control policy is of the “generalized bang-bang” variety which is obtained by solving at each instant in time a linear programming problem whose objective function parameters are determined by the “switching function” which is derived from the Hamiltonian function. The interpretation of the adjoint variables of the control problem and the dual evaluators of the linear programming problem demonstrates an interesting interaction of the cross section phase of the problem, which is characterized by linear programming, and the dynamic phase of the problem, which is characterized by control theory.     

The conjugate gradient technique for certain quadratic network problems

We consider a class of network flow problems with pure quadratic costs and demonstrate that the conjugate gradient technique is highly effective for large-scale versions. It is shown that finding a saddle point for the Lagrangian of an m constraint, n variable network problem requires only the solution of an unconstrained quadratic programming problem with only m variables. It is demonstrated that the number of iterations for the conjugate gradient algorithm is substantially smaller than the number of variables or constraints in the (primal) network problem. Forty quadratic minimum-cost flow problems of various sizes up to 100 nodes are solved. Solution time for the largest problems (4,950 variables and 99 linear constraints) averaged 4 seconds on the CBC Cyber 70 Model 72 computer.     

A network isolation algorithm

A set of edges D called an isolation set, is said to isolate a set of nodes R from an undirected network if every chain between the nodes in R contains at least one edge from the set D. Associated with each edge of the network is a positive cost. The isolation problem is concerned with finding an isolation set such that the sum of its edge costs is a minimum. This paper formulates the problem of determining the minimal cost isolation as a 0–1 integer linear programming problem. An algorithm is presented which applies a branch and bound enumerative scheme to a decomposed linear program whose dual subproblems are minimal cost network flow problems. Computational results are given. The problem is also formulated as a special quadratic assignment problem and an algorithm is presented that finds a local optimal solution. This local solution is used for an initial bound.     

A hybrid algorithm for the one machine sequencing problem to minimize total tardiness

In a recent paper, Hamilton Emmons has established theorems relating to the order in which pairs of jobs are to be processed in an optimal schedule to minimize the total tardiness of performing n jobs on one machine. Using these theorems, the algorithm of this paper determines the precedence relationships among pairs of jobs (whenever possible) and eliminates the first and the last few jobs in an optimal sequence. The remaining jobs are then ordered by incorporating the precedence relationships in a dynamic programming framework. Propositions are proved which considerably reduce the total computation involved in the dynamic programming phase. Computational results indicate that the solution time goes up less than linearly with the size (n) of the problem. The median solution time for solving 50 job problems was 0.36 second on UNIVAC 1108 computer.     

Generation and storage dispatch in electricity networks with generator disruptions

We present methods for optimizing generation and storage decisions in an electricity network with multiple unreliable generators, each colocated with one energy storage unit (e.g., battery), and multiple loads under power flow constraints. Our model chooses the amount of energy produced by each generator and the amount of energy stored in each battery in every time period in order to minimize power generation and storage costs when each generator faces stochastic Markovian supply disruptions. This problem cannot be optimized easily using stochastic programming and/or dynamic programming approaches. Therefore, in this study, we present several heuristic methods to find an approximate optimal solution for this system. Each heuristic involves decomposing the network into several single‐generator, single‐battery, multiload systems and solving them optimally using dynamic programming, then obtaining a solution for the original problem by recombining. We discuss the computational performance of the proposed heuristics as well as insights gained from the models. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 493–511, 2015     

Shortest path interdiction problem with arc improvement recourse: A multiobjective approach

We consider the shortest path interdiction problem involving two agents, a leader and a follower, playing a Stackelberg game. The leader seeks to maximize the follower's minimum costs by interdicting certain arcs, thus increasing the travel time of those arcs. The follower may improve the network after the interdiction by lowering the costs of some arcs, subject to a cardinality budget restriction on arc improvements. The leader and the follower are both aware of all problem data, with the exception that the leader is unaware of the follower's improvement budget. The effectiveness of an interdiction action is given by the length of a shortest path after arc costs are adjusted by both the interdiction and improvement. We propose a multiobjective optimization model for this problem, with each objective corresponding to a different possible improvement budget value. We provide mathematical optimization techniques to generate a complete set of strategies that are Pareto‐optimal. Additionally, for the special case of series‐parallel graphs, we provide a dynamic‐programming algorithm for generating all Pareto‐optimal solutions.     

Sequencing many jobs on a multi-purpose facility

Suppose a given set of jobs has to be processed on a multi-purpose facility which has various settings or states. There is a choice of states in which to process a job and the cost of processing depends on the state. In addition, there is also a sequence-dependent changeover cost between states. The problem is then to schedule the jobs, and pick an optimum setting for each job, so as to minimize the overall operating costs. A dynamic programming model is developed for obtaining an optimal solution to the problem. The model is then extended using the method of successive approximations with a view to handling large-dimensioned problems. This extension yields good (but not necessarily optimal) solutions at a significant computational saving over the direct dynamic programming approach.     

Bayesian selling problem with partial information

We introduce an optimal stopping problem for selling an asset when the fixed but unknown distribution of successive offers is from one of n possible distributions. The initial probabilities as to which is the true distribution are given and updated in a Bayesian manner as the successive offers are observed. After receiving an offer, the seller has to decide whether to accept the offer or continue to observe the next offer. Each time an offer is observed a fixed cost is incurred. We consider both the cases where recalling a past offer is allowed and where it is not allowed. For each case, a dynamic programming model and some heuristic policies are presented. Using simulation, the performances of the heuristic methods are evaluated and upper bounds on the optimal expected return are obtained. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

A note on the optimal scheduling of two parallel processors

The problem of minimizing mean flow time of two parallel processors is discussed. Prior results are briefly reviewed. A dynamic programming algorithm is presented which minimizes mean flow time for a set of n preordered jobs on two nonequivalent parallel processors. The algorithm is illustrated with an example problem. The computational experience is presented which illustrates the efficiency of the algorithm.     

Scheduling of depalletizing and truck loading operations in a food distribution system

This paper studies a scheduling problem arising in a beef distribution system where pallets of various types of beef products in the warehouse are first depalletized and then individual cases are loaded via conveyors to the trucks which deliver beef products to various customers. Given each customer's demand for each type of beef, the problem is to find a depalletizing and truck loading schedule that fills all the demands at a minimum total cost. We first show that the general problem where there are multiple trucks and each truck covers multiple customers is strongly NP‐hard. Then we propose polynomial‐time algorithms for the case where there are multiple trucks, each covering only one customer, and the case where there is only one truck covering multiple customers. We also develop an optimal dynamic programming algorithm and a heuristic for solving the general problem. By comparing to the optimal solutions generated by the dynamic programming algorithm, the heuristic is shown to be capable of generating near optimal solutions quickly. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Planning horizon procedures for machine replacement models with several possible replacement alternatives

This paper develops a forward algorithm and planning horizon procedures for an important machine replacement model where it is assumed that the technological environment is improving over time and that the machine-in-use can be replaced by any of the several different kinds of machines available at that time. The set of replacement alternatives may include (i) new machines with different types of technologies such as labor- and capital- intensive, (ii) used machines, (iii) repairs and/or improvements which affect the performance characteristics of the existing machine, and so forth. The forward dynamic programming algorithm in the paper can be used to solve a finite horizon problem. The planning horizon results give a procedure to identify the forecast horizon T such that the optimal replacement decision for the first machine based on the forecast of machine technology until period T remains optimal for any problem with horizon longer than T and, for that matter, for the infinite horizon problem. A flow chart and a numerical example have been included to illustrate the algorithm.     

Optimization of steerable sensor network for threat detection

A Markov chain approach to detecting a threat in a given surveillance zone by a network of steerable sensors is presented. The network has a finite number of predetermined states, and transition from one state to another follows a Markov chain. Under the assumption that the threat avoids detection, two game theoretic problems for finding an optimal Markov chain (two surveillance strategies) are formulated: the first maximizes the probability of threat detection for two consecutive detection periods, whereas the second minimizes the average time of detection for the worst‐case threat's trajectory. Both problems are reduced to linear programming, and special techniques are suggested to solve them. For a dynamic environment with moving noise sources, the optimal Markov chain changes at each detection period, and the rate of convergence of the Markov chain to its stationary distribution is analyzed. Both surveillance strategies are tested in numerical experiments and compared one with another. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Scheduling a maintenance activity on parallel identical machines

We study a problem of scheduling a maintenance activity on parallel identical machines, under the assumption that all the machines must be maintained simultaneously. One example for this setting is a situation where the entire system must be stopped for maintenance because of a required electricity shut‐down. The objective is minimum flow‐time. The problem is shown to be NP‐hard, and moreover impossible to approximate unless P = NP. We introduce a pseudo‐polynomial dynamic programming algorithm, and show how to convert it into a bicriteria FPTAS for this problem. We also present an efficient heuristic and a lower bound. Our numerical tests indicate that the heuristic provides in most cases very close‐to‐optimal schedules. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Dual-ascent methods for large-scale multicommodity flow problems

The capacitated multicommodity network flow problem presents itself in a number of problem contexts including transportation, communication, and production. To solve the large-scale multicommodity flow problems encountered in these fields, we develop dual-ascent heuristics and a primal solution generator. The dual-ascent solutions, in addition to determining lower bounds on the optimal objective function value, provide advanced starting solutions for use with primal-based solution techniques. The primal solution generator uses the dual-ascent solution to obtain heuristically primal solutions to the multicommodity flow problems. Computational experiments performed on three test problem sets show that the dual-ascent and primal heuristic procedures typically determine nearoptimal solutions quickly. In addition, by using the dual-ascent procedure to obtain advanced starting solutions, run times for optimal multicommodity flow procedures are reduced significantly and greatly improved solutions are obtained by the new primal solution generator. © 1993 John Wiley & Sons, Inc.     

A two product dynamic economic lot size production model with either-or production constraints

This paper considers the production of two products with known demands over a finite set of periods. The production and inventory carrying costs for each product are assumed to be concave. We seek the minimum cost production schedule meeting all demands, without backlogging, assuming that at most one of the two products can be produced in any period. The optimization problem is first stated as a nonlinear programming problem, which allows the proof of a result permitting the search for the optimal policy to be restricted to those which produce a product only when its inventory level is zero. A dynamic programming formulation is given and the model is then formulated as a shortest route problem in a specially constructed network.     

Algorithms for minimizing total cost,bottleneck time and bottleneck shipment in transportation problems

We consider the transportation problem of determining nonnegative shipments from a set of m warehouses with given availabilities to a set of n markets with given requirements. Three objectives are defined for each solution: (i) total cost, TC, (ii) bottleneck time, BT (i.e., maximum transportation time for a positive shipment), and (iii) bottleneck shipment, SB (i.e., total shipment over routes with bottleneck time). An algorithm is given for determining all efficient (pareto-optimal or nondominated) (TC, BT) solution pairs. The special case of this algorithm when all the unit cost coefficients are zero is shown to be the same as the algorithms for minimizing BT. provided by Szwarc and Hammer. This algorithm for minimizing BT is shown to be computationally superior. Transportation or assignment problems with m=n=100 average about a second on the UNIVAC 1108 computer (FORTRAN V)) to the threshold algorithm for minimizing BT. The algorithm is then extended to provide not only all the efficient (TC, BT) solution pairs but also, for each such BT, all the efficient (TC, SB) solution pairs. The algorithms are based on the cost operator theory of parametric programming for the transportation problem developed by the authors.     

Economic lot sizing with constant capacities and concave inventory costs

This article studies the classical single‐item economic lot‐sizing problem with constant capacities, fixed‐plus‐linear order costs, and concave inventory costs, where backlogging is allowed. We propose an O(T³) optimal algorithm for the problem, which improves upon the O(T⁴) running time of the famous algorithm developed by Florian and Klein (Manage Sci18 (1971) 12–20). Instead of using the standard dynamic programming approach by predetermining the minimal cost for every possible subplan, we develop a backward dynamic programming algorithm to obtain a more efficient implementation. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012