Precedence-constrained Knapsack Problem Research Articles

A Precedence Constrained Knapsack Problem with Uncertain Item Weights for Personalized Learning Systems

This paper studies a unique precedence constrained knapsack problem in which there are two methods available to place an item in the knapsack. Whether or not an item weight is uncertain depends on which one of the two methods is selected. This knapsack problem models students’ decisions on choosing subjects to study in hybrid personalized learning systems in which students can study either under teacher supervision or in an unsupervised self-study mode by using online tools. We incorporate the uncertainty in the problem using a chance-constrained programming framework. Under the assumption that uncertain item weights are independently and normally distributed, we focus on the deterministic reformulation in which the capacity constraint involves a nonlinear and convex function of the decision variables. By using the first-order linear approximations of this function, we propose an exact cutting plane method that iteratively adds feasibility cuts. To supplement this, we develop novel approximate cutting plane methods that converge quickly to high-quality feasible solutions. To improve the computational efficiency of our methods, we introduce new pre-processing procedures to eliminate items beforehand and cover cuts to refine the feasibility space. Our computational experiments on small and large problem instances show that the optimality gaps of our approximate methods are very small overall, and that they are even able to find solutions with no optimality gaps as the number of items increases in the instances. Moreover, our experiments demonstrate that our pre-processing methods are particularly effective when the precedence relations are dense, and that our cover cuts may significantly speed up our exact cutting plane approach in challenging instances.

A recursive time aggregation-disaggregation heuristic for the multidimensional and multiperiod precedence-constrained knapsack problem: An application to the open-pit mine block sequencing problem

A recursive time aggregation-disaggregation (RAD) heuristic is proposed to solve large-scale multidimensional and multiperiod precedence-constrained knapsack problems (MMPKP) in which a profit is maximized by filling the knapsack in multiple periods while satisfying minimum and maximum resource consumption constraints per period as well as precedence constraints between items. An important strategic planning application of the MMPKP in the mining industry is the well-known open-pit mine block sequencing problem (BSP). In the BSP, a mine is modeled as a three-dimensional grid of blocks to determine a block extraction sequence that maximizes the net present value while satisfying constraints on the shape of the mine and resource consumption over time. Large real-life instances of this problem are difficult to solve, particularly with lower bounds on resource consumption. The advantage of the time aggregation-disaggregation heuristic over a rolling-horizon-based time decomposition is twofold: first, the entire horizon is considered for the resource consumption from the first aggregation; and second, only two-period subproblems have to be solved. This method is applied to a well-known integer programming model and a variant thereof in which blocks can be extracted in parts over multiple periods. Tests on benchmark instances show that near-optimal solutions for both of the models can be obtained for extremely large instances with up to 2,340,142 blocks and 10 periods.

An efficient algorithm for the precedence constraint knapsack problem with reference to large-scale open-pit mining pushback design

ABSTRACT In this paper, a new Specific Breakpoint Algorithm (SBA), which can efficiently search appropriate breakpoints of parametric maximum-flow-related problems, is presented. The algorithm is used to solve Lagrangian Relaxed Precedence Constrained Knapsack Problem (LRPCKP) and Linear Programming Relaxed Precedence Constrained Knapsack Problem (LPRPCKP) in mine pushback design. The relaxed solutions are then processed through Rounded Topo-Sort (RoTS) heuristic to produce feasible solutions. The study results on seven bench mark datasets on Minelib for two approaches, referred here as LRPCKP-SBA and LPRPCKP-SBA, indicate that LRPCKP-SBA in spite of being faster, produces inferior quality solutions than well known BZ and CPLEX solutions. However, LPRPCKP-SBA produces a comparable quality of solutions as BZ in a computationally more efficient manner. Furthermore, the RoTS heuristics operated on relaxed solutions produce a better quality of feasible solutions than an existing technique, Expected Topo-Sort heuristic (ExTS).

Packet-size aware scheduling algorithms in guard band for time sensitive networking

As an emerging and promising technology, Time Sensitive Networking (TSN) can be widely used in many real-time systems such as Industrial Internet of Things (IIoT) and Cyber Physical System (CPS). TSN, while ensuring the bounded latency and jitter, exhibits the disadvantage of not being able to efficiently use the bandwidth resources in the guard band. In this paper, we propose an algorithm family named Packet-size Aware Shaping (PAS), which is inspired by abstracting the problem of utilizing the guard band to a classic Precedence-Constrained Knapsack Problem (PCKP). PAS works with the existing TSN standards, having achieved the goal of guaranteeing the end-to-end latency for scheduled time-sensitive applications while fully utilizing the available bandwidth in the guard band for others. Furthermore, we have proposed and implemented several hardware designs for both the current standard TSN scheduler and the programmable one. The simulation results show that the PAS family can achieve satisfying performance in maximizing the resource utilization in the guard band. The synthesis results on Xilinx Vivado show that our proposed Multi-group Push-In-First-Out (MPIFO) scheduler can achieve 100 Mpps scheduling rate for 1024 scheduling items, which is fast enough to support the high-speed TSN.

A methodology for the large-scale multi-period precedence-constrained knapsack problem: an application in the mining industry

This paper considers a generalization of the precedence-constrained knapsack problem known as multi-period precedence-constrained knapsack, in which the decision maker faces a horizon of several periods. Associated with each period is a capacity limit that cannot be exceeded by items chosen in that specific period. The motivation for studying this problem comes from a recognized problem in the mining industry, known as open pit mine production scheduling. An old, yet fast sequencing heuristic has been used in the literature to tackle similar combinatorial problems with precedence constraints. In this study, we first strengthen the LP relaxation formulation of the problem by adding inequalities derived from both precedence and knapsack constraints, and then use the LP solutions to generate efficient weights for the sequencing heuristic. Generating the heuristic's weights in this way significantly improves its output. Using this methodology, extremely large instances can be solved to near-optimum levels in minutes. The performance of this methodology is tested on a set of benchmark instances in the mining industry, where this problem is a major application.

Primal–Dual Algorithms for Precedence Constrained Covering Problems

A covering problem is an integer linear program of type $$\min \{c^Tx\mid Ax\ge D,\ 0\le x\le d,\ x \in \mathbb {Z}\}$$min{cTxźAxźD,0≤x≤d,xźZ} where $$A\in \mathbb {Z}^{m\times n}_+$$AźZ+m×n, $$D\in \mathbb {Z}_+^m$$DźZ+m, and $$c,d\in \mathbb {Z}_+^n$$c,dźZ+n. In this paper, we study covering problems with additional precedence constraints $$\{x_i\le x_j \ \forall j\preceq i \in \mathcal {P}\}$${xi≤xjźjźiźP}, where $$\mathcal {P}=([n], \preceq )$$P=([n],ź) is some arbitrary, but fixed partial order on the items represented by the column-indices of A. Such precedence constrained covering problems (PCCPs) are of high theoretical and practical importance even in the special case of the precedence constrained knapsack problem, that is, where $$m=1$$m=1 and $$d\equiv 1$$dź1. Our main result is a strongly-polynomial primal---dual approximation algorithm for PCCP with $$d\equiv 1$$dź1. Our approach generalizes the well-known knapsack cover inequalities to obtain an IP formulation which renders any explicit precedence constraints redundant. The approximation ratio of this algorithm is upper bounded by the width of $$\mathcal {P}$$P, that is, by the size of a maximum antichain in $$\mathcal {P}$$P. Interestingly, this bound is independent of the number of constraints. We are not aware of any other results on approximation algorithms for PCCP on arbitrary posets $$\mathcal {P}$$P. For the general case with $$d\not \equiv 1$$dź1, we present pseudo-polynomial algorithms. Finally, we show that the problem does not admit a PTAS under standard complexity assumptions.

The precedence constrained knapsack problem: Separating maximally violated inequalities

We consider the problem of separating maximally violated inequalities for the precedence constrained knapsack problem. Though we consider maximally violated constraints in a very general way, special emphasis is placed on induced cover inequalities and induced clique inequalities. Our contributions include a new partial characterization of maximally violated inequalities, a new safe shrinking technique, and new insights on strengthening and lifting. This work follows on the work of Boyd (1993), Park and Park (1997), van de Leensel et al. (1999) and Boland et al. (2011). Computational experiments show that our new techniques and insights can be used to significantly improve the performance of cutting plane algorithms for this problem.

Tailored Lagrangian Relaxation for the open pit block sequencing problem

A common strategic and tactical decision in open pit mining is to determine the sequence of extraction for notional three-dimensional production blocks so as to maximize the net present value of the extracted orebody while adhering to precedence and resource constraints. This problem is commonly formulated as an integer program with a binary variable representing if and when a block is extracted. In practical applications, the number of blocks can be large and the time horizon can be long, and therefore, instances of this NP-hard precedence-constrained knapsack problem can be difficult to solve using an exact approach. The problem is even more challenging to solve when it includes explicit minimum resource constraints. We employ three methodologies to reduce solution times: (i) we eliminate variables which must assume a value of 0 in the optimal solution; (ii) we use heuristics to generate an initial integer feasible solution for use by the branch-and-bound algorithm; and (iii) we employ Lagrangian relaxation, using information obtained while generating the initial solution to select a dualization scheme for the resource constraints. The combination of these techniques allows us to determine near-optimal solutions more quickly than solving the monolith, i.e., the original problem. We demonstrate our techniques to solve instances containing 25,000 blocks and 10 time periods, and 10,000 blocks and 15 time periods, to near-optimality.

A New Algorithm for the Open-Pit Mine Production Scheduling Problem

For the purpose of production scheduling, open-pit mines are discretized into three-dimensional arrays known as block models. Production scheduling consists of deciding which blocks should be extracted, when they should be extracted, and what to do with the blocks once they are extracted. Blocks that are close to the surface should be extracted first, and capacity constraints limit the production in each time period. Since the 1960s, it has been known that this problem can be cast as an integer programming model. However, the large size of some real instances (3–10 million blocks, 15–20 time periods) has made these models impractical for use in real planning applications, thus leading to the use of numerous heuristic methods. In this article we study a well-known integer programming formulation of the problem that we refer to as C-PIT. We propose a new decomposition method for solving the linear programming relaxation (LP) of C-PIT when there is a single capacity constraint per time period. This algorithm is based on exploiting the structure of the precedence-constrained knapsack problem and runs in O (mn log n) in which n is the number of blocks and m a function of the precedence relationships in the mine. Our computations show that we can solve, in minutes, the LP relaxation of real-sized mine-planning applications with up to five million blocks and 20 time periods. Combining this with a quick rounding algorithm based on topological sorting, we obtain integer feasible solutions to the more general problem where multiple capacity constraints per time period are considered. Our implementation obtains solutions within 6% of optimality in seconds. A second heuristic step, based on local search, allows us to find solutions within 3% in one hour on all instances considered. For most instances, we obtain solutions within 1–2% of optimality if we let this heuristic run longer. Previous methods have been able to tackle only instances with up to 150,000 blocks and 15 time periods.

Clique-based facets for the precedence constrained knapsack problem

We consider a knapsack problem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in manufacturing and mining, and also appears as a subproblem in decomposition techniques for network design and related problems. We present a new approach for determining facets of the PCKP polyhedron based on clique inequalities. A comparison with existing techniques, that lift knapsack cover inequalities for the PCKP, is also presented. It is shown that the clique-based approach generates facets that cannot be found through the existing cover-based approaches, and that the addition of clique-based inequalities for the PCKP can be computationally beneficial, for both PCKP instances arising in real applications, and applications in which PCKP appears as an embedded structure.

Large-scale multi-period precedence constrained knapsack problem: A mining application

We study an extension of the precedence constrained knapsack problem where the knapsack can be filled in multiple periods. This problem is known in the mining industry as the open-pit mine production scheduling problem. We present a new algorithm for solving the LP relaxation of this problem and an LP-based heuristic to obtain feasible solutions. Computational experiments show that we can solve real mining instances with millions of items in minutes, obtaining solutions within 6% of optimality.

A pegging approach to the precedence-constrained knapsack problem

The knapsack problem (KP) is generalized to the case where items are partially ordered through a set of precedence relations. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be packed in the knapsack. However, each item can be accepted only when all the preceding items have been included in the knapsack. The knapsack problem with these additional constraints is referred to as the precedence-constrained knapsack problem (PCKP). To solve PCKP exactly, we present a pegging approach, where the size of the original problem is reduced by applying the Lagrangian relaxation followed by a pegging test. Through this approach, we are able to solve PCKPs with thousands of items within a few minutes on an ordinary workstation.

Heuristic and Exact Algorithms for the Precedence-Constrained Knapsack Problem

The knapsack problem (KP) is generalized taking into account a precedence relation between items. Such a relation can be represented by means of a directed acyclic graph, where nodes correspond to items in a one-to-one way. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be included in the knapsack. However, each item can be adopted only when all of its predecessors have been included in the knapsack. The knapsack problem with such an additional set of constraints is referred to as the precedence-constrained knapsack problem (PCKP). We present some dynamic programming algorithms that can solve small PCKPs to optimality, as well as a preprocessing method to reduce the size of the problem. Combining these, we are able to solve PCKPs with up to 2000 items in less than a few minutes of CPU time.

Lifting valid inequalities for the precedence constrained knapsack problem

This paper considers the precedence constrained knapsack problem. More specifically, we are interested in classes of valid inequalities which are facet-defining for the precedence constrained knapsack polytope. We study the complexity of obtaining these facets using the standard sequential lifting procedure. Applying this procedure requires solving a combinatorial problem. For valid inequalities arising from minimal induced covers, we identify a class of lifting coefficients for which this problem can be solved in polynomial time, by using a supermodular function, and for which the values of the lifting coefficients have a combinatorial interpretation. For the remaining lifting coefficients it is shown that this optimization problem is strongly NP-hard. The same lifting procedure can be applied to (1,k)-configurations, although in this case, the same combinatorial interpretation no longer applies. We also consider K-covers, to which the same procedure need not apply in general. We show that facets of the polytope can still be generated using a similar lifting technique. For tree knapsack problems, we observe that all lifting coefficients can be obtained in polynomial time. Computational experiments indicate that these facets significantly strengthen the LP-relaxation.

Lifting cover inequalities for the precedence-constrained knapsack problem

This paper considers the polyhedral structure of the precedence-constrained knapsack problem, which is a knapsack problem with precedence constraints imposed on the set of variables. The problem itself appears in many applications. Moreover, since the precedence constraints appear in many important integer programming problems, the polyhedral results can be used to develop cutting-plane algorithms for more general applications. In this paper, we propose a modification of the cover inequality, which explicitly considers the precedence constraints. A combinatorial, easily implementable lifting procedure of the modified cover inequality is given. The procedure can generate strong cuts very easily. We also propose an additional lifting procedure, which is a generalization of the lifting procedure for cover inequalities. Some properties of the lifted inequality are analyzed and the problem of finding an optimal order of lifting is also addressed.

Polyhedral results for the precedence-constrained knapsack problem

A problem characteristic common to a number of important integer programming problems is that of precedence constraints: a transitive collection of constraints of the form x j ≤ x i with 0 ≤ x i ≤ 1, 0 ≤ x j ≤ 1, x i , x j integer. Precedence constraints are of interest both because they arise frequently in integer programming applications and because the convex hull of feasible integer points is the same as the region obtained by relaxing the integrality restrictions. This paper investigates the polyhedral structure of the convex hull of feasible integer points when the precedence constraints are complicated by an additional constraint or, more generally, by additional constraints defining an independence system. Sequential lifting for independence systems with precedence constraints is discussed and extensions of the cover and 1-configuration inequalities known for the knapsack polytope are presented. A general procedure for inducing facets called rooting is introduced and a variety of facets based on rooting are developed. The paper concludes by discussing a procedure for coalescing constraints related to minimal covers into facets.