Continuous Knapsack Research Articles

Robust knapsack ordering for a partially-informed newsvendor with budget constraint

This paper studies a robust version of the multi-item newsvendor problem with limited budget. The demand distribution belongs to an ambiguity set that contains all distributions that share the same range, mean and mean absolute deviation. The resulting optimization problem turns out to be solvable by a method reminiscent of the greedy algorithm for continuous knapsack problems, purchasing items in order of marginal effect on the total cost until the budget is spent.

The max-sum inverse median location problem on trees with budget constraint

The theory of inverse location involves modifying parameters in such a way that the total cost is minimized and one/several prespecified facilities become optimal based on these perturbed parameters. When the modifying parameters are grouped into sets, with each group's cost measured under the rectilinear norm and the overall cost measured under the Chebyshev norm, the resulting problem is known as the max-sum inverse location problem. This paper addresses the max-sum inverse median location problem on trees with a budget constraint, where the objective is to modify the vertex weights so that a specified vertex becomes a 1-median, while minimizing the max-sum objective within the available budget. To solve this problem, a uni-variable optimization problem is first induced, where the objective function for each specified value of the variable can be obtained through a continuous knapsack problem. Leveraging the monotonicity of the cost function, a combinatorial algorithm is developed, which solves the problem in O(nlog⁡n) time, where n denotes the number of vertices present in the tree.

Job scheduling for maximum revenue on uniform, parallel machines with major and minor setups and job splitting

We consider a revenue maximization scheduling problem where jobs can be split on parallel, uniform machines. By assuming that only major and minor setups exist between jobs and that these can be grouped into families, we reformulate the scheduling problem as a continuous knapsack problem with setup time. We then create the Revenue Rate Heuristic to solve the maximum revenue scheduling problem for both the partial and binary revenue cases. This makes use of a revenue rate – the ratio between total revenue that can be obtained from a job and the time required to achieve the revenue. The Revenue Rate Heuristic schedules the job with the highest revenue rate first and updates all the revenue rates accordingly based on existing scheduled families and remaining capacity. In cases where there are no setup times or if there is only one machine and an ordering of job revenue and production time, we are able to show that the Revenue Rate Heuristic is optimal. Numerical studies show that the Revenue Rate Heuristic performs well with an average optimality gap of 3.47% across a large set of parameters and when the setup times across families are different from each other. In a case study based on firm-level data, we also show that the average optimality gap remains small.

Cardinality-Constrained Continuous Knapsack Problem with Concave Piecewise-Linear Utilities

Cardinality-Constrained Continuous Knapsack Problem with Concave Piecewise-Linear Utilities

The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower’s Objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible packing maximizing his own profit. The leader’s aim is to optimize a linear objective function in the capacity and in the follower’s solution, but with respect to different item values. We address a stochastic version of this problem where the follower’s profits are uncertain from the leader’s perspective, and only a probability distribution is known. Assuming that the leader aims at optimizing the expected value of her objective function, we first observe that the stochastic problem is tractable as long as the possible scenarios are given explicitly as part of the input, which also allows to deal with general distributions using a sample average approximation. For the case of independently and uniformly distributed item values, we show that the problem is #P-hard in general, and the same is true even for evaluating the leader’s objective function. Nevertheless, we present pseudo-polynomial time algorithms for this case, running in time linear in the total size of the items. Based on this, we derive an additive approximation scheme for the general case of independently distributed item values, which runs in pseudo-polynomial time.

A Lagrangian heuristic for the multicommodity capacitated location problem with balancing requirements

We consider the multicommodity capacitated location problem with balancing requirements. This problem has applications in transportation and was introduced in the context of planning the movements of containers between ports and terminals, identified as depots in the problem formulation. We propose a Lagrangian relaxation based on a reformulation of the problem that allows the subproblem to be decomposed by depot, giving rise to a series of continuous knapsack subproblems. We optimize the Lagrangian dual by a bundle method and we develop a Lagrangian heuristic based on slope scaling. Numerical results are presented on a set of large-scale benchmark instances. We obtain good quality solutions within short computing times using the Lagrangian heuristic approach.

The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower’s problem. More precisely, adopting the robust optimization approach and assuming that the follower’s profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower’s reaction from the leader’s perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader’s objective function.

The sum of root-leaf distance interdiction problem by upgrading edges/nodes on trees.

Network interdiction problems by upgading critical edges/nodes have important applications to reduce the infectivity of the COVID-19. A network of confirmed cases can be described as a rooted tree that has a weight of infectious intensity for each edge. Upgrading edges (nodes) can reduce the infectious intensity with contacts by taking prevention measures such as disinfection (treating the confirmed cases, isolating their close contacts or vaccinating the uninfected people). We take the sum of root-leaf distance on a rooted tree as the whole infectious intensity of the tree. Hence, we consider the sum of root-leaf distance interdiction problem by upgrading edges/nodes on trees (SDIPT-UE/N). The problem (SDIPT-UE) aims to minimize the sum of root-leaf distance by reducing the weights of some critical edges such that the upgrade cost under some measurement is upper-bounded by a given value. Different from the problem (SDIPT-UE), the problem (SDIPT-UN) aims to upgrade a set of critical nodes to reduce the weights of the edges adjacent to the nodes. The relevant minimum cost problem (MCSDIPT-UE/N) aims to minimize the upgrade cost on the premise that the sum of root-leaf distance is upper-bounded by a given value. We develop different norms to measure the upgrade cost. Under weighted Hamming distance, we show the problems (SDIPT-UE/N) and (MCSDIPT-UE/N) are NP-hard by showing the equivalence of the two problems and the 0–1 knapsack problem. Under weighted l_1 norm, we solve the problems (SDIPT-UE) and (MCSDIPT-UE) in O(n) time by transforimg them into continuous knapsack problems. We propose two linear time greedy algorithms to solve the problem (SDIPT-UE) under unit Hamming distance and the problem (SDIPT-UN) with unit cost, respectively. Furthermore, for the the minimum cost problem (MCSDIPT-UE) under unit Hamming distance and the problem (MCSDIPT-UN) with unit cost, we provide two O(nlog n) time algorithms by the binary search methods. Finally, we perform some numerical experiments to compare the results obtained by these algorithms.

The Continuous Knapsack Problem with Capacities

We address a variant of the continuous knapsack problem, where capacities regarding costs of items are given into account. We prove that the problem is NP-complete although the classical continuous knapsack problem is solvable in linear time. For the case that there exists exactly one capacity for all items, we solve the corresponding problem in $$O(n\log n)$$ time, where n is the number of items.

Characterization of the optimal solution of the convex separable continuous knapsack problem and related problems

A separable convex continuous knapsack problem with a single equality constraint and bounded variables is considered in this paper. Necessary and sufficient condition (characterization) for a feasible solution to be an optimal solution to this problem is stated, and characterization theorem in terms of a relaxed problem is formulated and proved. Versions of this problem with a single inequality constraint of the form “less than or equal to” and “greater than or equal to” are also considered, and sufficient conditions for solving these problems are stated and proved, based on the characterization theorem for the original problem. Examples of some convex separable objective functions for the considered problems are presented.

A polynomial algorithm for a continuous bilevel knapsack problem

In this note, we analyze a bilevel interdiction problem, where the follower’s program is a parametrized continuous knapsack. Based on the structure of the problem and an inverse optimization strategy, we propose for its solution an algorithm with worst-case complexity O(n2).

A novel approach based on relaxation and reduction to solve the capacitated lot sizing problem

Capacitated lot sizing problem (CLSP) finds its applications in manufacturing, production as-well-as process industries. Typical size of the real life problems is much larger compared to the sizes that can be handled by commercial solvers, as CLSP is an NP-hard problem. Researchers have attempted different variants of CLSP in last few decades and provided many classical, exact, heuristic and metaheuristic solution approaches. In this paper, we attempt to solve the multi item capacitated dynamic lot sizing problem with backorders and setup times (CLSPBS). We apply Lagrangian relaxation on the material balance constraint, to reduce the problem into several single constraint continuous knapsack problems with upper bound on the real variables. Bounded variable linear program determines the values of real variables. For the negative or positive value of coefficients in the Lagrangian function, we fix the setup variables at zero or one respectively. Thus, the reduced problem is solved in polynomial time, as the main effort required is only sorting of numbers. We use the set up variable that are at one, to develop yet another good upper bound. Branch-and-bound procedure benefits immensely by using these bounds to make a warm start.

An approach to forming dashboards for business processes state analysis

There have been considered basic features of dashboards, their place and role in business process management concept, considered basic dashboards types, considered various recommendations of dashboards construction, and also considered basic visualization tools such as bar and line graphs, pie and scatter charts, bullet graphs and dials. An approach to formation of dashboards, used for the analysis of product supply business process state, has been proposed. Therefore, a set of performance indicators and a related system of scales have been defined. In order to solve the problem of dashboard design which means definition of number, type and place of visualization tools, mathematical models of unbounded and continuous knapsack problems have been applied. As a result of the proposed approach application, a prototype of a dashboard used for the analysis of product supply business process state has been developed.

Using Lagrangian Relaxation for Radio Resource Allocation in High Altitude Platforms

In this paper, we study radio resource allocation for multicasting in OFDMA based high altitude platforms (HAPs). We formulate and solve an optimization problem that finds the best allocation of HAP resources such as radio power, sub-channels, and time slots. The problem also finds the best possible frequency reuse across the cells that constitute the service area of the HAP. The objective is to maximize the number of user terminals that receive the requested multicast streams in the HAP service area in a given OFDMA frame. A bounding subroutine in a branch and bound algorithm can be obtained by decomposing it into two easier subproblems, due to its high complexity, and solving them iteratively. Subproblem 1 turns out to be a binary integer linear program of no explicitly noticeable structure and therefore Lagrangian relaxation is used to dualize some constraints to get a structure that is easy to solve. Subproblem 2 turns out to be a linear program with a continuous knapsack problem structure. Hence a greedy algorithm is proposed to solve subproblem 2 to optimality. The subgradient method is used to solve for the dual variables in the dual problem to get the tightest bounds.

On the solution of multidimensional convex separable continuous knapsack problem with bounded variables

A minimization problem with a convex separable objective function subject to linear equality constraints and box constraints (bounds on the variables) is considered. Necessary and sufficient optimality condition is proved for a feasible solution to be an optimal solution to this problem. Primal-dual analysis is also included. Examples of some convex separable objective functions for the considered problem are presented.

On the coupled continuous knapsack problems: projection onto the volume constrained Gibbs N-simplex

Coupled continuous quadratic knapsack problems (CCK) are introduced inthepresentstudy.ThesolutionofaCCKproblemisequivalenttotheprojectionofan arbitrary point onto the volume constrained Gibbs N-simplex, which has a wide range of applications in computational science and engineering. Three algorithms have been developed in the present study to solve large scale CCK problems. According to the numerical experiments of this study, the computational costs of presented algorithms scale linearly with the problem size, when it is sufficiently large. Moreover, they show competitive or even superior computational performance compared to the advanced QP solvers. The ease of implementation and linear scaling of memory usage with the problem size are the other benefits of the presented algorithms.

Continuous knapsack sets with divisible capacities

We study two continuous knapsack sets $$\mathcal {Y}_\ge $$ and $$\mathcal {Y}_\le $$ with $$n$$ integer, one unbounded continuous and $$m$$ bounded continuous variables in either $$\ge $$ or $$\le $$ form. When the coefficients of the integer variables are integer and divisible, we show in both cases that the convex hull is the intersection of the bound constraints and $$2^m$$ polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable. The latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the $$\ge $$ case. We also provide an extended formulation for the $$\le $$ case. It follows that, given a specific objective function, optimization over both $$\mathcal {Y}_\ge $$ and $$\mathcal {Y}_\le $$ can be carried out by solving $$m$$ polynomial size linear programs. A further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 (after scaling) in any non-trivial facet-defining inequality of the convex hull of such sets.

Lagrangian based approach to solve a two level capacitated lot sizing problem

AbstractTwo-level, multi-item, multi-period-capacitated dynamic lot-sizing problem with inclusions of backorders and setup times, TL_CLSP_BS, a well-known NP-hard problem, is solved using a novel procedure. Lagrangian relaxation of the material balance constraint reduces TL_CLSP_BS to a single-constraint continuous knapsack problem. Reduced problem is solved using bounded variable linear programs (BVLPs). We obtain promising bounds, which provides a better start to the branch-and-bound procedure. Limited empirical investigations are carried out on four problem sizes. In terms of computational time, the developed procedure is efficient than the CPLEX solver of GAMS. Further, while GAMS could not solve the largest problem size considered here, our procedure could solve the same in around one second time. This clearly highlights the efficacy of the developed procedure. The solution technique is applicable to any problem structure, which is reducible to the application of BVLPs.

The continuous knapsack set

We study the convex hull of the continuous knapsack set which consists of a single inequality constraint with $$n$$n non-negative integer and $$m$$m non-negative bounded continuous variables. When $$n=1$$n=1, this set is a generalization of the single arc flow set studied by Magnanti et al. (Math Program 60:233---250, 1993). We first show that in any facet-defining inequality, the number of distinct non-zero coefficients of the continuous variables is bounded by $$2^n-n$$2n-n. Our next result is to show that when $$n=2$$n=2, this upper bound is actually 1. This implies that when $$n=2$$n=2, the coefficients of the continuous variables in any facet-defining inequality are either 0 or 1 after scaling, and that all the facets can be obtained from facets of continuous knapsack sets with $$m=1$$m=1. The convex hull of the sets with $$n=2$$n=2 and $$m=1$$m=1 is then shown to be given by facets of either two-variable pure-integer knapsack sets or continuous knapsack sets with $$n=2$$n=2 and $$m=1$$m=1 in which the continuous variable is unbounded. The convex hull of these two sets has been completely described by Agra and Constantino (Discrete Optim 3:95---110, 2006). Finally we show (via an example) that when $$n=3$$n=3, the non-zero coefficients of the continuous variables can take different values.

A continuous knapsack problem with separable convex utilities: Approximation algorithms and applications

We study a continuous knapsack problem with separable convex utilities. We show that the problem is NP-hard, and provide two simple algorithms that have worst-case performance guarantees. We consider as an application a novel subsidy allocation problem in the presence of market competition, subject to a budget constraint and upper bounds on the amount allocated to each firm, where the objective is to minimize the market price of a good.