A successive approximation algorithm for the multiple knapsack problem

It is well-known that the multiple knapsack problem is NP-hard, and does not admit an FPTAS even for the case of two identical knapsacks. Whereas the 0-1 knapsack problem with only one knapsack has been intensively studied, and some effective exact or approximation algorithms exist. A natural approach for the multiple knapsack problem is to pack the knapsacks successively by using an effective algorithm for the 0-1 knapsack problem. This paper considers such an approximation algorithm that packs the knapsacks in the nondecreasing order of their capacities. We analyze this algorithm for 2 and 3 knapsack problems by the worst-case analysis method and give all their error bounds.     

Approximation Algorithms for the Multiple Knapsack Problem with Assignment Restrictions

Motivated by a real world application, we study the multiple knapsack problem with assignment restrictions (MKAR). We are given a set of items, each with a positive real weight, and a set of knapsacks, each with a positive real capacity. In addition, for each item a set of knapsacks that can hold that item is specified. In a feasible assignment of items to knapsacks, each item is assigned to at most one knapsack, assignment restrictions are satisfied, and knapsack capacities are not exceeded. We consider the objectives of maximizing assigned weight and minimizing utilized capacity.We focus on obtaining approximate solutions in polynomial computational time. We show that simple greedy approaches yield 1/3-approximation algorithms for the objective of maximizing assigned weight. We give two different 1/2-approximation algorithms: the first one solves single knapsack problems successively and the second one is based on rounding the LP relaxation solution. For the bicriteria problem of minimizing utilized capacity subject to a minimum requirement on assigned weight, we give an (1/3,2)-approximation algorithm.     

Building information modelling in construction: insights from collaboration and change management perspectives

A case study is used to obtain the experiences from a contractor and their subcontractors involved with constructing the landmark Perth Stadium, which required a building information model (BIM) to be delivered for the purpose of asset management. Insights about ‘how’ the adoption of a BIM influenced the practice of collaboration and change management within the project are obtained. It was revealed that having limited experience and knowledge to deliver a model for asset management often resulted the project team ‘muddling through a problem’. This was not necessarily due to a shortage of training, but a lack of BIM knowledge, which inadvertently influenced every day practice. The research presented builds on the extant body of works that have examined how the construction industry can effectively acquire the benefits of BIM for asset management. It also highlights the need to incorporate education and learning into a project’s BIM implementation strategy.     

Approximability of the subset sum reconfiguration problem

The subset sum problem is a well-known NP-complete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NP-hard, and is PSPACE-complete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in a reconfiguration. We show that this maximization problem admits a polynomial-time approximation scheme, while the problem is APX-hard if we are given a conflict graph.     

Make-or-break during production: shedding light on change-orders,rework and contractors margin in construction

A considerable amount of research has examined the cost performance of construction projects, yet there has been a paucity of studies that have examined the impact that client initiated change-orders and rework have on contractors. This paper seeks to add further clarity to this issue by replicating previous empirically-based research to establish the validity and reliability of the key issues influencing a contractor's cost performance. A total of 98 projects were used to examine the value of rework and change-orders and their influence on a contractor's margin. Only 65% of projects experienced a cost increase, though a mean rework cost of 0.39% of the contracted value was incurred. The difference between approved client change-orders and those by the contractor for subcontractors was 0.5% of the total costs incurred, which adversely impacted the organisation's profit. Margin losses may well have been higher as rework is seldom formally documented and reported.     

Approximation of knapsack problems with conflict and forcing graphs

We study the classical 0–1 knapsack problem with additional restrictions on pairs of items. A conflict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing these constraints is the use of conflict (resp. forcing) graphs. By modifying a recent result of Lokstanov et al. (Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms, SODA, pp 570–581, 2014) we derive a fairly complicated FPTAS for the knapsack problem on weakly chordal conflict graphs. Next, we show that the techniques of modular decompositions and clique separators, widely used in the literature for solving the independent set problem on special graph classes, can be applied to the knapsack problem with conflict graphs. In particular, we can show that every positive approximation result for the atoms of prime graphs arising from such a decomposition carries over to the original graph. We point out a number of structural results from the literature which can be used to show the existence of an FPTAS for several graph classes characterized by the exclusion of certain induced subgraphs. Finally, a PTAS for the knapsack problem with H-minor free conflict graph is derived. This includes planar graphs and, more general, graphs of bounded genus. The PTAS is obtained by expanding a general result of Demaine et al. (Proceedings of 46th annual IEEE symposium on foundations of computer science, FOCS 2005, pp 637–646, 2005). The knapsack problem with forcing graphs can be transformed into a minimization knapsack problem with conflict graphs. It follows immediately that all our FPTAS results of the current and a previous paper carry over from conflict graphs to forcing graphs. In contrast, the forcing graph variant is already inapproximable on planar graphs.     

Financial incentives for cost control under moral hazard

A distinguishing feature of contracted production is that the contractual terms must be agreed upon before the product exists. If the specified product is the result of an innovative and complex process, uncertainties as to its production cost estimates can be enormous. This renders the traditional cost-plus-fixed-fee and fixed-price contractual arrangements increasingly untenable. An alternative contractual arrangement that is currently gaining greater acceptance is the fixed-price-incentive contract, which may be as simple as a linear cost-sharing formula or as complex as a state-contingent reward penalty structure. The problem from the viewpoint of the contractee (principal) is to identify the optimal incentive structure in order to minimize his expected cost; the incentive affects the contractor (agent) through his risk aversion and his propensity for “moral hazard”, due to the co-insurance effect of cost-sharing incentives. Taking as premise the contractee's risk-neutrality, the problem is formulated as a constrained optimization problem, with the constraint arising from the returns to each of the parties to the contract. It is thereby demonstrated that the currently popular linear incentive mechanism is ineffective from the viewpoint of the contractee before the introduction of the problem of moral hazard. We then show that, the problem can be segregated into independent components of the contractor's risk aversion and his propensity for moral hazard. Having established this, we proceed to examine the incentive form as related to the “monitoring mechanism” instituted by the contractor during the performance of the contract. Acceptance of the conclusions necessitates that the contracting parties can interpret their objective as the maximization of the ratio of their expected profits to target profits, which is demonstrated as being equivalent to the minimization of the certainty equivalent of costs.     

A Novel Method for Solving Unbounded Knapsack Problem

Knapsack problem is one kind of NP-Complete problem. Unbounded knapsack problems are more complex and harder than general knapsack problem. In this paper, we apply QGAs (Quantum Genetic Algorithms) to solve unbounded knapsack problem and then follow other procedures. First, present the problem into the mode of QGAs and figure out the corresponding genes types and their fitness functions. Then, find the perfect combination of limitation and largest benefit. Finally, the best solution will be found. Primary experiment indicates that our method has well results.     

Implicit cover inequalities

In this paper we consider combinatorial optimization problems whose feasible sets are simultaneously restricted by a binary knapsack constraint and a cardinality constraint imposing the exact number of selected variables. In particular, such sets arise when the feasible set corresponds to the bases of a matroid with a side knapsack constraint, for instance the weighted spanning tree problem and the multiple choice knapsack problem. We introduce the family of implicit cover inequalities which generalize the well-known cover inequalities for such feasible sets and discuss the lifting of the implicit cover inequalities. A computational study for the weighted spanning tree problem is reported.     

Transforming an MRP plan into a short-term schedule

A new approach for transforming MRP orders, planned periodically, e.g. on a weekly base, into a detailed sequence of jobs is presented. In this model for a single machine environment, the jobs are partitioned into families and a family specific set-up time is required at the start of each period and of each batch, where a batch is a maximal set of jobs in the same family, that are processed consecutively. An integer program is formulated for both the problem of minimizing the number of overloaded periods and the problem of minimizing the total overtime. These programs generate benchmark results for the heuristic approach. A heuristic model is developed that constructs a schedule in which overloaded periods are relieved and set-up time is saved. In this approach, the job sequence is constructed by repeatedly solving a knapsack problem. The weights used in this knapsack problem relate to the preferred priorities of the jobs not yet scheduled and determine the quality of the final sequence. The different features of the heuristic model are compared using a large set of test problems. The results show that the quality of the final sequence depends on an appropriate choice for the weights.     

Strengthening a linear reformulation of the 0-1 cubic knapsack problem via variable reordering

The 0-1 cubic knapsack problem (CKP), a generalization of the classical 0-1 quadratic knapsack problem, is an extremely challenging NP-hard combinatorial optimization problem. An effective exact solution strategy for the CKP is to reformulate the nonlinear problem into an equivalent linear form that can then be solved using a standard mixed-integer programming solver. We consider a classical linearization method and propose a variant of a more recent technique for linearizing 0-1 cubic programs applied to the CKP. Using a variable reordering strategy, we show how to improve the strength of the linear programming relaxation of our proposed reformulation, which ultimately leads to reduced overall solution times. In addition, we develop a simple heuristic method for obtaining good-quality CKP solutions that can be used to provide a warm start to the solver. Computational tests demonstrate the effectiveness of both our variable reordering strategy and heuristic method.

     

The Knapsack Sharing Problem: An Exact Algorithm

In this paper, we propose an exact algorithm for the knapsack sharing problem. The proposed algorithm seems quite efficient in the sense that it solves quickly some large problem instances. The problem is decomposed into a series of single constraint knapsack problems; and by applying the dynamic programming and another strategy, we solve optimally the original problem. The performance of the exact algorithm is evaluated on a set of medium and large problem instances (a total of 240 problem instances). This algorithm is parallelizable and this is one of its important feature.     

Approximation algorithms on 0–1 linear knapsack problem with a single continuous variable

The 0–1 linear knapsack problem with a single continuous variable (KPC) is a natural generalization of the standard 0–1 linear knapsack problem (KP). In KPC, the capacity of the knapsack is not fixed, but can be adjusted by a continuous variable. This paper studies the approximation algorithm on KPC. Firstly, assuming that the weight of each item is at most the original capacity of the knapsack, we give a 2-approximation algorithm on KPC by generalizing the 2-approximation algorithm on KP. Then, without the above assumption, we give another 2-approximation algorithm on KPC for general cases by extending the first algorithm.     

Integrated production strategy and reuse scenario: A CoFAQ model and case study of mail server system development

One of the core problems in software product family (SPF) is the coordination of product building and core asset development, specifically the integration of production strategy decision and core asset scenario selection. In the current paper, a model of Cost Optimization under Functional And Quality (CoFAQ) goal satisfaction constraints is developed. It provides a systematic mechanism for management to analyze all possible products and evaluate various reuse alternatives at the organizational level. The CoFAQ model facilitates decision-makers to optimize the SPF development process by determining which products are involved in the SPF (i.e. production strategy) and which reuse scenario for each module should be selected to implement the SPF toward minimum total developing cost under the constraints of satisfying functional and quality goals. A two-phase algorithm with heuristic (TPA) is developed to solve the model efficiently. Based on the TPA, the CoFAQ is reduced to a weighted set-covering problem for production strategy decision and a knapsack problem for the reuse scenario selection. An application of the model in mail server domain development is presented to illustrate how it has been used in practice.     

Supply chain management: a supplier perspective

A new way of researching into supply chain management is introduced by adopting a supplier perspective. Details are given of mixed integer linear planning and simulation models. The planning model takes into consideration firm and forecast orders (customer's forecast purchasing orders) and the behaviour of the supplier's suppliers and suppliers' subcontractors. The simulation model takes into account the dynamic behaviour of the supply chain and includes the planning production behaviour of a supplying company based on the planning model. The supplier manager can use the simulation model to determine what kind of parameters most affect company performances and then propose new management rules. Quantitative results that prove the benefits of integrating forecast orders for an aeronautic supplier have been provided.     

A Semidefinite Programming Approach to the Quadratic Knapsack Problem

In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibilities to improve these basic relaxations by cutting planes are discussed. The cutting planes either arise from quadratic representations of linear inequalities or from linear inequalities in the quadratic model. In particular, a large family of combinatorial cuts is introduced for the linear formulation of the knapsack problem in quadratic space. Computational results on a small class of practical problems illustrate the quality of these relaxations and cutting planes.     

Algorithmic approaches to the multiple knapsack assignment problem

We consider a variant of the multiple knapsack problem in which some assignment-type side constraints have to be satisfied. The problem finds applications in logistics sectors related, e.g., to transportation and maritime shipping. We derive upper bounds from Lagrangian and surrogate relaxations of a mathematical model of the problem. We introduce a constructive heuristic and a metaheuristic refinement. We study the computational complexity of the proposed methods and evaluate their practical performance through extensive computational experiments on benchmarks from the literature and on new sets of randomly generated instances.     

In-house capability and supply chain decisions

This study considers an internal production option for a contractor and analyzes its effect on the supply chain decisions when the contractor has innovated and the subcontractor has an incentive for opportunistic behavior. In contrast to the single disclosure threshold in the benchmark scenario where the contractor lacks in-house capability, we find two thresholds in the referred scenario. When information misappropriation is possible and the contractor has in-house capability, the contractor will organize a coordinated supply chain only when innovations fall between the two thresholds. Compared to the benchmark scenario, in-house capability has a positive effect on the contractor's incentive to innovate and an ambiguous effect on the subcontractor's incentive to invest in the production process. When the contractor needs to incur an extra cost to build in-house capability, the contractor keeps the same levels of investment compared to the case of no additional in-house capability cost, whereas the subcontractor increases the levels of investment. Furthermore, we find that in the presence of potential misappropriation on the part of the subcontractor, the higher the level of in-house capability, the less likely the contractor will be to outsource innovative products that generate higher profitability. This study can explain why firms strategically outsource low-end products and produce high-end products themselves. This study provides new results on the effects of in-house capability on the strategic interactions of parties in supply chains and, hence, on supply chain efficiency.     

Bin packing with “Largest In Bottom” constraint: tighter bounds and generalizations

The (online) bin packing problem with LIB constraint is stated as follows: The items arrive one by one, and must be packed into unit capacity bins, but a bigger item cannot be packed into a bin which already contains a smaller item. The number of used bins has to be minimized as usually. We show that the absolute performance bound of algorithm First Fit is not worse than 2+1/6≈2.1666 for the problem, improving the previous best upper bound 2.5. Moreover, if the item sizes do not exceed 1/d, then we improve the previous best result 2+1/d to 2+1/d(d+2), for any d≥2. (Both previously best results are due to Epstein, Nav. Res. Logist. 56(8):780–786, 2009.) Furthermore, we define a problem with the generalized LIB constraint, where some incoming items cannot be packed into the bins of some already packed items. The (in)compatibility of the incoming item with the items already packed becomes known only at the arrival of the actual item, and is given by an undirected graph (and, as usual in case of online graph problems, we can see only that part of the graph what already arrived). We show that 3 is an upper bound for this general problem if some natural transitivity constraint is satisfied.     

The min-up/min-down unit commitment polytope

The min-up/min-down unit commitment problem (MUCP) is to find a minimum-cost production plan on a discrete time horizon for a set of fossil-fuel units for electricity production. At each time period, the total production has to meet a forecast demand. Each unit must satisfy minimum up-time and down-time constraints besides featuring production and start-up costs. A full polyhedral characterization of the MUCP with only one production unit is provided by Rajan and Takriti (Minimum up/down polytopes of the unit commitment problem with start-up costs. IBM Research Report, 2005). In this article, we analyze polyhedral aspects of the MUCP with n production units. We first translate the classical extended cover inequalities of the knapsack polytope to obtain the so-called up-set inequalities for the MUCP polytope. We introduce the interval up-set inequalities as a new class of valid inequalities, which generalizes both up-set inequalities and minimum up-time inequalities. We provide a characterization of the cases when interval up-set inequalities are valid and not dominated by other inequalities. We devise an efficient Branch and Cut algorithm, using up-set and interval up-set inequalities.