Stack-up Problem Research Articles

QUBO Models for the FIFO Stack-Up Problem and Experimental Evaluation on a Quantum Annealer

Quantum annealing has been applied to combinatorial optimization problems in recent years. In this paper we study the possibility to use quantum annealing for solving the combinatorial FIFO Stack-Up problem, where bins have to be stacked-up from a conveyor belt onto pallets. The problem is NP-hard and can be solved using linear programming approaches. We developed two QUBO (quadratic unconstrained binary optimization) objective functions based on a bin stack-up solution and a pallet stack-up solution for this problem suitable for a quantum annealer. The number of variables was minimized to increase the performance and their dependence on the number of bins and pallets was discussed. The performances of both methods were studied for various small problem sizes on a D-Wave quantum annealer. We found that only tiny instances could be solved and looked at the terms of the QUBO-formulations, which cause the quantum annealer to fail for larger problem sizes. Furthermore we compare the results to the performance of a classic computer using the same QUBO-formulations.

On the hardness of palletizing bins using FIFO queues

We study the combinatorial FIFO Stack-Up problem. In the delivery industry, bins have to be moved from conveyor belts onto pallets. Given is a list Q of k sequences of labeled bins and a positive integer p. The goal is to palletize the bins by iteratively removing the first bin of one of the k sequences and putting it onto a pallet located at one of p stack-up places. Each of these pallets has to contain bins of only one label, bins of different labels have to be placed on different pallets. After all bins of one label have been removed from the given sequences, the corresponding stack-up place becomes available for a pallet of bins of another label. The FIFO Stack-Up problem asks whether such a processing is possible. The FIFO Stack-Up problem is known to be NP-complete even if the sequences of Q contain together at most cQ=6 bins per pallet. This implies that the problem is also hard if all bins of every pallet are distributed to at most dQ=6 sequences. In this paper we strengthen the hardness to the cases cQ=5 and dQ=3. We also show the hardness for the practical case where the number of sequences is smaller than the number of pallets, and we point out differences to related problems like 3D Bin Packing and Train Marshalling.

On the complexity of the FIFO stack-up problem

We study the combinatorial FIFO stack-up problem. In delivery industry, bins have to be stacked-up from conveyor belts onto pallets with respect to customer orders. Given k sequences $$q_1, \ldots , q_k$$ of labeled bins and a positive integer p, the aim is to stack-up the bins by iteratively removing the first bin of one of the k sequences and put it onto an initially empty pallet of unbounded capacity located at one of p stack-up places. Bins with different pallet labels have to be placed on different pallets, bins with the same pallet label have to be placed on the same pallet. After all bins for a pallet have been removed from the given sequences, the corresponding stack-up place will be cleared and becomes available for a further pallet. The FIFO stack-up problem is to find a stack-up sequence such that all pallets can be build-up with the available p stack-up places. In this paper, we introduce two digraph models for the FIFO stack-up problem, namely the processing graph and the sequence graph. We show that there is a processing of some list of sequences with at most p stack-up places if and only if the sequence graph of this list has directed pathwidth at most $$p-1$$ . This connection implies that the FIFO stack-up problem is NP-complete in general, even if there are at most 6 bins for every pallet and that the problem can be solved in polynomial time, if the number p of stack-up places is assumed to be fixed. Further the processing graph allows us to show that the problem can be solved in polynomial time, if the number k of sequences is assumed to be fixed.

Least Cost Tolerance Allocation for Systems with Time-variant Deviations

This paper presents a methodology for the least cost tolerance allocation of systems with time-variant deviations, such as deformation, thermal expansion, mobility of parts and wear. By means of the approach, the product developer can identify the system's optimal tolerance design against the backdrop of the diverging requirements: wide tolerances to reduce costs vs. narrow tolerances to ensure functionality. Therefore, Particle Swarm Optimization – a global statistical optimization technique – is applied to the time-variant tolerance-optimization problem to determine the tolerance range as well as a certain mean shift for each non-ideal dimension.The practical use of the methodology is illustrated for a modified tolerance stack-up problem. We provide a comprehensive walkthrough of the methodology's application to the stack-up problem. Therefore, the essential mathematical background, the required functional dependencies and the considered parameters are detailed in a step-by-step procedure. So, we aim to establish the application of the statistical tolerance-cost-optimization in academia and industry and to motivate students and young graduates to apply this methodology by themselves.

Stack-up algorithms for palletizing at delivery industry

We study the following multi-objective combinatorial stack-up problem from delivery industry. Given a sequence q of labeled bins and two positive integers s and p. The aim is to stack-up the bins by iteratively removing one of the first s bins of the sequence and put it to one of p stack-up places. Each of these places has to contain bins of only one label, bins of different labels have to be placed on different places. If all bins of a label are removed from q, the corresponding place becomes available for bins of another label. We analyze the worst-case performance of simple algorithms for the stack-up problem that are very interesting from a practical point of view. In particular, we show that the so-called Most-Frequently on-line algorithm is (2,2)-competitive and has optimal worst-case on-line performance.

An approximation algorithm for the stack-up problem

We consider the combinatorial stack-up problem motivated by stacking up bins from a conveyor onto pallets. The stack-up problem is to decide whether a given list q of labeled objects can be processed by removing step by step one of the first s objects of q so that the following holds. After each removal there are at most p labels for which the first object is already removed from q and the last object is still contained in q. We give some NP-completeness results and we introduce and analyze a polynomial time approximation algorithm for the stack-up problem.

A Note on the Use of Discrete Convolutions

A convolution method is used to solve a tolerance stack-up problem. The input random variables can be of any form. A complete frequency distribution of the possible outcomes is generated in the process of obtaining the desired probability of exceeding the permissible tolerance. (This method has also proved valuable in the actuarial field; for example, in determining the probability that total insurance claims will not exceed a given level.)

A Note on the Use of Discrete Convolutions

A convolution method is used to solve a tolerance stack-up problem. The input random variables can be of any form. A complete frequency distribution of the possible outcomes is generated in the process of obtaining the desired probability of exceeding the permissible tolerance. (This method has also proved valuable in the actuarial field; for example, in determining the probability that total insurance claims will not exceed a given level.)