European Journal of Operational Research, Feb 1, 1994
ABSTRACT Martello and Toth are well known for their work on the 0–1 knapsack problem. Computation... more ABSTRACT Martello and Toth are well known for their work on the 0–1 knapsack problem. Computationally, their MT2 algorithm is one of the most efficient exact algorithms published in the OR/MS literature. For uncorrelated and weakly correlated data sets, Martello and Toth have shown that their MT2 algorithm can solve up to 10 000 variable problems very quickly. However, for strongly correlated data sets, even MT2 could only solve up to 100 variable problems without exceeding the time limit. In this note we take a closer look at the empirical performance of the Martello-Toth MT2 algorithm on strongly correlated data sets.
International journal of circuits, systems and signal processing, Nov 12, 2021
The 0-1 Multidimensional Knapsack Problem (MKP) is a NP-Hard problem that has important applicati... more The 0-1 Multidimensional Knapsack Problem (MKP) is a NP-Hard problem that has important applications in business and industry. Approximate solution approaches for the MKP in the literature typically provide no guarantee on how close generated solutions are to the optimum. This article demonstrates how general-purpose integer programming software (Gurobi) is iteratively used to generate solutions for the 270 MKP test problems in Beasley's OR-Library such that, on average, the solutions are guaranteed to be within 0.094% of the optimums and execute in 88 seconds on a standard PC. This methodology, called the simple sequential increasing tolerance (SSIT) matheuristic, uses a sequence of increasing tolerances in Gurobi to generate a solution that is guaranteed to be close to the optimum in a short time. This solution strategy generates bounded solutions in a timely manner without requiring the coding of a problem-specific algorithm. The SSIT results (although guaranteed within 0.094% of the optimums) when compared to known optimums deviated only 0.006% from the optimums-far better than any published results for these 270 MKP test instances.
To take maximum advantage of a new ingot mold stripping facility, which was installed in 1984 at ... more To take maximum advantage of a new ingot mold stripping facility, which was installed in 1984 at the Bethlehem Plant, we developed a two-phase, computer-based procedure for selecting optimal ingot dimensions and internal ingot mold dimensions. Phase 1 generated feasible ingot and internal ingot mold dimensions consistent with the new stripper's capability and with foundry, steelmaking, metallurgical, mill, and shipping yard constraints. Phase 2 then used a set-covering approach to select the optimal ingot and internal ingot mold sizes from among the feasible sizes. Based on model results and following trial mill tests, full production use of new mold sizes has affected the entire plant operation, resulting in over $8 million annual realized savings. B ethlehem Steel Corporation is the sec-greatly reduces costs and that has been ond largest steel producer in America adopted in Japan and Europe and more rewith major steel producing plants in Beth-cently in the United States to replace the lehem, Johnstown, and Steelton, Pennsyl-traditional ingot-based steelmaking techvania, and at Sparrows Point, Maryland, nology. While using continuous casters (inand Burns Harbor, Indiana. Continuous stead of ingot-based steelmaking) improves casting is a steelmaking technology that yield, productivity, and product quality.
The greedy heuristic for the weighted set covering problem is a ''column knowledge'' construction... more The greedy heuristic for the weighted set covering problem is a ''column knowledge'' construction heuristic where cost and row coverage information are used to insert columns into the solution. In this paper, we analyze the performance of construction heuristics that expand on the column knowledge functions described by Vasko and Wilson (1984) and row knowledge functions described by Ablanedo-Rosas and Rego (2010). If redundant columns are removed from solutions, then the basic greedy heuristic gives essentially the best results.
The multiple knapsack assignment problem (MKAP) is an interesting generalization of the multiple ... more The multiple knapsack assignment problem (MKAP) is an interesting generalization of the multiple knapsack problem which has logistical applications in transportation and shipping. In addition to trying to insert items into knapsacks in order to maximize the profit of the items in the knapsacks, the MKAP partitions the items into classes and only items from the same class can be inserted into a knapsack. In the literature, the Gurobi integer programming software has solved MKAPs with up to 1240 variables and 120 constraints in at most 20 min on a standard PC. In this article, using a standard PC and iteratively loosening the acceptable tolerance gap for 180 MKAPs with up to 20,100 variables and 1120 constraints, we show that Gurobi can, on average, generate solutions that are guaranteed to be at most 0.17% from the optimums in 43 s. However, for very large MKAPs (over a million variables), Gurobi's performance can be significantly improved when an initial feasible solution is provided. Specifically, using from the literature, a heuristic and 42 MKAP instances with over 6 million variables and nearly 90,000 constraints, Gurobi generated solutions guaranteed to be, on average, within 0.21% of the optimums in 10 min. This is a 99% reduction in the final solution bound (gap between the best Gurobi solution and the best upper bound) compared to the approach without initial solution inputs. Hence, a major objective of this article is to demonstrate for what size MKAP instances providing Gurobi with an initial heuristic solution significantly improves performance in terms of both execution time and solution quality.
International Transactions in Operational Research, Nov 1, 2009
P. Y. Wang's classic bottom-up two-dimensional cutting stock algorithm generates cutting patterns... more P. Y. Wang's classic bottom-up two-dimensional cutting stock algorithm generates cutting patterns by building rectangles both horizontally and vertically. This algorithm uses a parameter b 1 to tradeoff the number of rectangles generated by the algorithm and hence the quality of the cutting pattern solution obtained versus the amount of computer resources required. Several researchers have made relatively straightforward modifications to Wang's basic algorithm resulting in improved computational times. However, even with these modifications, Wang's approach tends to require large amounts of computer resources in order to optimally or near-optimally solve difficult two-dimensional guillotine cutting stock problems. In this paper, we present an iterative approach that judiciously uses Wang's basic algorithm (with some previously defined modifications) to obtain optimal cutting patterns to difficult two-dimensional cutting stock problems in reasonable computer run times.
In the mid-1990's, the Sparrows Point Steel Plant began an initiative to achieve 100% on time sla... more In the mid-1990's, the Sparrows Point Steel Plant began an initiative to achieve 100% on time slab availability to the hot strip mill. At that time, order dressing, heat building, slab application, and mill scheduling of strip product orders were manually intensive activities. It was decided to implement an integrated system to automate these activities. In this paper we discuss the central activity of this system-the application of slabs to strip product orders (used for appliance panels, steel storage sheds, food cans, etc.). A binary integer linear programming formulation will be given for the slab application problem. However, because of the size of typical problem instances, a robust and efficient hierarchical heuristic solution approach was developed that has been in daily use since 1999.
An important generalization of the classic 0-1 knapsack problem is the multi-demand multidimensio... more An important generalization of the classic 0-1 knapsack problem is the multi-demand multidimensional knapsack problem (MDMKP). In addition to being theoretically difficult to solve (it is NP-hard),...
A common application of linear programming is assisting with strategic planning decisions. In thi... more A common application of linear programming is assisting with strategic planning decisions. In this paper we describe a linear programming model that was used by a large US corporation to assist with decisions concerning one of its major product lines. In addition to standard capacity and market demand constraints, this model incorporated constraints termed "complete buildings" (CB) constraints that were dictated by the Sales Department. Essentially, the CB constraints ensured that the model product production levels were, within a tolerance, all the same percentage of their market demand. When Sales, with the help of Accounting, proposed opening a second (then idle) heavy products mill in an effort to capture more of the heavy products market demand, the Sales Department's own CB constraints within the strategic planning linear programming model told a different story. Operations, Marketing and Research were represented on the committee. Additionally, a manager of a corporate Operational Research group was a member of the committee and a young OR analyst was assigned as a technical consultant to the committee. The OR manager along with the support of most of the committee members suggested that a linear programming model be developed to assist the committee with its work of evaluating alternative operating and marketing strategies. The essential features of a basic linear programming model that was developed for the committee are given below.
In this paper, we study the performance of five population-based metaheuristics to solve a large ... more In this paper, we study the performance of five population-based metaheuristics to solve a large (393) number of comprehensive problem instances from the literature for the important (NP-Hard) multiple choice multidimensional knapsack problem (MMKP). The five metaheuristics are: teaching-learning-based optimisation (TLBO), artificial bee colony (ABC), genetic algorithm (GA), criss-cross optimisation algorithm (COA), and binary bat algorithm (BBA). All five of these metaheuristics are similar in that they transform a population of solutions in an effort to improve the solutions in the population and they are all implemented in a straightforward manner. Statistically (over all 393 problem instances), we show that COA, GA, and TLBO give similar results which are better than other published solution approaches for the MMKP. However, if we incorporate a simple neighbourhood search into each of these five metaheuristics, in addition to improved solution quality, there is now no statistically significant difference among the results for these five metaheuristics.
Journal of the Operational Research Society, Jun 1, 2011
Ant colony optimization (ACO) is a metaheuristic for solving combinatorial optimization problems ... more Ant colony optimization (ACO) is a metaheuristic for solving combinatorial optimization problems that is based on the foraging behavior of biological ant colonies. Starting with the 1996 seminal paper by Dorigo, Maniezzo and Colorni, ACO techniques have been used to solve the traveling salesperson problem (TSP). In this paper, we focus on a particular type of the ACO algorithm, namely, the rank-based ACO algorithm for the TSP. In particular, this paper identifies an optimal set of key parameters by statistical analysis applied to results of the rank-based ACO for the TSP. Specifically, for six frequently used TSPs available on the World Wide Web, we will solve a total of 27 000 instances for each problem.
So that Bethlehem Steel's Sparrows Point plant can produce narrow-width (NW) customer-plate o... more So that Bethlehem Steel's Sparrows Point plant can produce narrow-width (NW) customer-plate orders (typically 10′′ to 24′′) efficiently when its 60′′ plate mill is not operating, we developed a heuristic procedure to map these orders into mother plates for production on its 160′′ plate mill. Mother plates processed on the 60′′ plate mill are cut, using fairly simple, two-stage guillotine cutting patterns, into a few NW customer plates. Considerably more NW plates can be cut from the much larger mother plates processed on the 160′′ mill, and the number of possible mappings of NW customer plates into mother plates is much larger. Our heuristic procedure was implemented as a module in the plant's production planning and control system, and it is used daily to generate mother-plate dimensions and cutting patterns.
Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncap... more Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncapacitated facility location problem (UFLP). In this paper, we use his algorithm to solve large instances of an important subset of the UFLP; the set covering problem (SCP). In addition, we present further empirical evidence that a heuristic algorithm developed by Vasko and Wilson for the SCP is capable of quickly generating good solutions to large SCP's.
Welsh and Powell (10) , to construct large scale timetables. This program is capable of construct... more Welsh and Powell (10) , to construct large scale timetables. This program is capable of constructing schedules for as many as 960 events. For example, this program will schedule a total of 960 classes or examinations. It is also shown through linear regression
Typically, a women's collegiate basketball star player spends much of her time fine-tuning her sk... more Typically, a women's collegiate basketball star player spends much of her time fine-tuning her skills on the court. However, one member of the Kutztown University Women's Basketball Team (the team) decided to contribute both on and off the court. Specifically, she used her analytical skills to develop a mathematical model to optimally assign team members to positions. In this article, we will discuss several mathematical models developed to assign basketball team members to positions based on different assumptions and objectives. Model results using data based on the team will also be discussed. Several scenarios will be presented to illustrate how the coaching staff of a basketball team can make use of these math models to assist with tactical decisions courtside as well as pre-game strategy aids.
Journal of the Operational Research Society, Mar 1, 2000
Consider a replenishment problem in which several different rectangular sizes of material are sto... more Consider a replenishment problem in which several different rectangular sizes of material are stocked. Customers order rectangles of the material, but the rectangles ordered have a range on speci®ed width as well as on speci®ed length. To satisfy the customer requirements, the stock material can be cut once longitudinally in order to satisfy two customer requirements or not cut at all, that is, an entire stock piece of material is used to satisfy one customer requirement. If an exact match is impossible in the current planning period, the unused material must be returned to stockÐ an expensive and undesirable situation. In this paper, a nonbipartite weighted matching problem formulation will be given for determining the replenishment requirements of rectangular stock sizes. Then, a hybrid solution approach, capable of solving real applications (typically up to 3000 nodes) ef®ciently, will be discussed. This solution was implemented in September 1998 and has operated successfully since then.
International Journal of Industrial Engineering Computations, 2020
The 0-1 Multidimensional Knapsack Problem (MKP) is an NP-Hard problem that has many important app... more The 0-1 Multidimensional Knapsack Problem (MKP) is an NP-Hard problem that has many important applications in business and industry. However, business and industrial applications typically involve large problem instances that can be time consuming to solve for a guaranteed optimal solution. There are many approximate solution approaches, heuristics and metaheuristics, for the MKP published in the literature, but these typically require the fine-tuning of several parameters. Fine-tuning parameters is not only time-consuming (especially for operations research (OR) practitioners), but also implies that solution quality can be compromised if the problem instances being solved change in nature. In this paper, we demonstrate an efficient and effective implementation of a robust population-based metaheuristic that does not require parameter fine-tuning and can easily be used by OR practitioners to solve industrial size problems. Specifically, to solve the MKP, we provide an efficient adaptation of the two-phase Teaching-Learning Based Optimization (TLBO) approach that was originally designed to solve continuous nonlinear engineering design optimization problems. Empirical results using the 270 MKP test problems available in Beasley's OR-Library demonstrate that our implementation of TLBO for the MKP is competitive with published solution approaches without the need for time-consuming parameter fine-tuning.
European Journal of Operational Research, Feb 1, 1994
ABSTRACT Martello and Toth are well known for their work on the 0–1 knapsack problem. Computation... more ABSTRACT Martello and Toth are well known for their work on the 0–1 knapsack problem. Computationally, their MT2 algorithm is one of the most efficient exact algorithms published in the OR/MS literature. For uncorrelated and weakly correlated data sets, Martello and Toth have shown that their MT2 algorithm can solve up to 10 000 variable problems very quickly. However, for strongly correlated data sets, even MT2 could only solve up to 100 variable problems without exceeding the time limit. In this note we take a closer look at the empirical performance of the Martello-Toth MT2 algorithm on strongly correlated data sets.
International journal of circuits, systems and signal processing, Nov 12, 2021
The 0-1 Multidimensional Knapsack Problem (MKP) is a NP-Hard problem that has important applicati... more The 0-1 Multidimensional Knapsack Problem (MKP) is a NP-Hard problem that has important applications in business and industry. Approximate solution approaches for the MKP in the literature typically provide no guarantee on how close generated solutions are to the optimum. This article demonstrates how general-purpose integer programming software (Gurobi) is iteratively used to generate solutions for the 270 MKP test problems in Beasley's OR-Library such that, on average, the solutions are guaranteed to be within 0.094% of the optimums and execute in 88 seconds on a standard PC. This methodology, called the simple sequential increasing tolerance (SSIT) matheuristic, uses a sequence of increasing tolerances in Gurobi to generate a solution that is guaranteed to be close to the optimum in a short time. This solution strategy generates bounded solutions in a timely manner without requiring the coding of a problem-specific algorithm. The SSIT results (although guaranteed within 0.094% of the optimums) when compared to known optimums deviated only 0.006% from the optimums-far better than any published results for these 270 MKP test instances.
To take maximum advantage of a new ingot mold stripping facility, which was installed in 1984 at ... more To take maximum advantage of a new ingot mold stripping facility, which was installed in 1984 at the Bethlehem Plant, we developed a two-phase, computer-based procedure for selecting optimal ingot dimensions and internal ingot mold dimensions. Phase 1 generated feasible ingot and internal ingot mold dimensions consistent with the new stripper's capability and with foundry, steelmaking, metallurgical, mill, and shipping yard constraints. Phase 2 then used a set-covering approach to select the optimal ingot and internal ingot mold sizes from among the feasible sizes. Based on model results and following trial mill tests, full production use of new mold sizes has affected the entire plant operation, resulting in over $8 million annual realized savings. B ethlehem Steel Corporation is the sec-greatly reduces costs and that has been ond largest steel producer in America adopted in Japan and Europe and more rewith major steel producing plants in Beth-cently in the United States to replace the lehem, Johnstown, and Steelton, Pennsyl-traditional ingot-based steelmaking techvania, and at Sparrows Point, Maryland, nology. While using continuous casters (inand Burns Harbor, Indiana. Continuous stead of ingot-based steelmaking) improves casting is a steelmaking technology that yield, productivity, and product quality.
The greedy heuristic for the weighted set covering problem is a ''column knowledge'' construction... more The greedy heuristic for the weighted set covering problem is a ''column knowledge'' construction heuristic where cost and row coverage information are used to insert columns into the solution. In this paper, we analyze the performance of construction heuristics that expand on the column knowledge functions described by Vasko and Wilson (1984) and row knowledge functions described by Ablanedo-Rosas and Rego (2010). If redundant columns are removed from solutions, then the basic greedy heuristic gives essentially the best results.
The multiple knapsack assignment problem (MKAP) is an interesting generalization of the multiple ... more The multiple knapsack assignment problem (MKAP) is an interesting generalization of the multiple knapsack problem which has logistical applications in transportation and shipping. In addition to trying to insert items into knapsacks in order to maximize the profit of the items in the knapsacks, the MKAP partitions the items into classes and only items from the same class can be inserted into a knapsack. In the literature, the Gurobi integer programming software has solved MKAPs with up to 1240 variables and 120 constraints in at most 20 min on a standard PC. In this article, using a standard PC and iteratively loosening the acceptable tolerance gap for 180 MKAPs with up to 20,100 variables and 1120 constraints, we show that Gurobi can, on average, generate solutions that are guaranteed to be at most 0.17% from the optimums in 43 s. However, for very large MKAPs (over a million variables), Gurobi's performance can be significantly improved when an initial feasible solution is provided. Specifically, using from the literature, a heuristic and 42 MKAP instances with over 6 million variables and nearly 90,000 constraints, Gurobi generated solutions guaranteed to be, on average, within 0.21% of the optimums in 10 min. This is a 99% reduction in the final solution bound (gap between the best Gurobi solution and the best upper bound) compared to the approach without initial solution inputs. Hence, a major objective of this article is to demonstrate for what size MKAP instances providing Gurobi with an initial heuristic solution significantly improves performance in terms of both execution time and solution quality.
International Transactions in Operational Research, Nov 1, 2009
P. Y. Wang's classic bottom-up two-dimensional cutting stock algorithm generates cutting patterns... more P. Y. Wang's classic bottom-up two-dimensional cutting stock algorithm generates cutting patterns by building rectangles both horizontally and vertically. This algorithm uses a parameter b 1 to tradeoff the number of rectangles generated by the algorithm and hence the quality of the cutting pattern solution obtained versus the amount of computer resources required. Several researchers have made relatively straightforward modifications to Wang's basic algorithm resulting in improved computational times. However, even with these modifications, Wang's approach tends to require large amounts of computer resources in order to optimally or near-optimally solve difficult two-dimensional guillotine cutting stock problems. In this paper, we present an iterative approach that judiciously uses Wang's basic algorithm (with some previously defined modifications) to obtain optimal cutting patterns to difficult two-dimensional cutting stock problems in reasonable computer run times.
In the mid-1990's, the Sparrows Point Steel Plant began an initiative to achieve 100% on time sla... more In the mid-1990's, the Sparrows Point Steel Plant began an initiative to achieve 100% on time slab availability to the hot strip mill. At that time, order dressing, heat building, slab application, and mill scheduling of strip product orders were manually intensive activities. It was decided to implement an integrated system to automate these activities. In this paper we discuss the central activity of this system-the application of slabs to strip product orders (used for appliance panels, steel storage sheds, food cans, etc.). A binary integer linear programming formulation will be given for the slab application problem. However, because of the size of typical problem instances, a robust and efficient hierarchical heuristic solution approach was developed that has been in daily use since 1999.
An important generalization of the classic 0-1 knapsack problem is the multi-demand multidimensio... more An important generalization of the classic 0-1 knapsack problem is the multi-demand multidimensional knapsack problem (MDMKP). In addition to being theoretically difficult to solve (it is NP-hard),...
A common application of linear programming is assisting with strategic planning decisions. In thi... more A common application of linear programming is assisting with strategic planning decisions. In this paper we describe a linear programming model that was used by a large US corporation to assist with decisions concerning one of its major product lines. In addition to standard capacity and market demand constraints, this model incorporated constraints termed "complete buildings" (CB) constraints that were dictated by the Sales Department. Essentially, the CB constraints ensured that the model product production levels were, within a tolerance, all the same percentage of their market demand. When Sales, with the help of Accounting, proposed opening a second (then idle) heavy products mill in an effort to capture more of the heavy products market demand, the Sales Department's own CB constraints within the strategic planning linear programming model told a different story. Operations, Marketing and Research were represented on the committee. Additionally, a manager of a corporate Operational Research group was a member of the committee and a young OR analyst was assigned as a technical consultant to the committee. The OR manager along with the support of most of the committee members suggested that a linear programming model be developed to assist the committee with its work of evaluating alternative operating and marketing strategies. The essential features of a basic linear programming model that was developed for the committee are given below.
In this paper, we study the performance of five population-based metaheuristics to solve a large ... more In this paper, we study the performance of five population-based metaheuristics to solve a large (393) number of comprehensive problem instances from the literature for the important (NP-Hard) multiple choice multidimensional knapsack problem (MMKP). The five metaheuristics are: teaching-learning-based optimisation (TLBO), artificial bee colony (ABC), genetic algorithm (GA), criss-cross optimisation algorithm (COA), and binary bat algorithm (BBA). All five of these metaheuristics are similar in that they transform a population of solutions in an effort to improve the solutions in the population and they are all implemented in a straightforward manner. Statistically (over all 393 problem instances), we show that COA, GA, and TLBO give similar results which are better than other published solution approaches for the MMKP. However, if we incorporate a simple neighbourhood search into each of these five metaheuristics, in addition to improved solution quality, there is now no statistically significant difference among the results for these five metaheuristics.
Journal of the Operational Research Society, Jun 1, 2011
Ant colony optimization (ACO) is a metaheuristic for solving combinatorial optimization problems ... more Ant colony optimization (ACO) is a metaheuristic for solving combinatorial optimization problems that is based on the foraging behavior of biological ant colonies. Starting with the 1996 seminal paper by Dorigo, Maniezzo and Colorni, ACO techniques have been used to solve the traveling salesperson problem (TSP). In this paper, we focus on a particular type of the ACO algorithm, namely, the rank-based ACO algorithm for the TSP. In particular, this paper identifies an optimal set of key parameters by statistical analysis applied to results of the rank-based ACO for the TSP. Specifically, for six frequently used TSPs available on the World Wide Web, we will solve a total of 27 000 instances for each problem.
So that Bethlehem Steel's Sparrows Point plant can produce narrow-width (NW) customer-plate o... more So that Bethlehem Steel's Sparrows Point plant can produce narrow-width (NW) customer-plate orders (typically 10′′ to 24′′) efficiently when its 60′′ plate mill is not operating, we developed a heuristic procedure to map these orders into mother plates for production on its 160′′ plate mill. Mother plates processed on the 60′′ plate mill are cut, using fairly simple, two-stage guillotine cutting patterns, into a few NW customer plates. Considerably more NW plates can be cut from the much larger mother plates processed on the 160′′ mill, and the number of possible mappings of NW customer plates into mother plates is much larger. Our heuristic procedure was implemented as a module in the plant's production planning and control system, and it is used daily to generate mother-plate dimensions and cutting patterns.
Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncap... more Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncapacitated facility location problem (UFLP). In this paper, we use his algorithm to solve large instances of an important subset of the UFLP; the set covering problem (SCP). In addition, we present further empirical evidence that a heuristic algorithm developed by Vasko and Wilson for the SCP is capable of quickly generating good solutions to large SCP's.
Welsh and Powell (10) , to construct large scale timetables. This program is capable of construct... more Welsh and Powell (10) , to construct large scale timetables. This program is capable of constructing schedules for as many as 960 events. For example, this program will schedule a total of 960 classes or examinations. It is also shown through linear regression
Typically, a women's collegiate basketball star player spends much of her time fine-tuning her sk... more Typically, a women's collegiate basketball star player spends much of her time fine-tuning her skills on the court. However, one member of the Kutztown University Women's Basketball Team (the team) decided to contribute both on and off the court. Specifically, she used her analytical skills to develop a mathematical model to optimally assign team members to positions. In this article, we will discuss several mathematical models developed to assign basketball team members to positions based on different assumptions and objectives. Model results using data based on the team will also be discussed. Several scenarios will be presented to illustrate how the coaching staff of a basketball team can make use of these math models to assist with tactical decisions courtside as well as pre-game strategy aids.
Journal of the Operational Research Society, Mar 1, 2000
Consider a replenishment problem in which several different rectangular sizes of material are sto... more Consider a replenishment problem in which several different rectangular sizes of material are stocked. Customers order rectangles of the material, but the rectangles ordered have a range on speci®ed width as well as on speci®ed length. To satisfy the customer requirements, the stock material can be cut once longitudinally in order to satisfy two customer requirements or not cut at all, that is, an entire stock piece of material is used to satisfy one customer requirement. If an exact match is impossible in the current planning period, the unused material must be returned to stockÐ an expensive and undesirable situation. In this paper, a nonbipartite weighted matching problem formulation will be given for determining the replenishment requirements of rectangular stock sizes. Then, a hybrid solution approach, capable of solving real applications (typically up to 3000 nodes) ef®ciently, will be discussed. This solution was implemented in September 1998 and has operated successfully since then.
International Journal of Industrial Engineering Computations, 2020
The 0-1 Multidimensional Knapsack Problem (MKP) is an NP-Hard problem that has many important app... more The 0-1 Multidimensional Knapsack Problem (MKP) is an NP-Hard problem that has many important applications in business and industry. However, business and industrial applications typically involve large problem instances that can be time consuming to solve for a guaranteed optimal solution. There are many approximate solution approaches, heuristics and metaheuristics, for the MKP published in the literature, but these typically require the fine-tuning of several parameters. Fine-tuning parameters is not only time-consuming (especially for operations research (OR) practitioners), but also implies that solution quality can be compromised if the problem instances being solved change in nature. In this paper, we demonstrate an efficient and effective implementation of a robust population-based metaheuristic that does not require parameter fine-tuning and can easily be used by OR practitioners to solve industrial size problems. Specifically, to solve the MKP, we provide an efficient adaptation of the two-phase Teaching-Learning Based Optimization (TLBO) approach that was originally designed to solve continuous nonlinear engineering design optimization problems. Empirical results using the 270 MKP test problems available in Beasley's OR-Library demonstrate that our implementation of TLBO for the MKP is competitive with published solution approaches without the need for time-consuming parameter fine-tuning.
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Papers by Francis Vasko