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## using Cross-Entropy (2012)

### Citations

229 |
The Cross-Entropy Method: A Unified Approach to
- Rubinstein, Kroese
- 2004
(Show Context)
Citation Context ...ction 6. Applications of CE to combinatorial optimization problems that are less related to our problem are found in, e.g., [8] that solves a capacitated lotsizing problem with setup times. Moreover, =-=[35, 18, 38, 25]-=- solve traveling salesman problems (TSP) using CE, and [36, 18, 37] solve the max-cut problem using CE. In [36], a variant of CE is applied, with the updating rule (18), and applications to numerous c... |

171 | A tutorial on the cross-entropy method
- Boer, Kroese, et al.
- 2005
(Show Context)
Citation Context ...ction 6. Applications of CE to combinatorial optimization problems that are less related to our problem are found in, e.g., [8] that solves a capacitated lotsizing problem with setup times. Moreover, =-=[35, 18, 38, 25]-=- solve traveling salesman problems (TSP) using CE, and [36, 18, 37] solve the max-cut problem using CE. In [36], a variant of CE is applied, with the updating rule (18), and applications to numerous c... |

128 |
A genetic algorithm for the multidimensional knapsack problem
- Chu, Beasley
- 1998
(Show Context)
Citation Context ...vering problem is equal to (1a) and it has n constraints of the form: a1jx1 + a2jx2 + . . .+ amjxm ≥ dj j = 1, . . . , n. (28) The multidimensional knapsack instances are generated with the method of =-=[12]-=-. The values of aij are discrete uniform random numbers from [0, 1000]. Furthermore, dj = α ∑m i=1 aij , where α is a tightness ratio specifying approximately the fraction of the total number of items... |

102 | The cross-entropy method for combinatorial and continous optimization. Methodology and Computing
- Rubinstein
- 1999
(Show Context)
Citation Context ...ction 6. Applications of CE to combinatorial optimization problems that are less related to our problem are found in, e.g., [8] that solves a capacitated lotsizing problem with setup times. Moreover, =-=[35, 18, 38, 25]-=- solve traveling salesman problems (TSP) using CE, and [36, 18, 37] solve the max-cut problem using CE. In [36], a variant of CE is applied, with the updating rule (18), and applications to numerous c... |

45 |
The multidimensional 0-1 knapsack problem - Bounds and computational aspects
- Fréville, Hanafi
- 2005
(Show Context)
Citation Context ...CE aims to generate better solutions in the next iteration. We apply CE optimization, which is successfully applied to problems related to SUAH, like the multidimensional knapsack problem (see, e.g., =-=[22]-=-) and the capacitated facility location problem (see, e.g., [7]). The objectives of multidimensional knapsack and capacitated facility location are to select an ‘optimal’ set of items and an ‘optimal’... |

28 | Using the Cross-Entropy Method to Guide/Govern Mobile Agent’s Path Finding in Networks
- Helvik, Wittner
- 2001
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Citation Context |

25 |
2004, Optimal investment strategies for flexible resources, considering pricing and correlated demands
- Bish, Wang
(Show Context)
Citation Context ... staffing problem as in [33] determines the number of employees needed to cover a given workload, where employees are considered equal, whereas we consider employees with different contract-sizes. In =-=[6, 11, 39]-=- staffing problems in production planning settings are studied, which are modeled as two-stage processes. The first stage decides on the necessary capacity investment. After that, when demand realizat... |

23 |
Resource flexibility with responsive pricing
- Chod, Rudi
- 2003
(Show Context)
Citation Context ... staffing problem as in [33] determines the number of employees needed to cover a given workload, where employees are considered equal, whereas we consider employees with different contract-sizes. In =-=[6, 11, 39]-=- staffing problems in production planning settings are studied, which are modeled as two-stage processes. The first stage decides on the necessary capacity investment. After that, when demand realizat... |

23 |
On the Convergence of the Cross-Entropy Method
- Margolin
- 2005
(Show Context)
Citation Context ...ule (15) is replaced by v̂t,i = λ ∑N k=1 I{S(Xk)≤γ̂t}Xk,i∑N k=1 I{S(Xk)≤γ̂t} + (1− λ)v̂t−1,i, (17) for some 0 < λ < 1, in order to prevent the CE method from converging too quickly [18]. In addition, =-=[17, 31]-=- let λ depend on t, i.e., replace λ by λt in (17), to achieve this. We want to prohibit that CE converges too quickly, because once an entry of v̂t is fixed to 0 or 1, only sample solutions with (if t... |

20 |
Convergence Properties of the Cross-Entropy Method for Discrete Optimization
- Costa, A, et al.
- 2007
(Show Context)
Citation Context ...ule (15) is replaced by v̂t,i = λ ∑N k=1 I{S(Xk)≤γ̂t}Xk,i∑N k=1 I{S(Xk)≤γ̂t} + (1− λ)v̂t−1,i, (17) for some 0 < λ < 1, in order to prevent the CE method from converging too quickly [18]. In addition, =-=[17, 31]-=- let λ depend on t, i.e., replace λ by λt in (17), to achieve this. We want to prohibit that CE converges too quickly, because once an entry of v̂t is fixed to 0 or 1, only sample solutions with (if t... |

11 | The Cross-Entropy Method for Network Reliability Estimation
- Hui, Bean, et al.
- 2005
(Show Context)
Citation Context ... also found in this paper. These include, single-dimensional knapsack, multiple knapsack, bipartite matching, satisfiability, and counting the number of feasible solutions for integer programming. In =-=[1, 10, 26]-=- network re4 liability optimization problems are solved by using CE. A multi-objective optimization problem is solved in [5]. 3 Problem description Consider a workforce planning problem over a finite ... |

10 |
A case study of single shift planning and scheduling under annualized hours: a simple three-step approach
- Azmat, Widmer
- 2004
(Show Context)
Citation Context ... MILP approaches are proposed to solve the annualized hours problem for a fixed number of employees. Also the lower and upper bounds on the working hours per period are the same for all employees. In =-=[2, 3]-=- MILP approaches are proposed to solve an annualized hours problem, with demands expressed in the number of shifts, and constraints on the number and sequences of shifts employees can work. Again the ... |

10 |
Planning annualised hours with a finite set of weekly working hours and joint holidays
- Corominas, Lusa, et al.
(Show Context)
Citation Context ... in terms of number of hours and sequences of shifts they are allowed to work. In [34] shorter solving times are obtained after reformulating an MILP that models a problem closely related to ours. In =-=[13, 15, 16]-=- MILP approaches are proposed to solve the annualized hours problem for a fixed number of employees. Also the lower and upper bounds on the working hours per period are the same for all employees. In ... |

9 |
Scheduling a workforce under annualized hours
- Hung
- 1999
(Show Context)
Citation Context ...n shifts per period, under constraints on both the number and sequences of shifts employees can work, but without considering annualized hours. Variants of the annualized hours problem are studied in =-=[28, 27]-=-, where schedules are created explicitly, which differs from our problem. Also employees are considered equivalent in terms of number of hours and sequences of shifts they are allowed to work. In [34]... |

8 |
A multiple-shift workforce scheduling model under annualized hours
- Hung
- 1999
(Show Context)
Citation Context ...n shifts per period, under constraints on both the number and sequences of shifts employees can work, but without considering annualized hours. Variants of the annualized hours problem are studied in =-=[28, 27]-=-, where schedules are created explicitly, which differs from our problem. Also employees are considered equivalent in terms of number of hours and sequences of shifts they are allowed to work. In [34]... |

7 |
Using a MILP model to establish a framework for an annualised hours agreement
- Corominas, Lusa, et al.
(Show Context)
Citation Context ... in terms of number of hours and sequences of shifts they are allowed to work. In [34] shorter solving times are obtained after reformulating an MILP that models a problem closely related to ours. In =-=[13, 15, 16]-=- MILP approaches are proposed to solve the annualized hours problem for a fixed number of employees. Also the lower and upper bounds on the working hours per period are the same for all employees. In ... |

7 |
Du-Shean Fuh. Sizing and scheduling a fulltime and part-time workforce with off-day and off-weekend constraints
- Emmons
- 1997
(Show Context)
Citation Context ...orks is fixed per period. In addition, shortages are allowed, but lead to a penalty. The decisions in [29] are mainly about when 3 to hire and fire employees, without considering annualized hours. In =-=[20]-=- a staffing problem is solved where demands are expressed in shifts per period, under constraints on both the number and sequences of shifts employees can work, but without considering annualized hour... |

6 | Cross-entropy and rare events for maximal cut and partition problems
- Rubinstein
- 2002
(Show Context)
Citation Context ... are less related to our problem are found in, e.g., [8] that solves a capacitated lotsizing problem with setup times. Moreover, [35, 18, 38, 25] solve traveling salesman problems (TSP) using CE, and =-=[36, 18, 37]-=- solve the max-cut problem using CE. In [36], a variant of CE is applied, with the updating rule (18), and applications to numerous combinatorial optimization problems are also found in this paper. Th... |

5 |
Mixed integer programming to schedule a single-shift workforce under annualized hours
- Azmat, Hürlimann, et al.
(Show Context)
Citation Context ... MILP approaches are proposed to solve the annualized hours problem for a fixed number of employees. Also the lower and upper bounds on the working hours per period are the same for all employees. In =-=[2, 3]-=- MILP approaches are proposed to solve an annualized hours problem, with demands expressed in the number of shifts, and constraints on the number and sequences of shifts employees can work. Again the ... |

5 |
Using MILP to plan annualised working hours
- Corominas, Lusa, et al.
(Show Context)
Citation Context ... in terms of number of hours and sequences of shifts they are allowed to work. In [34] shorter solving times are obtained after reformulating an MILP that models a problem closely related to ours. In =-=[13, 15, 16]-=- MILP approaches are proposed to solve the annualized hours problem for a fixed number of employees. Also the lower and upper bounds on the working hours per period are the same for all employees. In ... |

5 |
Characteristics and classification of annualised working hours planning problems
- Corominas, Lusa, et al.
- 2004
(Show Context)
Citation Context ... and sequences of shifts employees can work. Again the number of employees is minimized, with the employees considered equivalent. A classification scheme for annualized hours problems is proposed by =-=[14]-=-. This paper considers cost-efficient staffing combined with annualized hours. 2.2 CE applied to combinatorial optimization problems CE is applied to a variety of combinatorial optimization problems. ... |

5 | Semi-iterative minimum cross-entropy algorithms for rare-events, counting, combinatorial and integer programming
- Rubinstein
- 2008
(Show Context)
Citation Context ...pacitated facility location are to select an ‘optimal’ set of items and an ‘optimal’ set of (supply) locations respectively. In [7] the capacitated facility location problem is solved using CE and in =-=[9, 19, 36]-=- CE is used to solve (variants of) knapsack problems. In addition to successful CE applications to related problems, metaheuristics, like tabu search and simulated annealing, have also shown to work w... |

3 |
Applying the cross-entropy method in multi-objective optimisation of dynamic stochastic systems
- Bekker
- 2013
(Show Context)
Citation Context ...unting the number of feasible solutions for integer programming. In [1, 10, 26] network re4 liability optimization problems are solved by using CE. A multi-objective optimization problem is solved in =-=[5]-=-. 3 Problem description Consider a workforce planning problem over a finite planning horizon T . Employing staff is subject to cost and availability constraints. Employees, indexed by i = 1, . . . ,m,... |

3 |
A cross entropy-based metaheuristic algorithm for largescale capacitated facility location problems
- Caserta, EQ
(Show Context)
Citation Context ...pply CE optimization, which is successfully applied to problems related to SUAH, like the multidimensional knapsack problem (see, e.g., [22]) and the capacitated facility location problem (see, e.g., =-=[7]-=-). The objectives of multidimensional knapsack and capacitated facility location are to select an ‘optimal’ set of items and an ‘optimal’ set of (supply) locations respectively. In [7] the capacitated... |

3 |
The generalized assignment problem with flexible jobs
- Rainwater, Geunes, et al.
- 2009
(Show Context)
Citation Context ... 27], where schedules are created explicitly, which differs from our problem. Also employees are considered equivalent in terms of number of hours and sequences of shifts they are allowed to work. In =-=[34]-=- shorter solving times are obtained after reformulating an MILP that models a problem closely related to ours. In [13, 15, 16] MILP approaches are proposed to solve the annualized hours problem for a ... |

2 |
A cross entropy-lagrangean hybrid algorithm for the multi-item capacitated lot-sizing problem with setup times
- Caserta, Rico
(Show Context)
Citation Context ...he Matlab File Exchange Central holds a script for this problem [32], see Section 6. Applications of CE to combinatorial optimization problems that are less related to our problem are found in, e.g., =-=[8]-=- that solves a capacitated lotsizing problem with setup times. Moreover, [35, 18, 38, 25] solve traveling salesman problems (TSP) using CE, and [36, 18, 37] solve the max-cut problem using CE. In [36]... |

2 |
A cross entropy based algorithm for reliability problems
- Caserta, Nodar
(Show Context)
Citation Context ... also found in this paper. These include, single-dimensional knapsack, multiple knapsack, bipartite matching, satisfiability, and counting the number of feasible solutions for integer programming. In =-=[1, 10, 26]-=- network re4 liability optimization problems are solved by using CE. A multi-objective optimization problem is solved in [5]. 3 Problem description Consider a workforce planning problem over a finite ... |

2 |
An integrated staff-sizing approach considering feasibility of scheduling decision
- Li, Chen, et al.
(Show Context)
Citation Context ...ment. After that, when demand realizations become known, the second stage decides on capacity allocation. These papers focus mainly on profits that are induced by production capacities and demand. In =-=[29]-=- a staffing problem is studied in which employees can be hired and fired per period. A fixed demand is given per period and the number of hours an employee works is fixed per period. In addition, shor... |

1 |
Multidimensional knapsack problem instances
- Beasley
(Show Context)
Citation Context ...taff or under-staff in a planning period. In total, we thus generate 10 · 10 · 30 · 10 = 30000 instances. We generate these instances instead of the instances of [12] that are made publicly available =-=[4]-=-, since the instances we generate better fit with our underlying practical application. In the SUAH problem it is typical to have up to 100 employees (variables) and up to 50 planning periods (constra... |

1 |
A cross entropy algorithm for the knapsack problem with setups
- Caserta, Rico, et al.
(Show Context)
Citation Context ...pacitated facility location are to select an ‘optimal’ set of items and an ‘optimal’ set of (supply) locations respectively. In [7] the capacitated facility location problem is solved using CE and in =-=[9, 19, 36]-=- CE is used to solve (variants of) knapsack problems. In addition to successful CE applications to related problems, metaheuristics, like tabu search and simulated annealing, have also shown to work w... |

1 |
A directed quantile crossentropy method for 0/1 knapsack problems
- Deng, Qiao, et al.
- 2009
(Show Context)
Citation Context ...o be prevented. There are two main approaches to deal with this. A first approach is to penalize infeasibility, see, e.g., [1, 10, 17]. A second approach is to repair infeasible solutions, see, e.g., =-=[9, 19]-=-. In our implementation, we choose to penalize infeasibility, which is discussed in detail in Section 5, next to a possible application of repair functions. 4.2 Solution strategy motivation This secti... |

1 |
Fathabadi and Mehdi Ghiyasvand. A new algorithm for solving the feasibility problem of a network flow
- Salehi
(Show Context)
Citation Context ... = w̄i i ∈M (19c) 0 ≤ w̄ij ≤ ūij i ∈M ; j = 1, . . . , n. (19d) Instead of an MILP we now have an LP, which is solvable by polynomial time algorithms. Moreover, (19) is a network feasibility problem =-=[21, 23]-=-, for which special purpose algorithms exist. 11 5.2 Initialization This section describes how our CE method is initialized. In many CE implementations that involve binary variables, v0 is set to ( 1 ... |

1 |
A new approach for computing a most positive cut using the minimum flow algorithms
- Ghiyasvand
(Show Context)
Citation Context ... = w̄i i ∈M (19c) 0 ≤ w̄ij ≤ ūij i ∈M ; j = 1, . . . , n. (19d) Instead of an MILP we now have an LP, which is solvable by polynomial time algorithms. Moreover, (19) is a network feasibility problem =-=[21, 23]-=-, for which special purpose algorithms exist. 11 5.2 Initialization This section describes how our CE method is initialized. In many CE implementations that involve binary variables, v0 is set to ( 1 ... |

1 |
Grabot and Agns Letouzey. Short-term manpower management in manufacturing systems: new requirements and DSS prototyping
- Bernard
(Show Context)
Citation Context ...ecisions on the contractmix and skill-mix of their workforce. In addition, some organizations have ∗corresponding author: Egbert.vanderVeen@ortec.com 1 adapted the annualized hours regime (see, e.g., =-=[24, 30]-=-), which allows them to measure working time per year, rather than per week or per month. This extra flexibility implies decision making on the distribution of workforce capacity throughout the year. ... |

1 |
Teso’s new composite distribution needed annual hours
- Why
- 1995
(Show Context)
Citation Context ...ecisions on the contractmix and skill-mix of their workforce. In addition, some organizations have ∗corresponding author: Egbert.vanderVeen@ortec.com 1 adapted the annualized hours regime (see, e.g., =-=[24, 30]-=-), which allows them to measure working time per year, rather than per week or per month. This extra flexibility implies decision making on the distribution of workforce capacity throughout the year. ... |

1 |
Multi-knapsack solver [www page]. http://www.mathworks.com/matlabcentral/fileexchange/ 20436-multi-knapsack-solver, Retrieved
- Paris
- 2012
(Show Context)
Citation Context ...similar to ours, see Section 3. Although literature does not report on CE applications to solve the multidimensional knapsack problem, the Matlab File Exchange Central holds a script for this problem =-=[32]-=-, see Section 6. Applications of CE to combinatorial optimization problems that are less related to our problem are found in, e.g., [8] that solves a capacitated lotsizing problem with setup times. Mo... |

1 |
Planning and Scheduling in Manufacturing and Services, 2nd Edition
- Pinedo
- 2009
(Show Context)
Citation Context ...2.1 Staffing under annualized hours literature This section first discusses literature on staffing problems. After that, annualized hours literature is discussed. The classical staffing problem as in =-=[33]-=- determines the number of employees needed to cover a given workload, where employees are considered equal, whereas we consider employees with different contract-sizes. In [6, 11, 39] staffing problem... |

1 |
Mieghem and Maqbool Dada. Price versus production postponement: Capacity and competition
- Van
- 1999
(Show Context)
Citation Context ... staffing problem as in [33] determines the number of employees needed to cover a given workload, where employees are considered equal, whereas we consider employees with different contract-sizes. In =-=[6, 11, 39]-=- staffing problems in production planning settings are studied, which are modeled as two-stage processes. The first stage decides on the necessary capacity investment. After that, when demand realizat... |