Variable metric approach for multi-skill staffing problem in call center

Transcription

1 Variable metric approach for multi-skill staffing problem in call center A. El Mouatasim Faculty of Science - Jazan University P.B Jazan Saudi Arabia. aelmouatasim@jazanu.edu.sa CIRO 10: Marrakech Mai 2010

2 Plan of Presentation 1 Introduction Algorithm of Variable Metrics 5

3 The aim of this research is minimizing the staffing costs of a multi skill call center subject to service-level requirements which are estimated by simulation or approximation, that give nonlinear integer programming [1]. In this research we propose random perturbation of variable metrics method (RPVM) via penalty function method for solving scheduling and staffing problem in call center.

4 The aim of this research is minimizing the staffing costs of a multi skill call center subject to service-level requirements which are estimated by simulation or approximation, that give nonlinear integer programming [1]. In this research we propose random perturbation of variable metrics method (RPVM) via penalty function method for solving scheduling and staffing problem in call center.

5 Call types k=1,...,k ; Agent types (or skill groups) i=1,...,i ; Periods p=1,...,p ; Shift types q=1,...,q. The vectors c, x and y are costs, decision variables and auxiliary variables respectively. Details

6 We have y = Ax where A is a block diagonal matrix with i blocks Ã, and element (p, q) of à is 1 if shift q covers period p, 0 otherwise. g k,p (y) = long-run service level for call type k in period p. g p (y) = aggregated long-run service level over period p. g k (y) = aggregated long-run service level for call type k. g(y) = aggregated long-run service level overall.

7 Scheduling Problem Minimize : c t x = I i=1 q=1 Q c i,q x i,q Subject to : Ax = y, g k,p (y) l k,p for all k, p, g p (y) l p for all p, g k (y) l k for all k, g(y) l, x 0, and integer. (1)

8 Staffing Problem Relaxation : forget about the admissibility of schedules ; just assume that any staffing is admissible. Costs : c = (c 1,1,.., c 1,P,.., c I,1,.., c I,P ) where c i,p = cost of an agent of type i in period p. Minimize : c t y = I P c i,p y i,p i=1 p=1 Subject to : g k,p (y) l k,p for all k, p, g p (y) l p for all p, g k (y) l k for all k, g(y) l, y 0, and integer. (2)

9 We consider the integer programming problem with nonlinear constraint in the form of scheduling problem (1). Let S sch and S sch be the sets defined by S sch = {x R IQ Ax = y, g k,p (y) l k,p, g p (y) l p, g k (y) l k for all k, p, g(y) l, x 0} S sch = {x x S sch, x i is integer, i = 1, 2,..., IQ}.

10 We consider the integer programming problem with nonlinear constraint in the form of staffing problem (2). Let S sta and S sta be the sets defined by S sta = {y R IP g k,p (y) l k,p, g p (y) l p, g k (y) l k for all k, p, g(y) l, y 0} S sta = {y y S sta, y i is integer, i = 1, 2,..., IP}.

11 Definition For an integer point x, the set N(x ) = {x : x x 1 5 } is called a 1 5 -cubic neighborhood of the integer point x.

12 Define the penalty function of scheduling problem φ sch (x, y, r, k) = c t x + r[ax y + K k=1 p=1 P max{0, l k,p g k,p (y)}+ + P max{0, l p g p (y)} + K max{0, l k g k (y)}+ p=1 k=1 + max{0, l g(y)}] k IQ cos 2πx i then the sample constraints global optimization problems associated to scheduling problem is i=1 min φ sch (x, y, r, k) (3) x R IQ+

13 Define the penalty function of staffing problem φ sta (y, r, k) = c t y + r[ K k=1 p=1 P max{0, l k,p g k,p (y)}+ + P max{0, l p g p (y)} + K max{0, l k g k (y)}+ p=1 k=1 + max{0, l g(y)}] k IP cos 2πy i then the sample constraints global optimization problems associated to staffing problem is where r and k are large enough. i=1 min y R IP+ φ sta (y, r, k), (4)

14 Lemma Suppose that x is a global minimizer of optimization problem (3) in S sch and if x is in a 1 5- cubic neighborhood of an integer point x S sch, then x is a solution of the scheduling problem (1).

15 Lemma Suppose that y is a global minimizer of optimization problem (4) in S sta and if y is in a 1 5- cubic neighborhood of an integer point ȳ S sta, then ȳ is a solution of the staffing problem (2). [3].

16 Lemma Suppose that y is a global minimizer of optimization problem (4) in S sta and if y is in a 1 5- cubic neighborhood of an integer point ȳ S sta, then ȳ is a solution of the staffing problem (2). [3].

17 Algorithm of Variable Metrics Stochastic Perturbation We denote that ϕ(x) = φ(x, r, K ). Step 0. Parameters : α 1 = 1.5, α 2 = 0.8, α 3 = Data : x, ρ,, G. Step 1. Initialization. Set s = 1,, l = 0, B = I. Step 2. Function and subgradient calculation : ϕ = ϕ(x), h ϕ(x). Step 3. If h G, then stop. Step 4. Subgradient normalization : h = h/ h.

18 Algorithm of Variable Metrics Stochastic Perturbation Step 5. Set h d = h. Step 6. Set h b = B t h, h m = Bh b, d s = hm h b. Step 7. Set x p = x, ϕ p = ϕ. Step 8. Set Step 9. If x x p, then stop. Step 10. Function computation : ϕ = ϕ(x). x = ω(x) = x ρd s. (5) Step 11. If ϕ < ϕ p, then set l = l + 1 ; if s = 1, then set ρ = 1.5ρ ; if l > 1 and s > 1, then set ρ = α 1 ρ ; go to Step 7.

19 Algorithm of Variable Metrics Stochastic Perturbation Step 12. If l = 0, then set ρ = α 2 ρ. Step 13. Set h ϕ(x). Step 14. If h G, then stop. Step 15. Subgradient normalization : h = h/ h. Step 16. Set B = B + α 3 (h d h t B + hhd t B). Step 17. Set x = x p, ϕ = ϕ p. Step 18. If l = 0, then h = h d. Step 19. Set l = 0, s = s + 1. Step 20. Go to Step 7.

20 Algorithm of Variable Metrics Stochastic Perturbation The main difficulty remains the lack of convexity : if the objective function is not convex (e.g., g is not concave), the Kuhn- Tucker points may be not correspond to global minimum. In the sequel, we shall improve this point by using an appropriate random perturbation.

21 Algorithm of Variable Metrics Stochastic Perturbation The sequence of real numbers {x k } k 0 is replaced by a sequence of random variables {X k } k 0 involving a random perturbation P k of the deterministic iteration (5). A simple way for the generation of a convenient sequence of perturbations {P k } k 0 is where P k = ζ k Z k 1 {ζ k } k 0 is non increasing sequence of strictly positive real numbers converging to 0 and such that ζ {Z} k 0 is a sequence of random vectors taking their values on R n,

22 Algorithm of Variable Metrics Stochastic Perturbation The sequence of real numbers {x k } k 0 is replaced by a sequence of random variables {X k } k 0 involving a random perturbation P k of the deterministic iteration (5). A simple way for the generation of a convenient sequence of perturbations {P k } k 0 is where P k = ζ k Z k 1 {ζ k } k 0 is non increasing sequence of strictly positive real numbers converging to 0 and such that ζ {Z} k 0 is a sequence of random vectors taking their values on R n,

23 Algorithm of Variable Metrics Stochastic Perturbation and a simple strategy consists in X 0 = x 0 ; k 0 X k+1 = Q k (X k )+P k (6) Equation (6) can be viewed as perturbation of the descent direction d k, which is replaced by a new direction D k = d k + P k /µ k and the iterations (5) become X k+1 = X k + µ k D k.

24 Algorithm of Variable Metrics Stochastic Perturbation and a simple strategy consists in X 0 = x 0 ; k 0 X k+1 = Q k (X k )+P k (6) Equation (6) can be viewed as perturbation of the descent direction d k, which is replaced by a new direction D k = d k + P k /µ k and the iterations (5) become X k+1 = X k + µ k D k.

25 Algorithm of Variable Metrics Stochastic Perturbation The procedure generates a sequence U k = f (X k ). By construction this sequence is decreasing and lower bounded by l = min x R +φ (x, r, k). k 0 : l U k+1 U k (7) Thus, there exists U l such that U k U for k + [2] Theorem Let Z k = Z, where Z is a random variable following N(0, σid), σ > 0 and let a ζ k = (8) log(k + d) where a > 0, d > 0 and k is the iteration number. Then, for a large enough, U = l almost surely.

26 Algorithm of Variable Metrics Stochastic Perturbation The procedure generates a sequence U k = f (X k ). By construction this sequence is decreasing and lower bounded by l = min x R +φ (x, r, k). k 0 : l U k+1 U k (7) Thus, there exists U l such that U k U for k + [2] Theorem Let Z k = Z, where Z is a random variable following N(0, σid), σ > 0 and let a ζ k = (8) log(k + d) where a > 0, d > 0 and k is the iteration number. Then, for a large enough, U = l almost surely.

27 Example A topologically small center In our first example, there are 5 call types and 4 skill groups. The arrival rates and service rates per hour are : (λ j ) 5 j=1 = (1290, 659, 350, 202, 673) and (µ j) 5 j=1 = (6.2, 6.2, 16.6, 14, 5.1). We have target service-levels (l j ) 5 j=1 = (.80,.80,.80,.75,.75), agent skill (s i) 4 i=1 = (2, 2, 1, 2). The routing policy is as follows. Overflow routing data are : R 1 = {2}, R 2 = {1}, R 3 = {3, 4, 2}, R 4 = {4} and R 5 = {1}. The RPVM approach furnish the following optimal vector solution see also Table 1, staff opt = (252, 229, 4, 24), CPU time = 2284, k sto = 40.

28 Example A medium-size center There are 7 call types and 10 skill groups. The arrival rates and service rates per hour are (λ j ) 7 j=1 = (200, 133, 323, 760, 95, 10, 380) and (µ j) 7 j=1 = (7.7, 7.7, 7.5, 7.7, 15, 7.7, 15). We have target service-levels (l j ) 7 j=1 = (.80,.80,.80,.75,.60,.60,.60), agent skill (s i ) 10 i=1 = (2, 1, 2, 2, 1, 2, 1, 2, 2, 1). Overflow routing data are : R 1 = {1}, R 2 = {1, 3}, R 3 = {2, 4}, R 4 = {5, 4, 3, 6}, R 5 = {7, 6, 8, 9}, R 6 = {9} and R 7 = {10, 8}. The initial point x 0 = (42, 40, 15, 25, 80, 15, 5, 10, 5, 25). The RPVM approach furnish the following optimal vector solution see also Table 1, staff opt = (43, 37, 13, 24, 76, 10, 2, 8, 3, 21) CPU time = 4545, k sto = 35.

29 A.N. Avramidis, W. Chan, M. Gendreau, P. L Ecuyer and O. Pisacaneand, Optimizing daily agent scheduling in a multiskill call center, European Journal of Operational Research 200, , A. El Mouatasim, R. Ellaia and J.E. Souza de Cursi, Random perturbation of variable metric method for unconstraint nonsmooth nonconvex optimization, Applied Mathematic and Computer Science, 16(4), , G. Zhang, A note on : A continuous approach to nonlinear integer programming, Applied Mathematics and Computation 215, , 2009.

30 Merci pour votre attention

31 Call types k=1,...,k ; Agent types (or skill groups) i=1,...,i ; Periods p=1,...,p ; Shift types q=1,...,q. The vectors c, x and y are costs, decision variables and auxiliary variables respectively. The shift specifies the time when the agent starts working, when he/she finishes, and all the lunch and coffee breaks. c = (c 1,1,.., c 1,Q,.., c I,1,.., c I,Q ) where c i,q = cost of an agent of type i having shift q. x = (x 1,1,.., x 1,Q,.., x I,1,.., x I,Q ) where x i,q = number of an agent of type i having shift q. y = (y 1,1,.., y 1,P,.., y I,1,.., y I,P ) where y i,q = number of an agent of type i in period p. Return