Multiple Criteria Secretary Problem: A New Approach

Transcription

1 J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and S. P. Kane, K. T. H. M. College, ashik, India R. T. M. agpu Univesity, agpu, India Received: Dec. 03, Revised: May 04, Accepted: 5 May 04 Published online: Jul. 04 Abstact: In multiple citeia secetay poblem selection of a unit is based on two independent chaacteistics. The units, appeaed befoe an obseve ae known (say. is consideed as best ank of a unit. A unit is selected, if it is bette with espect to eithe fist o second o both the chaacteistics. In this pape, joint pobability distibutions along with maginal distibutions (of eal anks of both the chaacteistics and the position at which the selection is made ae deived systematically using simple and explicable method. Futhe these maginal distibutions ae used to deive expected cost of inspection and expected value of eal ank of selected unit. A new citeion fo selecting the best unit based on expected eal ank is developed. Keywods: Secetay Poblem, joint distibution, maginal distibution, eal anks, selection citeion Intoduction The Secetay Poblem deals with the sequential decision pocedue. Accoding to the classical secetay poblem, andomly aanged units ae to be obseved one afte anothe with the aim of stopping at a suitable position and selecting a unit appeaing at that position such that the pobability of selecting the best unit is maximum. This is done with the condition that the units once obseved and ejected ae not allowed to be called back by the obseve at any time in futue, and moe ove if the obseve eaches the last, the th one, then it must be accepted. Many solutions and vesions of the poblem ae available in liteatue. Multiple citeia optimal selection poblems wee intoduced in moe geneal fom, with obsevations in a patially odeed set and with an abitay payoff utility by Beezovskii, Geninson and Rubchinskii (980 and Stadje (980. The multiple citeia poblem of optimum stopping of the selection pocess was solved by Gnedin (98. Such a poblem may be consideed as a genealization of the classical secetay poblem (one citeion best choice poblem as found in Gilbet and Mostelle (966. Poblems in which the unit selected is said to be the best if it is optimal with espect to a social choice function, fo example Paeto optimal, wee teated by Beezovskii and Gnedin (98, Gnedin (983, Bayshnikov, Beezovskii and Gnedin (984. Bayshnikov and Gnedin (986, Samuels and chotlos (987 discussed the poblem whee the goal of the obseve is to minimize the expectations of the sum of the anks of the unit selected, ank one being the best. Feguson (99 genealized the poblem pesented by Gnedin by allowing dependencies between the attibutes, and showed that the optimal policy has the same theshold fom as the standad single attibute Classical Secetay Poblem (CSP. Samuels and Chotlos (987 extended the ank minimization poblem of Chow et al. (964. They sought an optimal policy fo minimizing the sum of two anks fo independent attibutes. Beaden et al. (004 poposed a multi-attibute (o multidimensional genealization of genealized secetay poblem, pesenting a method fo computing its optimal policies, and testing it in two expeiments with incentive-compatible payoffs. In the pesent study of multiple citeia secetay poblem andom vaiables ae eal anks of both the chaacteistics and the position at which the selection is made. The joint distibution and maginal distibutions of these andom vaiables ae deived using simple algeba given in section. In section 3, expected values of these andom vaiables ae found to obtain expected eal ank of the selected unit and cost incued in the selection pocess. In section 4, optimality citeion based on pobabilistic appoach is discussed. A new optimality citeion based on expected eal ank is developed and its usefulness ove pobabilistic appoach is evealed in the last section. Coesponding autho alakapadhye@gmail.com, svpkane@yahoo.co.in c 04 SP atual Sciences Publishing Co.

2 30 A. Padhye, S. P. Kane : Multiple Citeia Secetay Poblem: A ew... Multiple Citeia Secetay Poblem Thee ae known units. Assume that each unit can be anked with espect to two obsevable chaacteistics which ae common fo all units. The best unit is given ank, the second best ank (-, etc. and the wost ank. It is assumed that these two chaacteistics ae independent. The selection pocedue is as follows:. Obseve the fist units without selecting any.. Select i th unit if it is bette than the best of the fist units with espect to the fist o the second o both the chaacteistics then stop (+ i If none of the (- units is selected, then th unit is to be selected. Let R k (i (k, ; i,,..., be the eal ank of the i th unit with espect to the k th chaacteistic. Let X and X be the eal anks of Y th unit with espect to the fist and second chaacteistics espectively. ote that X, X and Y ae discete andom vaiables having the anges X,,..., ; X,,..., and Y The total numbe of pemutations of R (, R (,..., R ( and R (, R (,..., R ( ae (! and each pai is assumed to be equally likely. otations:. α (x, y (x y (. P(x, x, y, P(X x, X x, Y y, is the pobability that the obseve stops afte examining y units and the selected unit has eal anks x,x with espect to chaacteistic and chaacteistic espectively fo fixed values of and. 3. P(x, x,, is the joint pobability distibution of X and X. 4. P X (x, is the maginal pobability distibution of X. 5. Px (x, is the maginal pobability distibution of X. 6. P y (y, is the maginal pobability distibution of Y. To analyze this poblem we need a esult deived by Kane S. P. (988 fo one chaacteistic, is as given below: Joint distibution of X and Y: Result: The pobability of (X x, Y y is given by ( α(x,y + y ; y x P(x, y, ( y ; x 0, othewise. Poof : Poof is divided into two pats. When Y - Let j be the maximum eal ank of the fist units. The eal anks of the units in y- positions must be less than j. The unit with eal ank j can be at any of the positions. Futhe, j can be at most y-and it cannot exceed x-. Theefoe, the numbe of pemutations qualifying such condition is equal to ( j ( (-y! Theefoe, P(x,y, ( y!! Using the esult fom A. fom Appendix and simplifying, we get P (x, y, ( y! (y! (x (y ( (y P (x, y, ( y! (y! (x (y ( x ( j ( jy P (x, y, α(x,y, y x, + y - When Y i If Y and X,,..., -, then it is obvious that the best unit has aleady been appeaed in the fist units. Hence excluding the best unit which is in fist units and the last inspected unit, the emaining (- units can appea in (- ways. The best unit can be at any one of the fist positions theefoe the numbe of pemutations qualifying this condition is (-. Hence the pobability of this event is P (x, y, (!! Thus, c 04 SP atual Sciences Publishing Co.

3 J. Stat. Appl. Po. 3, o., 9-38 (04 / 3 P (x, y, (, x ii If Y and X, then it is obvious that the second best unit has aleady been appeaed in the fist units. Hence, the pobability of this event is (!!. P (,, ( Fo othe pais of X and Y, P (x, y, s ae zeo. Thus we have, ( α(x,y, + y ;y x P(x, y, (, y ; x 0, Othewise. Lemma: The pobability that the units in the positions fom ( + though y ae not elatively bette than the best of the fist units (x is eal ank of the unit at the y th position, denoted by P (x, y is given by P (x, y ( α(,y α (x, y, if + y, x, Poof: Let j be the eal ank of the fist units. j must be at least x + and at the most. Since j can be in any one of the positions, the numbe of pemutations ae: ( j ( (-y! ( P (x, y P(x,y ( y!! ( y! (!! ( y! (x! (x y! α(,y α (x, y, if + y, x, ( j ( jx+ Remak: Using the above lemma and the esult by Kane S. P. fo one chaacteistic, joint pobability distibution in multiple citeia secetay poblem is deived.. Joint Pobability Distibution of X, X and Y Theoem : The joint pobability distibution of (X, X, Y is given by, α (x (, yα(, y+α (x, yα(, y α (x, yα (x, y, + y ; y x, x P(x,x,y, ( (, y ; x, x 0, othewise Poof: Following ae the fou mutually exclusive and exhaustive events, defined by, A, A, A 3, A 4 associated with the Y th unit : A : Y th unit is elatively bette with espect to X but it is not bette with espect to X. A : Y th unit is elatively bette with espect to X but it is not bette with espect to X. A 3 : Y th unit is elatively bette with espect to both chaacteistics. A 4 : Y th unit is not elatively bette with espect to both chaacteistics. The joint pobabilities of (X, X, Y with espect to above fou events ae as follows: c 04 SP atual Sciences Publishing Co.

4 3 A. Padhye, S. P. Kane : Multiple Citeia Secetay Poblem: A ew... ( α(x, yα(, y α(x, y, i f y x ; x ; + y, unde A ( α(x, yα(, y α(x, y, i f y x ; x ; + y, unde A ( α(x, y α(x, y, i f y x ; y x ; + y, unde A 3 (, i f y; x, x, unde A 4 Combining the above fou pobabilities, joint pobability distibution of X, X and Y as given in ( is obtained.. Joint Distibution of X and X Coollay : The joint distibution of (X, X is P(x, x,, + ( ( α(x, yα(, y+α(x, yα(, y α(x, yα(x, y, x, x 0, othewise ( Poof: P(x,x,, ( + (.3 Maginal Distibutions of X, X and Y P(x,x,y, P(x,x,y,+P(x,x,, α(x, yα(, y+α(x, yα(, y α(x, yα(x, y Coollay: The maginal pobability distibutions of X and X ae espectively given by, + ( + y(y ( y α(x, y P x (x, i f x (3 0, othewise c 04 SP atual Sciences Publishing Co.

5 J. Stat. Appl. Po. 3, o., 9-38 (04 / 33 ( + P x (x, 0, othewise y(y + ( y α(x, y i f x (4 Poof: P x (x, x P(x,x,, ( ( + ( α(x, yα(, y+ α(, y x y α(x, y ( ( + ( α(x, yα(, y+ ( ( y α(x, y x y ( y ( α(x, y ( y ( (by using A.3 fom Appendix ( α(x, y ( P x (x, ( + y(y + ( y α(x, y, i f x Since P(x,x,, is symmetic in X and X, the maginal pobability distibution of X is given by (4. Coollay 3: Maginal pobability distibution of Y is, (y y (y, i f + y P y (y, (, i f y. (5 0, othewise Poof: The poof is given in two pats: Case i + y - P y (y, x x P(x,x,y, c 04 SP atual Sciences Publishing Co.

6 34 A. Padhye, S. P. Kane : Multiple Citeia Secetay Poblem: A ew... ( ( (y y (y Case ii y x x ( y α(x, yα(, y+ α(x, yα(, y α(x, yα(x, y fom ( ( y ( (y ( + ( (y ( ( ( P y (y, y ( y ( x x ( fom ( P(x,x,, 3 Expected Values of X, X and Y Coollay 4: The mathematical expectation of X is given by { E (X, (+ ( + } y(y + (y+y(y (6 Poof: x x ( + E(X, x x P x (x, y(y + ( y α(x, y f om( (+ ( + (+ (+ ( (+ ( E (X, (+ + (+ { + (+ y(y + ( y(y + ( ( + x ( x y(y + ( y(y + (y+y(y x ( + y+ ( y ( x y ( Remak: Fom (3 and (4, it may be seen that, E (X, E (X,. E(X, is computed fo vaious combinations of (fom to - and (0, 5, 0 as listed in Table. } ( x y c 04 SP atual Sciences Publishing Co.

7 J. Stat. Appl. Po. 3, o., 9-38 (04 / 35 TABLE: Computation of E (X, showing the values of E(X, fo some E(X,0 E(X, 5 E(X, 0 E(X, 5 E(X, Fom the above table, it is obseved that E(X, attains maximum at some fo given. Expected value of Y: Coollay 5: Expectation ( of Y is given by E(Y, (7 (+ ( Poof: E(Y, yp y (y, y P y (y, +P y (y, (y y(y + ( + ( + (+ + + ( 4 Optimality Citeion fo Selection It may be ecalled that denotes the numbe of units that ae passed without selection. In oiginal Secetay Poblem, the usual citeion fo the choice of optimum is to maximize the pobability that the best unit is selected. On the same lines, we suggest the following optimality citeion fo selection of. c 04 SP atual Sciences Publishing Co.

8 36 A. Padhye, S. P. Kane : Multiple Citeia Secetay Poblem: A ew... Select such that PX o X, is maximum. We know that, P x, P x,. ( + y(y + ( + (+ + + ( ( y(y f om(3 P(,, ( + ( ( (y f om ( (+ ( Theefoe, PX o X, P x, + P x, P,, ( (+ ( PX o X, is computed fo vaious combinations of (fom to - and (0, 5, 0 as listed in Table. TABLE : Computation of PX o X, R P( X o X P( X o X P( X o,0,5 X, Fom the above table it may be noted that PX o X, attains maximum at a value of (say 0 fo given. This 0 is the optimum value of unde said optimality citeion. In this connection we have the following coollay. Coollay 6: Fo given, 0, satisfies the following inequalities: ( 0 + k 0 + k ( 0 k 0 k (8 Poof: We have noticed ealie that PX o X, is maximum at 0. So fo this 0, we must have PX o X 0 -, PX o X 0, PX o X 0 +, The fist half of the above inequality leads to c 04 SP atual Sciences Publishing Co.

9 J. Stat. Appl. Po. 3, o., 9-38 (04 / 37 ( 0 ( ( ( 0 ( ( 0 + ( On simplifying, we get ( 0 k k 0...( PX o X 0, PX o X 0 +, leads to 0 ( + 0 ( ( ( 0 + ( + ( 0 + On simplifying, we get ( ( Fom (* and (** we get, ( 0 + k 0 + k...( ( 0 + k 0 + k ( 0 k k 0 These inequalities ae too complex to give 0 explicitly. Howeve fo lage, using Eule s summation fomula in (8, we get a good appoximate solution fo 0 which is appoximately equal to the integal pat of /. And the maximum value of PX o X, PX o X 0, is appoximately equal to /. This fact was also noticed by Gnedin (98 using diffeent appoach. If 0 is chosen to be integal pat of /, then E(Y/ 0, is appoximately equal to 3/4.That is 75% of total numbe units ae expected to be obseved which is quite lage. 5 ew Appoach fo Optimum The aim is to select a unit such that it should be sufficiently good as pe eithe citeion. This means that select a unit such that E (X, o E(X, is maximum. Theefoe, we popose the following citeion. Choose such that E(X, is maximum. Since E(X, E(X,, the optimum, denoted by should satisfy E(X, E(X,, fo all Such * exists in view of the discussion in section 3. Theefoe, fo optimum *, we have the following inequalities: E (X -, E(X, E(X +,... (9 Putting the expessions in (9 and simplifying, we get ( (3 ( + ( B A ( + (3 + ( + ( + B (0 Whee A ( and B y(y + (y+y(y These inequalities ae also too complex to povide explicit expession fo. In the following table, Expected values of X and Expected values of Y ae given fo 0 and, fo some selected values of. c 04 SP atual Sciences Publishing Co.

10 38 A. Padhye, S. P. Kane : Multiple Citeia Secetay Poblem: A ew... TABLE 3. VALUES OF E (Y,, E(Y 0,, E (X, and E(X 0, FOR SOME SET OF (, 0, 0 E(Y 0, E(Y, E(X 0, E(X, Conclusion Fom the above table, it may be noted that, in the citeion, which consides the maximization of pobability of selecting the best unit, attention is not paid to the eal anks of the selected units if the pocedue fails to select the best unit. Howeve, if we select the new appoach of optimality citeion based on anks, we can see that E(X, of the selected unit is lage. Futhe it can be obseved that is always less than 0. Hence, the expected cost of inspection coesponding to the scheme which allows to inspect fist units only, without selecting any fom them is less than the expected cost of inspection coesponding to the scheme which allows to inspect 0 units without selecting any fom them. This, theefoe, suggests that it is moe appopiate to choose optimum value of as it is going to educe the obsevation cost, and at the same time ank of the selected unit is appoximately 0.7. APPEDIX A. : n ( n(n (n...(n +. A. : b na n (x x+ (b+ (x+ a (x+ A.3 : b n ( na( x b+ ( x+ a x+ A.4 : ( x y x y+ y ( x Refeences Bayshnikoy, Yu. M., Beezovskiy, B, A. and Gnedin, A.V. (986. On class of best choice poblems, Infom Sci. 39, -7. Batoxzynski, R. and Govindaajulu, Z. (978.The secetay poblem with inteview cost, Sankhya Se. B 40, Beaden, J.., Muphy, R.O.(004. On Genealized secetay poblems. Kluwe Academic Publishes. Pinted in ethelands.td Document.tex; Beaden, J.., Muphy, R.O. and Rapopot, A (004. A multi-attibute extension of the secetay poblem: Theoy and expeiments. Pepint submitted to Jounal of Mathematical Psychology. 5 Chow, Y.S., Robbins, H. Moiguti, S. and Samuels, S.M. (964. Optimal selection based on elative ank (the secetay poblems ; Isael J. Math, Chow, Y.S., Robbins, H. and Selgmund, D. (97 Geat Expectations: The Theoy of optimal Stopping. Houghton Mifflin, Boston. 7 Feguson, T.S. (989 Who solved the secetay poblem? Statistical science 4, Gilbet, J. and mostelle, F. (966 Recognizing the maximum of sequence. Jounal of Ameican Statistical Association.6, Gnedin, A.V. (98 Multiciteial poblem of optimum stopping of the selection pocess, (tanslated fom Russian. Automation and Remote Contol 4, Samuels, S.M. and Chotlos, B. (966. A multiple citeia optimal selection poblem. Adaptive Statistical Pocedues and Related Topics, (J. Van Ryzin, ed. I.M.S. Lectue notes- Monogaph Seies, vol. 8. Stadje, W. (980 Efficient stopping of a andom seies of patially odeed points Multiple Citeia Decision Making Theoy and Applications Lectue notes in Econ.and Math.Syst.vol.77, Spinge-Velag, c 04 SP atual Sciences Publishing Co.