The stochastic linear continuous type knapsack problem: A generalized P model

l l g

The stochastic linear continuous type knapsack problem: A generalized P mode:

Hiroaki ISHII and Toshio N I S H I D A Facz,',ty of Engineering. Department of Applied Physi,.'s, Osaka University, Suita, Osaka 56 5, Japan

Abstract. This paper considers a stochastic version of the linear continuous type knapsack problem in whicla the cost coefficients are random variables. The problem is to find an optimal solution and an optimal probability level of the chance constraint. This problem P0 is first transformed into a de- terministic equivalent problem P. Then a subprob- lem with a positive parameter is introduced and a close relation between P and its subproblem is shown. Further, an auxiliary, problem of the sub- problem is introduced and a direct relation be- tween P and the auxiliary problem is derived through a relation connecting the mbproblem and it.,; auxiliary problem. Fully utilizing these rela- tions, an efficient algorithm is proaosed that finds an optimal solution of P in at most O(n 4) compu- tational time where n is the nun,ber of decision variables. Finally, further r,:search problems are discussed.

of the change constraint. The problem P0 is to determine an optimal solution and an optimal probability level a minimizing a linear function of the original variables x and a. That is, Po is the LKP version of a generalized P model considered in lshii et al. [3].

Section 2 formulates the p rob lemP o a_nd gives its deterministic equivalent problem Po. Po is fur- ther transformed into the simplified problem P by variable transformations, which is the main prob- lem in this paper. Section 3 introduces subproblem p0 with a positive parameter q and clarifies a close relation between P and Pq. An auxiliary problem Pq of P'~ with positive parameter R is also intro- duced and a relation between Pq and Pg is known. From these relations, a direct relation connecting P and P~ is derived. Based on the relation, Section 4 proposes a parametric type algorithm that finds an optimal solution of P in at most O(n 4) compu- tational time. Finally Section 5 discusses further research problems.

2. Problem formulation

Ke.~vords: ('ombinatorial anaiys, s, probabilistic pr:~gramming, polynomial time a~gorithm

1. Introduction

The linear continl,ous type knaosack problem. i.e. the continuous version of the knapsack prob- lem (abbreviated as LKP) has been well studied and many efficient algorithms are known. For example, works of Balas and Zemel [1] and Jobn- son and Mizoguchi [4] are O(n) algorithms where n is the number of decision variables.

This paper generalizes LKP and considera a stochastic version Pu of LKP with stochastic cost ,,:oefficients and a controllable Frobabillt,: level a

Received X fa~ 1983: revised Janua.~ 1984

North-Holland

European Journal of Operattonal Research 19 (1985 j 118-- ] 24

Ti~e following stochastic programming problem P, is considered.

(P0) Minimize

subject to

d - yoe,

Pr (':x~ ~ d ) a,

l # a ) i,., ~ e j . r ;=b. i=1

where Pr denotes probability. The following is assumed"

(i) 0 < a , ~ b . b , > 0 . j = l . 2 . . . . . n. (ii) Y'" a~b > -h ,y>O. " - ' j I , -

(lii) Each c j ~ a rzmdom ~ariabIe acc~rdiag a; the normal distribution N(.~.,, 6j 2) with :,,aea:a ,ft., and variance d,-' > 0. and they are mutua:liy inde- pendent.

P~, is equivalent to the following -~ , ~aeterministic

0377-2217 / 8 5 / $3.30 , 1985 Elsevier S,-,enc.. ,7,~,,:~,,:.,. .... .-,, I] v ,'\:~.rth-!-I,,l'. ,:.,,j~-:

H. lshii, T. Nishida / The stochastw linear continuous type knapsack problem 119

problem E,. (See Ishii et al. [3].)

,, 1 / 2

(Po) Minimize ~j.~:~ + q a f x 7 = l , - - - I ,,

- " / F ( q ),

subject to Y'. a jx , = b, j = l

0 ~ < x ~ b j , j = l , 2 . . . . . n,

q > 0 ,

where F(-) is the cumulat ive distr ibution function of the s tandard normal distribution N(0, 1). By variable t ransformat ion,

A 1,. A

tl

subject to ~V" )) = b, 1 = - I

O < ) ) 4 b , , j = l . 2 . . . . . n.

As is easily shown, pu is a strictly convex program- ruing problem with a single constraint and upper bounded variables, so it has a unique opt imal solution yq.

Theorem 1. An optimal solution (y* . q*) of P satisfies

I , / 2

q * = log y 2 , " r E o f ( y y " ) I' / - 1

i f q* > 0 , or

and

" 6)2 %" ~= / a ,. j = 1 , 2 . . . . . n,

~'u is further t ransformed into the following prob-

lem P.

(P) Minimize

subject to

II

j = l

+ q / Z I o~-.Ij

- T F ( . ) . #l

Y~. I) = h. / ~ 1

0 ~ y, ~ h .

q > O .

/ = 1 ,2 . . . . . n.

in the sequal, it is assumed without any los~ of

gmzerality that

p.I ~P.2 ~< . . . ~< p.,,.

3. The subproblem Pq and its auxiliaD' problem Pq

In order to solve P. the following subproblem P'; with a po~,itive parameter q is introduced.

y * : ) , )*=b~, j = l , 2 . . . . . k * , j *

= t , - t,,. (2) j = l

) , * = 0 . j = k - * 4- 1 . . . . . n ,

,t

/]" q* = 0. where k * = max { / [ b # £',_ ,, h, }.

Proof. If q* = 0. v* is an opt imal solution of the ordinary linear continuous type knapsack problem po as follow,<

( P " ) Minimize .~ if, >,. I I

,,u~-,ject t~, ~ y, = t , . /

0-~.y, d h , . / = 1. 2 . . . . . n

So v* given in (2) is really optimal for P -,into P-1 --<- " ' " <'- ,- ",, . . . . . . If q* --. 0 Theorem g (~f [,hii et al.

[31 shows that (y*. q* ; ~,atisfies ~ l ). 7J

Now the following auxiliary probl,:m l'i] ,>f P" with a positive paramet,er /,* i~, intr.,>duvcd

pt tl

{Pj~) Miaimizc R ~ f f ,~ , - ! q ~ ~ ' , ! 1 ¢ !

t1

subject to ~ 3, =' t,.

(P'~) Minimize p.j)) + q d,-.V/ ! I ,/ _1 t

_ .,,,=(~, )

0 < " .I-. b I---- I ,

Let >?i d~no te an op,'.in:ai .~o!u,~oI~ ~.,! P,,.: "~ i~v:~ tn~- ..

,q. k.a >.. [o lh,v. ing rclat,on, c(,i~.nc:.::.;ng ,~'';, :~.. I~ h ." '

120 H. Ishi+, T. Nishtda / The ::tochastic linear continuous type knapsack problem

Theorem 2. I f R 2 = Y"7- ,Oj'( )'~ )~. then .V[4 becomes an opt imal solution yq o f Pq.

Proof. See Ishii et al. [31.

Theorem 3. }'l r = YqR and y l R sat is fying

I1 ")

q-'r ~= I2 o)(y~);

i~ an opt imal solution yq of Pq.

Theorem 4. I f q* > O, q* satisf ies

+ : R +, : q - ; , .v,+

1=I

and

I ( ( ..... ..I q * = icg 3,2/ 2 e r e "f'(v:]2] 1

'. J= , + ] I

1/2

Proof. Due to Theorem 1 and Theorem 3.

(6)

(7)

Proof. First note that Pq is a strictly convex pro- gram.ming problem and so yg is unique. Especially, y~ is they-part of the sdut ion of the Kuhn-Tucker =ondition KTPR l given as follows.

• 1 I,.tP]+ ) %2)) + R/t , -- u, - uj + X = C

j = 1, 2 . . . . . n , ( 3 ) t l

E y , = b , j = i

0~<))~<6,, u,>/O, % >I0,

j = 1 ,2 . . . . . n, (4)

,,,y, = 0, , , ( y , - b,) = 0,

j = 1 .2 . . . . . n. (5)

t Multiplying (3) by q :- 0 and se.ting u, & qu, , c'j & qt,,. j = 1.2 . . . . . n. it follows that X' g qX changes KTP,~ into the foliove ng form.

qo,-)) + q R u , + u, - t , + X" = O,

J = l , 2 . . . . . n , /7

y, = b.

(Y)

t O ~ y ~ < ~ b , u I>1'), % >10,

j = 1, 2 . . . . . n,

,';y, = 0. ,+;(.,', - b,) = 0.

(4 ')

j = 1 . 2 . . . . . , . ( 5 ' )

Fiae con.straints ~3't ( 5 ' ) together con,;ist of KTPIS+

q is the v-part of the solution of of P:,'~ and so vu~

(3 ' ) - (5 ' ) . This shows that v ~ ." . r, = -},it" Substituting qR and .~']~ for R and .,"qR respec-

tively in Theorem _'~ _~ives q 2 R 2 = ~'".1. 1%-~ ( ):tel )7, ~

. q for Pq. [] which ,s the optimality concition o f ) r

TLeorem 4 implies that y* exists among y~ if q* > 0, i.e. check of all PR ~ is sufficient to solve P.

4. A n a lgor i thm for so lv ing P

For ease of exposition, P~ is denoted with Pr, KTP~ with KTP R, and y~ with y t r ) respectively. Now PR and KTP r are written again.

I (Pt~) M i n i m i z e R Y" t~,y~ + ~ E %Y,,2 2 1=1 J ~ l

subject to ~ ),, = b,

0 ~<.vj ~ b,, j = l , 2 . . . . . n"

(KTPr) o , 2 y , + % - t , j + X = - R / , , ,

j = 1, 2 . . . . . n. 11

E ),; = b, j = l

O ~.v~ ~ bj, % > I0 .

j = 1, 2 . . . . . n .

vj)) = O, u , ( yj - b, ) = O,

j = 1, 2 . . . . . n .

The solution of KTP R. (),~/+,.u{r~ t,tr} h<e)) can be constructed as follows, using similar ideas as Helgason [2]. (To show dependency of v. < R ~ t+ 1/+j and u I R ' J = 1, 2 . . . . . n, on X, these expressions are used.)

, ,~ ' ( ,X )= m ~ ( - , ° , ~ , , - ~ - ¢ b , . o). ( s )

• " + ' ( x ) = m..,~,( x + R~ , , 0) . (9) Uj

H. Ishii. 7". Nishida / Tire stochastic linear continuous type knapsack problem 12 i

=

'bj for ( X ~ < - b j ~ ? - - R / t j ) , (10)

( - RI~S - X )/os 2 for

( - b s o s 2 - R t t s < h < - R/z,), (11)

0 for (X > - gF,,), (12)

n

E g " ) ( x ) = b. (~3) j = l

Now let g ( a ) ( h ) & Y:~=~y)a)(X). Then g(R)(h) is a nonincreasing function of X. Thus y(R) can be found as y (R~ = y(R)(h) satisfying g ( R ) ( h ) = b. De- noting -b jo j 2 - Rtts and -Rp . j with 5 n and s~ a respectively and arranging different §R and s~ m the increasing order for each R, we let

wff < w2 R < - - - < w,,,~R)

where r e (R) is the number of them. Each ,)R is a point of ~ where type of y~ R ) ( X ) changes. Further. calculate R such that r, R= f f (i e j ) and r~R = i"/~ (k < / ) and denote them with R,, and R~.t respec- tively. Sorting different positivc R,~ and R~.t, we have

z$ Ro~-O<R~ < . . . < R , , , < R , , , + ~ = M

where M is a sufficient large positive number and m is the number of them. Note that Rt is R where an order of wf changes.

Now we are ready to describe our algorithm for solving P. In the algorithm, let m , ~ m((R, + R,+ i ) /2 ) and solutions R ~1~, R ~ of E~ are given as follows.

R(~)=

R ( " ) =

1 / 2 • )2 b', +{(b~' +4c'[(q ~ - a',)}

2( q : - a~)

2 b~, - {(b~, + 4c~ (q 2 - a,')}

2(q2 _ a~,)

I/2 (q2 =/: a:,,),

R( ' )= R '2)= -c'~/b'~ ( q2 = a[ )

where a~,.. b~, and c~ are determined by

i i 3

2o;{.; x)} ./--- 1

t -~- L, l, ~ - el. ,,

moreover, superscripts i and subscripts k are sup- pressed in the algorithm (As above R ~'~ and R ~'-~) for ease of exposition if confusion does not occur.

Algori~_m~: f o r P

Step 1. Calculate Ro . - ' - ,R , , , , Solve p0 and con- s t ructy ° as in (2). Set i ~ 0, h* ,--- h( .r t'~. 0), q* <-- 0 and y* ~ y0. Go to Step 2. Step 2. Calculate w ~ , . . , w n for each R • rl'l,

[ R,, R, + ~ ] and set S ~ [ R,, R, ~, ~ ]. Go to Step 3. Step 3. Set k ,-- 0 and go to Step 4. Step 4. For R ~ S and X =~ [w~, wff+ l l, calculate y(R)(h) and g'R'(h), and solve the equation g'"J(x) = b with respect to X. Let this solution be h'k & dr. R a h~,. Go to Step 5. Step 5. Sot', .: the inequality wff ,%< X'~. ~< w re. . , with respect to R ~ S and find its ~o!utior~ ~c.', S.' g',,:,: Appendix 1). Go to Step ,5. Step 6. If S~ =,0, go to Step 9. Otherwise. 5,olve the

, . - . ~ 2 2R2 following equation Ea Y_..,=~%-{ ).~t~(X)} = q with respect to R and find its solutions R ~ . R 'e'. Solve tb.e inequality R '~' or R ' - ~ S~ v,-ith respect to q and find its solution set Q~. (see Appendix 2). Step 7. If Q~, =,0, go to step 9. If Q'~ ¢,,0, solve the equation

y 1 qexp{'~q2} = (2.,=.) ~/'- R " '

o r

, ~ ? 1 q exp{ _~q- } . . . . 7~--' ;Sv

{2".:) '" I<- '

with respect to q=-QI,. If any ,,(4u',~m, ~."~,, denote them by q~. q2 and q~ (~,ce app=r~c"J,-, 3j. (,,~ to Step 8. ()therwise. :,_() t~ Step 9. Step 8. If

m a x ( h ( y q ' , q ~ ) , h ( v u . q : ) . h ( r '':.q~))

h(y '~', q~) -- h~.

go to Step 9. Otherv,i,,,e. set q* ,- q,.. y* ' - ~:~ and h* ~ h( v q', q~). Go to Step 9.

• ~ . t ) . Step 9. Set k-'---k + 1. If l , = m oo to Step 1" Otherv,ise. return to Step 4. Step 10. Set ~ , - . , # 1. If i = m + 1. ter.minat:-. Other~:ise. return to Step 2.

Theorem 5. The a&oridm7 fimh an ~)pttmc,,/ ~o/::t:,,: '. ~ . ¢.1 ~ (p! ( t [ i i t ( s . k [ \ . , , ( I t " ) ( ~ , ~ ? / ~ ' ? f C U ' r , i ' r ; [ ~ , , ; ~ " ' . '.'

eqttatiotl

7 ] q ¢~pl c/'/9~, - ! -

( "~ i~' ~ .}

122 H. Ishii, T. Nishida / The stochastic linear continuous t)Te knapsack problem

or

q exp(q2 /2} = - - (2,:r) ~/" R t2~

can be solved in a constant time.

Proof (Validity). By the Theorems 1-4, y* exists among )'~ R ~, satisfying

Sl

Y" 0,2( , J R ' ( X ) } 2 = q 2 R Z , (14) j= l

q2 exp{ q2} = y;/2,rrR 2, (15)

q>~0, R>_-0, (16)

where v ~R~ can be found from ?~ satisfying (13) by the rules (10)-(12). The algorithm divides all posi- tive R's into ( r e + l ) regions [R , ,R ,+I] , i = O, 1 . . . . . m, and X into (m, + 1) regions [w~, w/,'~ + 1], , t-=0, 1 . . . . . m. for each R ~ [ R , , R,+~]. Then it checks all these regions. Finitenes,., of th~ number of them assures finiteness of the algorithm. For each loop i and each subloop k, the Steps 4-7 solve the system of equations (14)-(16) and Step 8 updates a current optimal solution. Thus, the validity of the algorithm is clear.

(Complexity). Step 1 takes O(w: log n) compu- tational time. Step 3 is executed at most O ( n ~) times and each execution takes 3(n) computa- tional time. Thus the computational burden of Step 8 is O(n 4 ). The other loop of, from Step 2 to Step 10 is executed at most Ot n 2 ) times and inner loop of k from Step 3 to Step 9 is e~ecuted at most O(n) times. Therefore, the total nt mber of execu- tions from Step 2 to Step 10 is O(n ~). The compu- tational burden of each iteration ,0,ith respect to i ar, d k is as follows.

h'~ , S~, Q~" O( n ) t totaiO~n). q,. qz. q3: O(1) i

Tiros computational burden of the ~teps 2-8 in all is O(n'~). Summarizing above re~,.ults, computa- tional complexity of the algorithm is O(n4).

5. Discussion

We have considered the LKP version of a gen- eralized P model in [3] and proposed a parametric type algorithm with time C,31ilpiCXiL 3 Ci,,q';). Ja,J~-

ing from the graphs of the functions t exp( t } and

(bk +_bt, (bl, + 4 c ' ~ ( t - a k + 2 c ' k ( t - a , , )

4,n'( C~. ) 2

(see Appendix 3), we suspect that equation (15) has at most two solutions though we could not prove it. Moreover,we suspect Q~. has a more simple structure. Other further research problems are investigations in the following cases. (a) The case in which the random variables are not necessarily independent. (b) The case in which the random variables are distributed according to non-normal probability distributions.

Appendix 1. S~

First note that h~, ~- O. (i) Case d~, < O.

S~=

{ R 1( wff - h; ) / d ; ~ R >i (,,,t~, - i,'A ) /d~ } n S .

(ii) Case d~, = O.

Is (w ' , ~ , l >~tt~>~ ), S~ = ,0 (otherwisc).

(iii) Case d~, >/0.

S~ = { R I( ' " L , - hl )/d'~ >1 R >i ( w F - h'~ )/d'~ }

N S .

Appendix 2. Q~

! . I Let S k = { p ~ 4 R 4 r ~ } . Note that 0 < p ~ , < t k and a~,. c~, > 0. In the sequal of tbis Appendix 2. superscripts i and subscripts k are :;uppressed (ex- cept in Q~) for ease of exposition. For example, b means b~,. Further, let us define

b'~ = a + ( t,r + c ) / r 2

U s = a + ( b / 2 r ) ,

Case I. b >10.

tZ = a + ( l,p + ¢ ) / p .

U4 = b / 2 r .

= ( q t '"2 q >. u , ,2 ).

Note that in this case U~, U 2 > a.

H. lshii, 7". Nishida ,/ The stochmtic linear cu,:tmuou.i (~pe ,,,,upmck' . . . . . .moblem "'~"t_.~

Case II. b < O. Subcase l la. r < - 2 c / b .

Q~. = A, U {( A 2 U A 3 U A 6 ) O A 7 }.

Subcase l ib . r >i - 2 c / b and v < - 2c /b .

Qi~=A~ U { ( A 4 t O A 6 ) O A 7 } .

Subcase Ilc. p >i - 2 c /b .

Q~ = A, u { A s ~ A 7 }.

The above-mentioned sets are defined as follows.

Note that substituting t = a into (A3) yields t exp{t} = y2b2/2vc2 if b ~ 0. and this is indeed the case R ~ = RCx~= - b / c . Thus a special treat- ment of the case R ~ = R ~-'~ is tmnecessary.

Equation (A 3). Let us define

b 2 - b {/)2 X ( t ) = t e x p ( t } 4,rrc2

+ 4 c ( t a)} 1/2 ] - + 2 c ( t - a ) for t>~0.

Then its derivative

A1---

A2=

A3=

A4=

A s =

/16=

AT=

{q

(q

( q

U~ /2 >~ q >~

(max(O, U~

(max(0, U-,

{ q (max(O, /--/2

( q (max(O. U,

U11/2

))1/2

))'/~

)),/-'

and q >1 a 1/2 },

>_-q>_-(maxt0. U4))1/2},

>/q >/(max(0, /-/1 ))l ,z},

>_-q>_. (max(0, ~)))1/2}.

>_-q>_-(max(0, Uz))~/2}.

{ q (max(0, rain(U,, U3))) '/2

>~q> (max(0, U4)) ' /2},

{ q l a 1/2 > q >I 0}.

Appendix 3. qt, qz and q3

This appendix shown that there exist at most three solutions for the equation (AI) or (A2):

q exp{ q2 /2} = 3' 1 (A1) (2,l.i.)1/2 R +1)

or

q exp{ q2 /2} = "/ 1 (A2) ( 2¢r ) I , ' 2 RC-'~ "

Let t " q- > 0. Superscripts i and subscripts k are suppressed for ease of exposition. (A1) and (A2) are transformed into (A3) and (A4) respectiveb.

t e x p { t } = 72 [ t,:_ b{i,~ + 4, . ( , - a)} ''" 4 ,,rrc 2

+ 2 c ( t - a j j , ( a 3 i

= __ b 2 + b { b 2 + 4 d ( r - a ) } 1''2 t e-:p { t } z, ~.2

+ 2 c ( t - a ) } . (A4)

X ' ( t ) = (t + l) exp{t }

I 2"rrc 1 - {b 2 + 4 c ( t - a ) } ' ' 2

Further

X " ( t ) = (t + 2) exp{t }

2 h _ ' t ' {b 2 + 4 c ( t - a ) } - ' ' 2

,"2"

and

' " ( t ) = (t + 3) exp{t }

2 I, c r 2 { t , : + 4 ~ ( t - a ) }

Case I. h ~ O. Clearly

s n ( ! . , . . ,

X t) 0 (,'~::;

holds. Thus ( A 5 J ~ X ' t t ) : strictl,, tncrca,~ng X'(t} has at most one vanishing point = X(t) has at most one local minimumai point = (A3) has at most two solution in the positive interval. (_'a.~e II. b > 0. Then

x ( a ) = a exp{ a },

7 2 , " {A6) x(o)= - { ~ , - t f , : - 4 . ~ ) - ' I . 0

C" - ' 7 ~

X ' ( t ) ~ 0 f o r t ~ a A7

and

~< "" ( t ) -~0

(A6) ar.d tA7)~cquat~.on (?:3) h.:>: e~,:'.,~_:l', .~,~.~ . . " A O ' soluticr, in the in~erva! 0 < i < a. (=~-o) ~ X '( r i b.a.,,

two vanisking points in tile inter-al t > ,q, v,hcrc the smaItcr one i,,, the local maximum point and

124 H. lshii, T. Nishida / The stochastic linear continuous type knapsack problem

the other is lhe local minimum in (A6) =* equat ion (A3) has at most two solutions in the interval (a, ~c).

Equation (,44). Note that R ~2) > 0 only if t < a and b < 0. Define

- ~ b 2 + b { b 2 ~o(t) e t exp{t} 4"~c 2

+ 4 c ( t - a ) } ' / 2 + 2 c ( t - a ) ] fora>~t>~0.

Then its derivative satisfies

o~'(t)=(t+ 1) exp{t }

2~rc l + - - I

{ + 4 c ( , . - a)}

Care I lL b < 0. Then

,,,'(t) > 0

holds, since

,5 | _ -

{ b 2 + 4 c ( t - a ) } ;/2

( l ÷ 1) e x p { t ) > 0.

< 0 and

(A9)

(A9) =, o:( t ) has exactly one ' ,anishing point in the interval 0 < t < a, since

~(0) 1 [ b + { b ~ - + 4 c ( t - a ) } l / 2 ] 2 = - ~ < 0 8,.ffC 2

and

6 o ( a ) = a e x p { a } > 0 .

The cases I - I I I show that there exist at most three solutions for the equat ion (A3) or (A4), i.e. for the equat ion (A1) or (A2).

References

[11 Balas, E. and Zemel, E., "An all~orithm for large zero-one knapsack problem", Operations Research 28 (1980) 1132- 1154.

[21 Helgason, R., Kennington. J. and Lall, H.. "A pol 3 nomially bounded algorithm for a singly constrainted quadratic pro- gram", Mathematical Programming 18 (1980) 338-343.

[3] Ishii, H., Nishida, T. and Nanbu, Y., "A generalized chance constrained programming problem", Journal of the Opera- tions Research Society of Japan 21 (1978) 124-145.

[4] Johnson, D.B. and Mizoguchi, q'., "Selecting the k-th ete- ment in X+ Yand X ~ . . . . + X,", SIAM Journal of Com- puting 7 (1978) 147-153.