Stanisław Gędek
Methods of Considering Risk in
Programming Models Used in
Agriculture
Annales Universitatis Mariae Curie-Skłodowska. Sectio H, Oeconomia 18,
363-379
A N N A L E S
U N I V E R S I T A T I S M A R I A E C U R I E - S K L O D O W S K A
L U B L I N — P O L O N I A
VOL. X V III, 21 SECTIO H 1984
M ięd zy w y d zia ło w y In stytu t E k on om iki i O rganizacji R oln ictw a A kadem ia R oln icza w L u b lin ie
S t a n i s ł a w G Ę D E K
M etods ef C o n sid e rin g R isk in P ro g ra m m in g M odels U sed in A g ric u ltu re M etody u w zględ n ian ia ryzyka w m odelach op tym alizacyjn ych
sto so w a n y ch w ro ln ictw ie
М етоды учета риска в оптим ализационны х м оделях, прим еняем ы х в сельском хозяй стве
F a rm o rg a n iz a tio n p la n n in g calls fo r ta k in g in to ac co u n t m a n y v a r ia n ts of possib le so lu tio n s of th e p ro b lem as w ell as fo r a d ju s tm e n t to m a n y c o n s tra in ts im posed b y n a tu r a l an d econom ic co nd ition s. D ue to th is fa ct, lin e a r p ro g ra m m in g is recog n ized as an e ffic ie n t in s tru m e n t of o p tim izin g p ro d u c tio n a n d in v e s tm e n t plans, a lth o u g h n o t fre e fro m d efects. O ne of th e m is th a t c o n v e n tio n a l lin e a r p ro g ra m m in g ca n n o t a d e q u a te ly cope w ith flu c tu a tio n s of crop y ield s, p rice s a n d o f o th e r ,p a ra m e te rs . C o n se q u e n tly , th e re w as a lo n g -p re v a ilin g o p inion th a t lin e a r p ro g ra m m in g could be u sed in v e ry ra re cases o nly. O v e r th e la st s e v e ra l y e a rs, h o w e v er, s ig n ific a n t p ro g ress h as b ee n m ad e in th e so -called sto ch astic p ro g ra m m in g , esp ecially in its th e o ry . N u m e ro u s m e th o d s also a p p e a re d w h ich could be, an d in d e e d w e re, ap p lied in a g r i c u ltu re . I t w o u ld be u se fu l to p re s e n t a t le a st som e of th e m o st im p o r t a n t fo rm u la tio n s . F o r th e in te re s t in lin e a r p ro g ra m m in g is re la tiv e ly h ig h w h ile th e re is little in fo rm a tio n in P o lish scien tific lite r a tu r e on th e m eth o d s of risk con sid erin g .
I. FORM ULA TIO N OF THE PROBLEM
A s ta n d a rd v ersio n of th e lin e a r p ro g ra m m in g p ro b lem is th e fo llo w ing:
m ax im ize m Tx, su ch th a t: A x < b an d x > 0
3 6 4 S. G çdek w h e re:
m T — a co lu m n v e c to r of o b jectiv e fu n c tio n p a r a m e te r m ean v alu es, x — v e c to r of ac tiv ities,
A — an in p u t- o u tp u t co e ffic ie n ts m a trix ,
b — v e c to r of a v a ila b le a m o u n ts of scarce re so u rc e s.
W e h a v e to a ssu m e t h a t th e v e c to r m a n d also th e m a tr ix A are s u b je c t to flu c tu a tio n b ec a u se flu c tu a tio n s of crop y ie ld s an d p rice s c a n n o t be e x c lu d e d . In som e cases th e v e c to r b h a s to b e c o n sid ered as w ell: th e a m o u n t of a v a ila b le la b o u r in re s p e c tiv e p e rio d s of d iffe re n t y e a rs can d iffe r d u e to c h a n g in g w e a th e r cond ition s.
II. R ISK C O N SID ER IN G IN OBJECTIVE FUN CTIO N PAR AM ETERS If w e a ssu m e th a t p ric e s are th e o n ly so u rc e of flu c tu a tio n s of a g r i c u ltu r a l p la n n in g p a r a m e te r s or th a t a ll cro p s g ro w n on th e fa rm a re cash crops, it is s u ffic ie n t to c o n c e n tra te on o b je c tiv e fu n c tio n p a r a m e te rs only. T h is a ssu m p tio n , a p p a r e n tly a rtific ia l in f a rm co nd itio ns, is u s e fu l to th e e x te n t th a t it p e r m its to see th e a p p ro a c h to th e p ro b le m of ris k in o b je c tiv e fu n c tio n p a ra m e te rs . T h e e x te n sio n of ch a n ce ac tio n u p o n o th e r e le m e n ts of th e lin e a r p ro g ra m m in g m odel, t h a t is an in p u t- - o u tp u t c o e ffic ie n t m a trix a n d a rig h t-h a n d sid e v ecto r, does n o t in a n y w a y a ffe c t th e a p p ro a c h to th e in tro d u c tio n of risk in to th e o b jectiv e fu n c tio n . In th e tw o o ld est an d b e s t-k n o w n m e th o d s of c o n sid e rin g ris k in o b je c tiv e fu n c tio n p a ra m e te rs , fo rm u la te d b y M a rk o w itz (14) an d F re u n d (10), th e m e a su re of flu c tu a tio n is th e to ta l v a ria n c e of o b je c tiv e fu n c tio n :
VmTx = xTD x,1 w h e re :
D — v a ria n c e -c o v a ria n c e m a trix of o b je c tiv e fu n c tio n p a r a m e te rs,
x T, x — c o lu m n an d ro w v e c to rs of ac tiv ities, re s p e c tiv e ly , VmTx — to ta l v a ria tio n of o b je c tiv e fu n c tio n .
T h e M a rk o w itz m e th o d (14) w as o rig in a lly m e a n t fo r choosing
1 Or another w ay: V mTx = Z x ia f + 2 S x i x j a lj5 w h ere j = i »=1
Xi — i-th a ctiv ity , Xj — j-th a ctiv ity ,
a — varian ce of o b jectiv e fu n ctio n p aram eters of the i-th a ctiv ity ,
M ethods of C onsidering R isk in P rogram m ing M odels 3 6 5
a stocks com bination, hence its n am e of ’’p o rtfo lio selec tio n ” . It is fo u n d e d on th e a ssu m p tio n th a t th e goal^of fin a n c ia l a c tiv ity is to m a x i m ize th e p ro fit, w h ich , tra n s la te d in to fo rm u la s of lin e a r p ro g ra m m in g , jn e a n s a m a x im iz a tio n of m Tx o b jectiv e fu n c tio n , w ith A x ^ b and x > 0. T he A x ^ b c o n s tra in ts a re n e c e s s a ry b ecau se th e a m o u n t of m o n ey th a t could be s p e n t fo r stocks by a n y in d iv id u a l or c o m p an y is lim ite d ju s t as is th e a m o u n t of a single f ir m ’s stock s av a ila b le on th e m a rk e t. It also follow s fro m th is assu m p tio n th a t th e m a x im iz a tio n sh o u ld be su ch th a t th e to ta l v a ria tio n of p ro fit does n o t ex c eed a c e r t a i n v alu e, w h ich co u ld be a c cep ted b y a d e c isio n -m a k e r. T his m ean s
th a t an a d d itio n a l c o n s tra in t h as to be im po sed on m Tx, th a t is x TD x ^ < a, w h e re a th e m a x im u m ad m issib le v a lu e of o b jectiv e fu n c tio n v a r ia tion. S in ce th is is an e n tire ly s u b je c tiv e v a lu e an d it is d iffic u lt to a s su m e a n y re la tio n b e tw e e n a and m Tx in ad van ce, th e m o st co n v e n ie n t w a y of so lving th is p ro b le m is to u se p a ra m e tric p ro g ra m m in g , w ith th e p r o b le m fo rm u la te d as follow s: m axim ize: m Tx, su ch th a t A x ^ b x TD x ^ a x ^ 0, w h e re: m T, x, A, b, D an d a as above.
S u ch p ro b lem s could n o t be solved in th e e a rly fiftie s w h e n th e ’’p o rtfo lio s e le c tio n ” m e th o d w as fo rm u la te d . T he co n v erse p ro b lem , th a t of m in im iz a tio n of o b jectiv e fu n c tio n v a ria tio n , w ith th e assu m p tio n th a t th e m e a n v a lu e of th e p ro fit w ill no t d ec rea se below a c e rta in v alu e, could a lre a d y be solved ow ing to th e e a rlie r w o rk by K u h n a n d T u c k e r (12). Its m a th e m a tic a l so lu tio n tu r n e d ou t to be id e n tic a l w ith th e o rig in al p ro b le m . T h e fin a l v e rs io n of th e M ark o w itz m e th o d can th u s be fo r m u la te d as follow s: m inim ize: x TDx, su ch th a t: A x < b, m Tx ^ |3, x ^ 0, w h e re m, x, A, b, D as above (3 — p a r a m e te r d e te rm in in g th e m in im u m a c c e p ta b le p ro fit.
T he so lu tio n to th is p ro b le m a re p a irs of m e a n p ro fit v a lu e a n d p ro fit v a ria tio n s, a n d a c o rresp o n d in g set of v alu es of each a c tiv ity inv olved .
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Ś. G ędekA choice is m a d e ac co rd in g to in d iv id u a l p re fe re n c e s of p ro fit h e ig h t an d its v a ria n c e . In o th e r w o rd s, so lu tio n s a f te r th e ’’p o rtfo lio se le c tio n ” m e th o d p ro v id e in fo rm a tio n th a t w ith a g iv en m ean p ro f it v alu e, v a r ia tio n e q u a l to x TD x c a n n o t be avo ided , an d th a t in th is case a ll a c tiv itie s h a v e to asu m e th e v a lu e s as in th e o p tim u m so lu tio n to th e fo reg o in g p ro b le m .
F re u n d fo u n d e d his m e th o d on th e ’’u tility th e o r y ” fo rm u la te d b y vo n N e u m a n a n d M o rg e n ste rn (9). T h e c e n tr a l p o in t of th is th e o ry is th e a sse rtio n of a d e c re a se in m o n ey v a lu e fo llo w in g its a c q u isitio n u n c e rta in ty . T h is m ean s th a t of tw o fa rm e n te r p ris e s w ith th e sam e a m o u n t of p ro fit, th e one w ith a lo w e r p ro fit v a ria tio n is ’’m o re u s e fu l” . M oreover, tw o e n te rp ris e s w ith d iffe re n t p ro fits an d w ith a d iffe re n t r a te of p ro fit v a ria tio n h a v e e q u a l ’’u t i li t y ” if th e e n te r p ris e w ith a h ig h e r p ro fit v a ria tio n o b tain s th is p ro f it h ig h e r b y a d e fin ite am o u n t. T his v a lu e v a rie s w ith e v e ry in d iv id u a l f a rm o p e ra to r. T h e re la tiv e m e a s u re of th is v a lu e is r e f e r r e d to as a ’’ris k av e rsio n c o e ffic ie n t” . T h e re la tio n b e tw e e n p ro f it h e ig h t an d its v a r ia tio n an d p ro f it u tility is ca lle d u tility fu n c tio n .2 T h e one p ro p o se d b y F re u n d fo r fa rm e rs h as th e fo llo w in g fo rm :
f(u) = 1 — e~ ar, w h e re e — n a tu r a l lo g a rith m base, a — ris k a v e rs io n co e fficien t, r — p ro f it h e ig h t.
T h e b ig g e r a is, th e less re a d ily a fa rm o p e ra to r w ill ta k e u p risk , an d th e h ig h e r p ro f it h a s to b e o b ta in e d to le v e l h ig h e r v a ria tio n in a lte r n a tiv e a c tiv itie s. A ssu m in g r to h a v e a n o rm a l d is trib u tio n , th e e x p e c te d u t ili ty v a lu e w ill b e as fo llow s:
E(u) = [x—ao2/2, w h e re : (i — m e a n p ro fit v a lu e , a — ris k a v e rs io n co e fficien t, g2 — p ro f it v a ria tio n . T ra n s la te d in to lin e a r p ro g ra m m in g , th is m eans: m ax im ize: m Tx — j x TD x su ch th a t: A x < b x > 0
2 F reund ca lled th is rela tio n the ’’u tility of m on ey fu n ctio n ”. In other papers, the term ’’u tility of fu n c tio n ” can be en cou n tered .
M ethods of C onsidering Risk in P rogram m ing M odels 3 6 7
S o lu tio n of th e above p ro b lem s, w h e re ris k h a s b ee n d e a lt w ith ac co rd in g to th e tw o p re s e n te d m etho ds, re q u ire s q u a d ra tic p ro g ra m m ing. A v a ila b le c o m p u te rs solve th a t p ro b le m easily, n e v e rth e le ss, q u a d ra tic p ro g ra m m in g is fa r less c o n v e n ie n t th a n lin e a r p ro g ra m m in g , m a in ly b ec au se th e size of p ro b le m s is th e n m u ch m o re lim ited . H en ce th e re w e re a tte m p ts to m o d ify and a d a p t th e p o rtfo lio selec tio n an d F re u n d m e th o d to th e sim p le x p ro c e d u re , an d to lin e a riz e th e o b jectiv e fu n c tio n .
T he b e st-k n o w n lin e a riz a tio n of th e p o rtfo lio selec tio n m e th o d is th e so -called M OTA D p ro posed by H azell (11). Its g u id in g id ea is to re p la c e v a ria tio n by a b so lu te dev iatio n . H azell assu m es f u r th e r th a t it is su ffic ie n t to ta k e in to a c co u n t n eg a tiv e d e v ia tio n s only. T h e re s u ltin g fo rm u la is as follow s: n m in im ize ^ yr- su ch th a t: i= 1 Ax < b m Tx > {3, w h e re
yj- — n e g a tiv e ab so lu te d ev iatio n of j - t h a c tiv ity fro m its m e a n p ro fit valu e.
C hen an d B a c k e r (7) p ro p o sed a lin e a riz a tio n of o b je c tiv e fu n c tio n
a
E(u) = m x —— x 1 D x, fo u n d e d on th e assu m p tio n th a t no a c tiv ity can be a c tiv a te d b e y o n d th e p o in t w h e re its m a rg in a l u tility a ssu m es a zero v alu e. T his m e a n s th a t th a t th e v a lu e %of a n y a c tiv ity can be in c re a se d as long as its in c re a se adds a n y th in g to th e su m of to ta l u tility . If th is lim it is ex c eed e d , to ta l u tility decreases. T his m a rg in a l u tility equ als:
d E ( u ) a
^ SljXj, where:
1 j='l
Sjj — co v a rian ce b e tw e e n th e i-th an d j - t h a c tiv itie s 3.
3 The v a lu e of con varian ce b etw een the o b jective fu n ctio n param eters of the i-th and i-th a ctiv ity is th e variance of ob jective fu n ction p aram eter of the i-th a c ti v ity , w ith the m argin al u tility thus b ein g as fo llo w s:
oE(u) 2 ;
= m, = aSjXi4-2a 2, SijXi, w here S 2 is of course the variance
<TXi j = i
C o n se q u e n tly , th e p ro b le m to be solved is as follow s: m ax im ize: mTx , su ch th a t Ax < b n a
m> - - r 2 u SijXj
z j=i w h e re i = 1 ... n, j = 1 ... n n x th e n u m b e r of a c tiv itie s O r a n o th e r w ay: m ax im ize: mTx , su ch th a t: Ax < b 2 Dx < — m a x > 0U n fo rtu n a te ly , so sim p ly fo rm u la te d a p ro b le m can be solved o n ly if s assu m es a p o sitiv e valu es. If a n y a c tiv ity of w h ic h th e x v e c to r consists, say x k, assu m es a zero v a lu e , it co u ld t u r n o u t th a t th e c o n s tra in t:
a n ’
mk- — ^ skjx / > 0
I j=ii)
is re s tr ic tiv e to o th e r a c tiv itie s a lth o u g h x k sh o u ld n o t h a v e a n y in flu en c e o n th e o p tim u m so lu tio n , b ec au se it is an id le a c tiv ity . C h en and B a c k e r d e v e lo p e d a m u lti-s ta g e a lg o rith m fo r th is p u rp o se , w h ich g ra d u a lly re m o v e s all . id le a c tiv itie s an d th e ir c o rre sp o n d in g con s tr a in ts w h ich e n su re th e a ssu m p tio n of its n o n -n e g a tiv e m a rg in a l u t i
lity . T he a lg o rith m is 'th e fo llo w in g :
1. F in d an o p tim u m so lu tio n of a p a r a m e tric L .P . p ro b le m : m ax im ize: mTx , s u b je c t to: Ax < b 2 Dx <$ — m u x > o w h e re: a — a p a r a m e te r a ssu m in g v a lu e s fro m + o o to 0.
M ethods of C onsidering .Risk in P rogram m ing M odels 369
2. R eco rd th e w h o le set of so lu tio n s an d th e ir o b jectiv e fu n c tio n v alu es, if n one of th e d u a l so lu tio n s associated w ith th e c o n s tra in ts w h ich are to p re s e rv e n o n -n e g a tiv e Xj u tility , assu m es a p o sitiv e v alu e.
3. R em o ve fro m th e x v e c to r all th e ac tiv ities w h ich a re n o t in th e basis and all th e c o rre sp o n d in g c o n stra in ts e n su rin g n o n -n e g a tiv e Xj u tility .
4. F in d a n ew se t of solution s. C om e b a c k to ste p 2 an d re c o rd o nly so lu tio n s w ith a lo w e r m Tx v a lu e th a n p re v io u sly o b tain ed .
A n o th e r w a y of th e lin e a riz a tio n of o b jectiv e fu n c tio n in th e M ark o w itz m e th o d is ’’S e p a ra b le P ro g ra m m in g ” (27).
I t co n sists in th e divisio n of th e x TD'x fu n c tio n in to a su m of sin g le a rg u m e n t fu n c tio n s, w h ich p e rm its, th e ir sp a tia l lin e a riz a tio n .
T he th ird a p p ro a c h to th e p ro b lem of o b jectiv e fu n c tio n flu c tu a tio n s is fo u n d e d on th e th e o ry of gam es. In th e P o lish econom ic lite r a tu r e th is a p p ro a c h h a s b e e n d esc rib ed in d e ta il by T. M arszalk o w icz (15). It a p p e ars, h o w e v er, th a t a f u r th e r d iscu ssion w ill be m o re lu cid if th e
isic te n e ts of th e th e o ry are ex p la in e d a t th is p o in t.
In th e fa rm o rg a n iz atio n p la n n in g or o th e r decision m ak in g , th e s-o-called gam es w ith n a tu r e are selec ted o u t of a n u m b e r of gam es co v ered b y th is th e o ry . T h ese gam es h av e su ch a p ro p e rty th a t th e o p p o n e n t in th e gam e — n a tu r e — a lth o u g h ru th le ss , is n o t s p ite fu l. It is th e re fo re a ssu m ed th a t a p la y e r — in th is case a decision m a k e r — faces m p o ssib ilities, each of th e m h a v in g n re a liz a tio n s of th e v a lu e u n d e r c o n sid eratio n . T h e p ro b le m is to selec t one o u t of m p o ssib ilities, th e choice in no w a y a ffe c tin g th e o p p o n e n t’s actio n. T he selec tio n n eed n o t be lim ite d to th e choice of one p o ssib ility , w h ich is c a lle d ’’p u re s t r a te g y ” . T h is can also be an y co m b in a tio n of p o ssib ilities, w h ic h is th e n c a lle d ’’m ix ed s tr a te g y ” . M ean v a lu e s or v a ria n c e s as a c rite rio n of choice c a n n o t be ap p lie d as th e y c a n n o t be c a lc u la te d b e c au se n o th in g is k n o w n ab o u t th e p ro b a b ility of an y m re a liz a tio n s .4 T he o n ly in fo rm a tio n w e h a v e is th e se t of v a lu e s w h ich e v e ry m p o ssib ility ca n assum e. To deal w ith th is re a lly d iffic u lt situ atio n , th e m in im a x ru le is ad o p te d if th e re a liz a tio n s of m are costs, an d th e m a x im in ru le if th e re a lia - tio n s of m a re incom es.
T he m in im a x ru le con sists in th e choice of such a p u re or m ix ed s tr a te g y th a t h as th e lo w e st m a x im u m cost v a lu e of a ll m ix ed an d p u re stra te g ie s. B y an alo g y , th e m a x im in r u le selec ts su ch a m ix ed or p u re
4 T he m e a n v a lu e a s a c r ite r io n o f s e le c t io n is c a lle d th e L a p la c e c r ite r io n . It is b a sed o n th e a s s u m p tio n th a t if th e p r o b a b ility o f n o m r e a liz a tio n s c a n b e d e te r m in e d , it is n e c e s s a r y to a s s u m e th a t th e p r o b a b ility of e a c h r e a liz a tio n is th e sa m e . T h is a p p ro a c h h a s b e e n c r itic iz e d in p a p e r (24).
3 7 0 S. Ggdek
s tra te g y th a t h as th e h ig h e s t m in im u m v a lu e of in com e of all m ix ed or p u r e stra te g ie s.
A n e x a m p le w ill se rv e as a b e tte r illu s tra tio n . In T ab le 1 are sho w n th e y ie ld s of fo u r oats v a rie tie s in th e co u rse of fiv e y ea rs. T h e re s u lts a re g iv en in p o u n d s p e r ac re (th e e x a m p le w a s d ra w n fro m H ead y, P e s e k and W a lk e r (29).
A cco rd in g to th e m a x im in rule^ th e B v a r ie ty is th e best b ecau se its lo w est y ie ld o b ta in e d in th e firs t y e a r of th e e x p e rim e n t is h ig h e r th a n th e lo w est y ie ld of a n y o th e r v a r ie ty u n d e r co n sid eratio n .
C hoosing a m ix e d s tr a te g y is m u c h m o re co m plex. F o r it is im po ssible to m a k e a se t of all co m b in a tio n s sin ce th e n u m b e r of p ro p o rtio n s of each v a r ie ty in su ch a c o m b in a tio n is in fin ite , w h e re a s th e m ix ed s ta - te g y is su pposed to h a v e su c h p ro p o rtio n s of each v a r ie ty th a t a co m b i n a tio n w ith a h ig h e r m in im u m y ie ld cou ld n o t b e fo u n d . It is th e re fo re n e c e s sa ry to solve th e fo llo w in g L .P . p ro b lem :
m ax im ize: x 5, su ch th a t: 1472xi + 1 5 6 8 x 2+ 1440x3+ 1 5 5 2 x 4 —x 5 ^ 0 2112x1+ 1 9 8 4 x 2+ 2 3 6 8 x 3 + 2688x4- x 5 > 0 1920x1 + 1824x2+ 2 4 9 6 x 3+ 2 7 8 4 x 4- x 5 > 0 3620x1 + 3104x2+ 3 5 5 2 x 3 + 0x4 —x 5 ^ 0 3 0 7 2 x i+ 3328x2+ 2 8 4 8 x 3+ 3200x4 — x5 > 0 X i + x 2+ x 3+ x 4 = 1 Xi > 0 x 2 > 0 x 3 > 0 x 4 ^ 0 x 5 > 0
T h e so lu tio n to th e above p ro b le m is a m ix ed s tra te g y co n sistin g of 56% of B v a r ie ty an d 4 4 % of C v a rie ty . T h e w h o le p ro b le m of d e te rm in in g a m ix ed s tra te g y can be g e n e ra lized as follow s: x p m a x im iz a tio n , su ch th a t n ijx — Xp ^ 0 m 2x x p ^ 0
M ethods of C onsidering Risk in P rogram m ing M odels 371 m nx —x p ^ 0 n < d i = 1 x ^ 0, w h e re: x p — v a lu e of a gam e, nij ... m n — v e c to rs of m re alizatio n s, x — a c tiv ity v ecto r,
d — v a lu e w h ich th e su m to ta l of a c tiv ity v a lu e s c a n n o t ex ceed (m ost o ften 1 or 100%),
Xi — c o n s titu e n t a c tiv itie s of th e x v ecto r.
A fte r th is th e o re tic a l discussion, it is n e c e ssa ry to r e tu r n to th e a p p li c a tio n of th e m e th o d in th e c o n stru c tio n of an L P m a trix w h ic h is to d e te rm in e th e o p tim u m p ro g ra m of p ro d u c tio n an d p o ssible in v e stm e n ts. T he se t of c o n s tra in ts c o n stitu tin g th e m ix ed s tra te g y c o n ta in s an
n
e le m e n t. In th e m a trix c o n s tru c te d fo r th e d esc rib ed ta sk , th e = i
n
V x ; < d is 're p la c e d b y th e w h o le in p u t-o u tp u t co efficien ts m a trix . T he
i = 1 p ro b le m can th u s be fo rm u la te d as follow s: m ax im ize: x p, su ch th a t: A x ^ b rn^x - Xp ^ 0 m nx — x p ^ 0 x > 0
T h e c rite rio n of choice u sed in th e fo reg o in g exam ple, is n o t th e o n ly one, a lth o u g h th e m ost p o p u la r. A d e ta ile d an a ly sis of a ll c r ite ria ca n be fo u n d in A d a m u s (1).
A s im ila r ap p ro a c h to o b je c tiv e fu n c tio n flu c tu a tio n s as in th e th e o ry of gam es can be fo u n d in th e ’’s a f e ty -firs t” m eth o d . T h e id ea of ’’s a f e ty f ir s t ” w as w o rk e d o u t b y R oy (26) and T esler (28). It w as f u r th e r develo p ed an d a p p lied to L P by M a ru y a m a (16) a n d b y P e tit an d 2 4*
3 7 2 S. G çdek
B o u ssard (21). A cco rd in g to th is m eth o d , a fa rm sh o u ld be o p e ra te d in su ch a w a y th a t th e p ro fit e v e ry y e a r cou ld be h ig h en o u g h fo r th e fa rm to m a in ta in its existen ce. T his m e a n s th a t th e f a r m ’s incom e h as to e n s u re a t le a s t a social m in im u m fo r th e fa rm e r an d h is fam ily , an d to p a y fo r a ll th e c h a rg e s (d e b et in s ta llm e n t p a y m e n ts , in te re sts , ta x e s etc.) e v e ry y e a r irre s p e c tiv e of w e a th e r co n d itio n s an d p ric e flu c tu a tio n s. It is n o t e n o u g h to h a v e a h ig h m e a n incom e b ec au se it can be sp e n t if ’’b ad h a r v e s t” is n o t e x p e cted . M oreo ver, p re v io u s incom es do n o t n e c e ssa rily im p ly th a t th e y w ill be s im ila r in ^th e fu tu re . A t b est, it o n ly fo llo w s th a t su ch an d su ch incom es, p ro fits, o r y ield s w ill be o b ta in e d in th e f u tu re . I t is im p o ssib le to k n o w ho w o fte n th is w ill h a p p e n fo r th e sa m p le is too sm a ll to in fe r a n y th in g fro m , th e m o re so t h a t th e o b se rv a tio n s fro m th e p re v io u s y e a rs c a n n o t p o ssib ly be r e cognized as d ra w n o u t b y lot.
T h e re fo re , th e L P m a tr ix sh o u ld be su ch as to p re v e n t a s itu a tio n w h e re th e m e a n p ro fit or inco m e is h ig h , b u t its s ta b ility is n o t s u ffi cien t, w h ich lead s to a f a rm fa ilu re . In M a ru y m a ’s a lre a d y -c ite d w o rk , th is p ro b le m is so lv ed by: m x m ax im iz atio n , su ch th a t: A x <C b nijx < d m nx <C d x ^ 0, w h e re : m — m e a n o b je c tiv e fu n c tio n p a r a m e te rs v ec to r,
m i ... m n — o b je c tiv e fu n c tio n p a r a m e te rs in each of n y ea rs,
A — in p u t- o u tp u t co e fficien t m a trix ,
d — th e le v e l b elo w w h ic h inco m e (p ro fit) c a n n o t d ro p in
a n y y ea r.
II I . I N T R O D U C T I O N O F R I S K I N T O I N P U T - O U T P U T C O E F F I C I E N T S M A T R I X
F lu c tu a tio n s of p la n n in g p a r a m e te rs a re cau sed e ith e r b y p ric e flu c tu a tio n or y ie ld c h a n g e . P ric e flu c tu a tio n s , in te rm s of L P , a ffe c t o b je c tiv e fu n c tio n p a r a m e te rs only. O n th e o th e r h an d , y ie ld f lu c tu a tions a ffe c t also in p u t-o u tp u t coefficien ts. If fa rm p la n n in g is to
M ethods of C onsidering Risk in P rogram m ing M odels 3 7 3
b e c o n siste n t w ith re a lity , th is p ro b le m m u st be ta k e n in to ac co u n t as w ell.
O ne of th e m eth o d s of co n sid erin g flu c tu a tio n s of in p u t-o u tp u t co e ffi cien ts is th e so-called ’’C h a n c e -c o n stra in e d P ro g ra m m in g ” (5). T he assu m p tio n s of th is m e th o d a re th e follow ing: if in som e c o n s tra in ts th e re a re p a r a m e te rs su b je c t to ra n d o m flu c tu a tio n s , th e s e c o n s tra in ts c a n n o t be m e t w ith a 100% p ro b a b ility . To p u t it in a n o th e r w ay, w e can assu m e th a t th e ris k -a ffe c te d c o n s tra in t sho u ld be m e t w ith a p ro b a b ility of no less th a n fo r in sta n c e 0.90, 0.95 or 0.99. U sing th e l a t t e r a p p ro a c h as th e s ta rtin g p o in t, it is n e c e ssa ry to add th e 90% , 95% or 9 9 % co n fid en ce in te rv a l to th e su m of th e p ro d u c ts of p a r a m e te rs m ean v a lu e s b y th e v a lu e of th e ir c o rre sp o n d in g a c tiv ities. T hus, if th e d e te rm in is tic fo rm u latio n of th e p ro b le m is th e follow ing:
a kx ^ bk, w h e re :
a k — v e c to r of in p u t-o u tp u t co e fficien ts v ector, x — a c tiv ity v ec to r,
bk — th e m in im u m v a lu e of akx e n su rin g th e co h e re n c e of th e p ro
g ram ,
th e n it is n e c e ssa ry to re p la c e a kx by: akx --- — i/'V akx , w h e re
a *
t — sta n d a rd iz e d co n fid en ce in te rv a l, V akx — a kx v a ria tio n ;
if th e c o n s tra in t is to be m e t w ith th e re q u ir e d p ro b a b ility . F u r th e r :
V akx = x TG kx, w h e re:
G k — v a ria n c e -c o v a ria n c e m a trix of th e a k v ec to r. TJie w h o le e q u a tio n can th u s be p re s e n te d as follow s:
akx i/x G kx , a v an d it h a s to b e m o re th a n or e q u a l to b k. T h en th e w h o le p ro b le m is as follow s: m ax im iz e: m Tx 5, su ch th a t: A x ^ b akx —t x TG kx ^ b x > 0.
D The ob jective fu n ctio n has b een form u lated in a d eterm in istic w a y to sim p lify the notation. There is no ob stacle to form u latin g it in any other w ay.
3 7 4 S. G^dek
Tw o d iffic u ltie s a re c o n n e c te d w ith th is p ro b le m . F irs t, in o rd e r to be u se fu l, it h a s to b e re so lv a b le an d th e r e m u s t be an a lg o rith m of th e so lu tio n . A lth o u g h th is a lg o rith m is a v a ila b le (32), it h a s a n u m b e r of d efects. N ot le a st is its s m a ll e ffe c tiv e n e ss an d v e r y h ig h re s tric tio n s on th e size of th e p ro b le m . T h e o th e r d iffic u lty w ith th e c h a n c e -c o n s tra in e d p ro g ra m m in g is th e a ssu m p tio n of a n o rm a l d is trib u tio n of flu c tu a tio n s of in p u t- o u tp u t co efficien ts, w h ich is n o t a lw a y s te n a b le . T his in c o n v e n ie n c e can be av oid ed b y u sin g th e T sh e b y sh e v in e q u a lity (24), in th is case a c o n sid erab le in c re a se of th e t a p a r a m e te r h as to be ta k e n in to a c c o u n t.6'
v
To avoid all th e s e in co n v en ie n ces, a tte m p ts w e re m ad e to sim p lify th is m eth o d . M e rill (17) a n d C hen (6) d ev e lo p e d m e th o d s co n sistin g in th e in te rc h a n g e of th e o b je c tiv e fu n c tio n an d th e c o n s tra in t a ffe c te d b y in p u t-o u tp u t p a r a m e te r flu c tu a tio n s , w h e n o n ly one c o n s tra in t is s u b je c t to th e m . R a h m a n an d B e n d e r (24) fo rm u la te d a rp e th o d a p p li cable in a s itu a tio n w h e re c o v a ria n c e b e tw e e n in p u t-o u tp u t p a r a m e te rs does n o t ex ist o r ca n be ig n o red , A m o re g e n e ra l an d sim p lified m e th o d w a s W orked o u t b y W ic k s an d G u ise (30). I t p e rm its th e u s e of L P b ec au se it is fo u n d e d on a b so lu te d e v ia tio n r a th e r th a n s ta n d a rd d e v ia tio n as a m e a s u re of flu c tu a tio n .
M a d a n sk y ’s m e th o d (13) h as an e n tir e ly d iffe re n t b a c k g ro u n d as it is d e riv e d fro m th e th e o ry of g am es. T he m e th o d assu m es th a t, if th e re is a n y p a r a m e te r a ffe c te d b y flu c tu a tio n s in th e c o n stra in t, th e co n s tr a in t h as to be m e t in ea ch situ a tio n . In te rm s of th e L P u se d in fa rm o rg a n iz a tio n p la n n in g , th is m e a n s th a t th e c o n s tra in ts u n d e r c o n sid e ra tio n h av e to be m e t ea ch y e a r w h ic h is an in fo rm a tio n sou rce. T hus, if th e c o n s tra in t h a s a d e te rm in is tic fo rm u la :
a^x <C bi,
th e n in th e case of th e aj v e c to r flu c tu a tio n s a n d u sin g th e M ad a n sk y assu m p tio n , th is n o ta tio n s h o u ld b e p re s e n te d as follow s:
6 The T sh eb y sh ev in e q u a lity is: [(Px„ —x) < to 2] > 1 w h ich m eans that the
t2
prob ab ility that the n -th rea liza tio n w ill not not d ev ia te from m ean by no m ore than t tim es of <5 is h igher th an 1 .Thus, if the con strain t is to be m et w ith
t2 __
th e p robability o f not less than 1 —a, th en 1 —a - i - h ence t = 4 / — . For
t2 V a
the p rob ab ility eq u al 0.95, t t&AA, w h ich ijs m ore than tw ic e of t0 05. The sm aller th e a p aram eter, the bigger that disproportion is.
M ethods of C onsidering Risk in P rogram m ing M odels
375
ajjx ^ b ai2x <c: b
a inx <c; bi, w h e re:
ajj — re a liz a tio n of th e aA v e c to r in each so u rc e -o f-in fo rm a tio n y ea r. If bi w e re also su b je c t to flu c tu a tio n s, th e above n o ta tio n co u ld b e m o d ified as:
a u x < bu
a inx ^ bin, w h e re :
by — re a liz a tio n of th e bi p a ra m e te r.
In tro d u c tio n of w h a t h as b e e n p re v io u sly ac h ie v ed in to th e L P m odel is a lre a d y obvious:
m axim ize: m Tx 7, such th a t: A x <C b
a n x < b „
a lnx < bln
**knx
3 7 6 S. G^dek
IV. SUM M ARY: E V A LU A TIO N OF M ETHOD U SE FU LN E SS
M ost of th e p re s e n te d m e th o d s h a v e th e ir p ra c tic a l ap p lica tio n . T h e m e th o d s d ev e lo p e d b y F r e u n d an d b y M a rk o w itz a re m o st f r e q u e n tly e m p lo y e d (2, 3, 4, 8, 25, 31), b u t th o se fo u n d e d on th e th e o ry of g am es a re also app lied . H o w e v er, th e re a re no stu d ie s w h a ts o e v e r th a t w o u ld c o m p a re all th e m e th o d s in q u estio n . M ore o ften , w e ca n e n c o u n te r c r i ticism of a p a r tic u la r m eth o d , w ith its w e a k p o in ts an d d e fe c ts em ph asized .
A f r e q u e n t o b je c t of c ritic ism is th e F re u n d m eth o d . A cco rd in g to P e tit an d B o u ssard (21), th e fu n d a m e n ta l o b jectio n to th e m e th o d is th a t it re q u ire s an a ssu m p tio n of th e n o rm a l d is trib u tio n of y ie ld s an d p ric e s in o rd e r to o b ta in th e o b je c tiv e fu n c tio n . T his h a s n o t b e e n p ro v e d so fa r w h e re a s o n ly ab so lu te c e r ta in ty w o u ld ju s tif y tjrfis assu m p tio n . F u rth e rm o re , th e r e a re r e p o rts th a t th e d is trib u tio n of cro p y ield s an d p ric e s of fa rm p ro d u c ts is n o t n o rm a l or ev en n o t sy m m e tric . P e tit an d B o u ssa rd a f te r D a y (9). A n o th e r o b je c tio n co n c e rn s th e ris k av e rsio n coefficient, w h ich is d iffe re n t fo r e v e ry d ec isio n -m a k in g fa r m e r an d h as to b e d e te rm in e d b e fo re o p tim iz a tio n p ro c e d u re s. T his m u st be d e te r m i n ed b y e x p e rim e n t, w h ic h is critic iz e d b y M o scard i an d d e J a n v r y (18) b e c au se th e co e ffic ie n t v a lu e so d e fin e d w ill be a ffe c te d b y th e f a r m e r ’s a ttitu d e to w a rd s g am b lin g .
A lth o u g h fr e e fro m th e fo reg o in g o bjectio n s, th e M ark o w itz m eth o d h as also its ow n d efects, th e m ost serio u s b ein g th a t a d u a l so lu tio n s is n o t p o ssib le (20).
T he above d isa d v a n ta g e s of th e tw o m e th o d s c a n be f u r th e r s tr e n g th ened b y th e fa c t th a t th e y re q u ir e q u a d ra tic p ro g ra m m in g , w h ich is m o re re s tr ic tiv e as to th e siz e of th e p ro b le m , w h ile th e in fo rm a tio n on w h ich th e m e th o d s a re based, th a t is m e a n Values a n d v a ria n c e , is r a r e ly cred ib le. In o rd e r to o b tain su ch fig u re s, th e d a ta co v e rin g fa r m o re th a n te n y e a rs sh o u ld be used. T h ese d a ta a re n o t a lw a y s av a ila b le ; m o reo v e r, th e p ic tu re can be d is to rte d b y y ie ld ch a n g es o v er a lo n g e r p e rio d d u e to n e w d e v e lo p m e n ts in tech n o lo g y , u n le s s w e h a v e th e d a ta o b tain ed fro m e x p e rim e n ts. P ric e s can also be a ffe c te d b y su c h s y ste m a tic changes.
A ll th e s e d e fe c ts of th e tw o m e th o d s also h o ld fo r th e ir m o d ifica tions, e x c e p t fo r t h a t re s u ltin g fro m th e u se of q u a d ra tic p ro g ra m m in g . M eth o d s fo u n d e d on th e th e o ry of g am es h a v e th e ir ow n d e fe c ts as w ell. F o r ex a m p le , W ickas a n d G u ise (30) ra is e an o b jectio n th a t a p p li catio n of th e th e o ry of gam es in c re a se s th e m a trix size. T h is is an essen tia l o b jectio n since th e L P m a tric e s em p lo y ed in o p tim izin g fa rm p ro d u c tio n and in v e s tm e n t p la n s a lre a d y h a v e c o n sid erab le sizes. A n o th e r
M ethods of C onsidering Risk in P rogram m ing M odels 3 7 7
o b jectio n W icks an d G uise d iscuss is th a t w h ile ap p ly in g th e th e o ry of gam es to risk co n sid eratio n , w e im p lic itly assu m e th a t th e f a r m e r ’s a tti tu d e to w a rd s ris k can be d esc rib ed b y th a t th e o ry . T h e re is no ev id en ce to su p p o rt th is assu m p tio n . S till one m o re o b jectio n can be ad d e d th a t in fo rm a tio n d ra w n fro m th e p ast, esp e cially like th a t u sed in th e th e o ry of gam es, c o n trib u te s v e ry little to p lan n in g . M oreover, w ith th e selec tio n of d a ta fro m p re v io u s y ea rs, an un con scio u s assu m p tio n is m ad e th a t o nly th o se y e a rs an d n o n e o th e r are re p re s e n ta tiv e a n d th e ir n u m b e r is su ffic ie n t as th e in fo rm a tio n basis. .
N one of th e d iscussed m e th o d s seem s to be fre e fro m d efects. S uch a m e th o d is d iffic u lt to im agine, e sp e cially u n til th e h a rm fu ln e s s of ris k is defin ed . A n a tte m p t to d eal w ith th e p ro b le m in th a t w a y is p a r t of th e s a f e ty -firs t m eth o d b u t it is d iffic u lt to a p p ly it in th e case of flu c tu a tio n s of in p u t-o u tp u t co effien ts.
Tab. 1. Crop yields of four oats varieties in lbs. per acre in 1953—1957
Y ear V ariety 1953 1954 1955 1956 1957 A 1472 2112 1920 3520 3072 B 1568 1984 1824 3104 3328 C 1440 2368 2496 3552 2848 D 1952 2688 2784 O1 3200
1. The D crop w as destroyed by hail in 1956. This is the slo w e st- -g ro w in g of the four v a rieties tested . H ail, w h ich norm ally occurs after harvest, a ffected this v a riety in 19l56 due to a prolonged v e g eta tio n period.
Source: O. L. W alker et al., A pplication of G am e T heoretic M odel to D ecision M aking, A gronom y Journal, no 2, 1964.
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S T R E S Z C Z E N I E
C elem p rzedstaw ionej tu pracy jest opis i porów nanie m etod u w zględ n ian ia ryzyka w m odelach op tym alizacyjn ych stosow an ych w roln ictw ie. P rzedm iotem opisu b y ły przede w szystk im zagadnienia teoretyczne, a w ięc zarów no strona for- m aln o-m atem atyczn a p rezen tow an ych m etod, jak i form u jący je zesta w założeń ekonom icznych.
O pisyw ane w n in iejszej pracy m etody u w zględ n ian ia w ahań losow ych para m etrów fu n k cji celu oparte są na teorii użyteczności bądź na teorii gier, a służące do uw zględ n ian ia w ah ań param etrów techniczn o-ek on om iczn ych rów n ież m ają uza sad n ien ie teoretyczn e w teorii gier oraz na tak zw anych ograniczeniach losow ych (chance constraints). Próba oceny w yk azała, iż w ięk sze n ad zieje n ależy w iązać z grupą m etod opartych na teorii gier. Do czasu ustalen ia na czym polega szk o d li w ość ekonom iczna ryzyka trudno jest jednak w yd aw ać jednoznaczne oceny.
Р Е З Ю М Е Ц ель настоящ ей работы — описать и сопоставить методы уч ета риска в оптим ализационны х м оделях, применяемы х в сельском хозя й стве. Предметом описания были п р е ж д е всего теоретич еские вопросы, в том чусл е как ф ор м ал ь н о-м атем атическая сторона представляем ы х методов, так и ф орм ирую щ ий их комплекс эконом ических предпосы лок. Описанные в настоящ ем исследовании методы учета случайны х к олеба ний параметров ф ун к ц и и цели опираю тся на теорию п олезности или на теорию игр, а методы, сл уж ащ и е для учета колебаний техн и к о-эк он ом и ческ и х п ар а метров, теоретически обоснованы т ак ж е теорией игр и, кроме того, так назы ваемыми случайны ми ограничениями (chance constraints). Попытка оценки обн а р уж и л а, что больш е н а д еж д подае ттруппа методов, опираю щ ихся на сеорию игр. Однако до установления, в чем состоит экономическая вредность риска, ф о р мировать однозн ачны е оценки представляется затруднительны м .
