Multiple-Criteria Decision-Making Techniques and Their ... · dgricultural Systems 15 (1984) 23-49 Multiple-Criteria Decision-Making Techniques and Their Role in Livestock Ration

dgricultural Systems 15 (1984) 23-49

Multiple-Criteria Decision-Making Techniques and Their Role in Livestock Ration Formulation

Tahir Rehman

Department of Agriculture and Horticulture, University of Reading, Reading, Great Britain

&

Carlos Romero

Departmento De Economia Y Sociologia Agraria, University of Cordoba, Cordoba, Spain

S U M M A R Y

The livestock ration formulation problem is postulated within the framework of multiple-criteria decision-making techniques. This exercise is motivated by theJact that the ordinary least-cost approach can, and does, generate solutions that either cannot be implemented or supply nutritionally undesirable levels of various nutrients. This originates from using cost as the criterionJbr selecting the ingredients of the diet. But in fact, diet Jormulation problems invoh'e several criteria and should, therejbre, be solved using techniques designed for modelling such problems. This paper is an attempt at introducing these techniques to agricultural systems modellers and then demonstrating their use in livestock ration jormulation. The multiple-criteria decision-making techniques covered include goal programming and its variants such as weighted and lexicographic approaches and multiple-objective programming.

I N T R O D U C T I O N

Of various mathematical optimisation techniques, linear programming (LP) is probably the most commonly used technique for establishing the

23 Agricultural Systems 0308-521X/84/$03'00 ~ Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain

24 Tahir Rehman, Carlos Romero

least cost combination of foods that will meet a specified level of nutritional requirements for livestock. Waugh (1951) was perhaps first in applying LP to problems of animal nutrition. Subsequently, several successful attempts have been made to explain the underlying methodology and its practical applicability (for an excellent treatment see Dent & Casey, 1967).

Despite its widespread use, the theoretical development of LP as a means of formulating livestock rations seems to have remained static over the years. The original analytical framework has been extended somewhat by such attempts as: (a) the introduction of probabilistic restraints (Rahman & Bender, 1971; Chen, 1973); (b) the investigation of the relationship between bulk and cost (Mohr, 1972) and (c) the use of the technique within a practical environment (Chappell, 1974; Crabtree, 1982).

The main weakness of the LP paradigm when applied to ration formulation lies in the exclusive reliance on cost as the only decision criterion. This is a very rigid assumption, as more often than not the decision-maker is interested in an economically optimal ration that achieves a compromise amongst several conflicting objectives such as minimisation of cost, imbalances of nutrient supplies and food bulk, and the satisfaction of certain conditions like the calcium/phosphorus ratio. A further weakness of LP is the rigidity with which given nutritional requirements have to be met. In many real life situations these requirements do not impose nutritional constraints that are absolutely binding.

We believe that these methodological weaknesses of LP can be overcome, and its application to diet formulation can be substantially improved if the problem is analysed within the relatively new framework of multiple-criteria decision-making techniques. Such an approach offers the prospect of important practical advantages.

This paper introduces the applicability of multiple-criteria decision- making techniques to agricultural systems modellers. The techniques to be explored in this context are: goal programming (GP) and its variants: weighted goal programming (WGP) and lexicographic goal programming (LGP), and multi-objective programming (MOP).

To our knowledge no previous attempt exists that analyses livestock ration formulation as a problem involving multiple objectives and goals; except, perhaps, the works of Arthur & Lawrence (1980) where the blending problems--closely related to the diet problem--is analysed

Multiple-criteria decision-making techniques and livestock ration Jbrmulation 25

using LGP, and Masud & Hwang (1981) who use interactive sequential goal programming for problems in human nutrition. Recently, Anderson & Earle (1983) have used WGP to minimise nutrient imbalances in human diets in the Third World.

It should be remembered, however, that our main purpose is to illustrate the applicability of a new method; therefore, it is quite possible that some of the rations obtained may be unimplementable, and/or the problems postulated in this paper are not particularly realistic or interesting from a nutritional point of view. At this stage we wish to explain and illustrate a new method rather than solve specific animal feeding problems.

LIVESTOCK RATION FORMULATION EXAMPLE INVOLVING MULTIPLE-CRITERIA DECISIONS

This section presents a ration formulation problem published in this journal (Crabtree, 1982) which is used as a basis for discussion in this paper. Crabtree's example has been chosen as it has been derived from the most widely accepted sources of information on animal nutrition and also because it seems to have many of the problems that lend themselves to analysis by multiple-criteria decision-making techniques.

Crabtree's LP model analyses the ration formulation of a 600 kg dairy cow with a milk yield of 30kg/day and no liveweight change. The available ingredients, their compositions, and nutritional requirements of the animal are given in Table 1. However, for our own purposes we have relaxed somewhat the upper limits on total dry matter intake and the amount of swedes and distillers' wet grain in the ration as well as the proportion of dry matter derived from silage and straw.

Most of Table 1 is self explanatory but the constraint (14) means that at least 20 ~o of the dry matter intake must be silage and straw, while the constraint (16) implies that the dry matter intake from distillers' wet grains must not exceed 20 ~o of the total dry matter.

The least-cost diet for this problem is comprised of the following:

Silage (xl) = 4.981 kg. Straw (x2) = 0-0 kg. Distillers' wet grains (x3) = 3.436 kg. Swedes (x4) --- 2-0 kg.

Barley (Xs) = 2.778 kg. Dairy compound (x6) = 5-069 kg.

The diet reported in the above solution and the subsequent ones discussed in this paper are all given as units of kg DM/day.

TA

BL

E

1

Cra

btr

ee's

Mo

dif

ied

Mat

rix

I'O

O

~

Sdag

e St

raw

D

isti

ller

s' w

et

Swed

es

Bar

ley

Dai

ry c

ompo

und

Tran

sfer

(k

g D

M)

(kg

DM

) gr

ains

(kg

DM

) (k

g D

M)

(kg

DM

) (k

g D

M)

acti

vity

48.2

26

.8

69.8

92

.6

105"

6 16

2.5

0"0

=Z

min

(£

/ton

ne)

10.3

6.

5 10

.3

12.8

13

.0

12.8

0.

0 10

8.0

32.0

13

8.6

97.2

10

5.3

107.

2 0.

0 19

.0

8.0

59.4

10

.8

11.7

52

.8

0.0

3.9

2-7

1.7

3.6

0.5

11.4

0.

0 3.

3 0.

9 3.

7 3.

2 4.

0 8.

5 0-

0 >

1.6

0.7

1.4

1.2

1.3

3.5

0.0

> 3.

8 1.

1 0.

9 2.

6 0.

2 5.

4 0-

0 >

5-7

3.2

10.0

3.

8 4.

8 19

.0

0.0

> 0.

09

0.04

0.

2 0.

04

0-04

1.

50

0.0

>_

1.0

1.0

1.0

1.0

1.0

1.0

0.0

<

1.0

1.0

1.0

1.0

1.0

1.0

- 1.

0 =

0.56

0.

35

0.56

0.

70

0.71

0.

70

-0-6

37

> 0.

56

0.35

0.

56

0.70

0.

71

0.70

-0

-642

<

-0.8

-0

.8

0.2

0-2

0.2

0.2

0-0

< 0

0 0

1.0

0 0

0.0

< -0

.2

-0.2

0.

8 -0

-2

-0-2

-0

.2

0.0

< 1.

0 0

0 0

0 0

0.0

< > 21

3.3

Met

abol

isab

le e

ner

gy

--M

E (

MJ)

(1

) >

1 66

3.3

Rum

en D

egra

dabl

e P

rote

in--

RD

P (

g)

(2)

.~.

> 62

0.5

Und

egra

dabl

e P

rote

m--

UD

P (

g)

(3)

r%

> 73

.6 C

alci

um--

-Ca

(g)

(4)

~-

67-8

Ph

osp

ho

rus-

-P (

g)

(5)

24.0

Mag

nes

ium

--M

g (

g)

(6)

" 26

.9 S

od

ium

--N

a (g

) (7

) 18

0-0

Co

pp

er--

Cu

(m

g)

(8)

2.0

Co

bal

t--C

o (

rag)

(9

) 19

.0 D

ry m

atte

r in

tak

e--D

M (

kg)

(10)

0.

0 T

ie l

ine

(I 1

) 0-

0 M

etab

olis

abil

ity

min

imum

q

(12)

0.

0 M

etab

olis

abil

ity

max

imum

q +

r

(13)

0.

0 L

ong

roug

hage

(~o

) (1

4)

2.0

Sw

edes

(kg

) (1

5)

0.0

Dis

till

ers'

gra

in (

%)

(16)

6.

5 Si

lage

(kg

) (1

7)

Sou

rce:

Cra

btre

e (1

982,

p.

301)

wit

h m

odif

icat

ions

to r

ight

-han

d si

de p

aram

eter

s fo

r dr

y m

atte

r an

d sw

edes

and

the

pro

port

iona

te a

mou

nt o

f dis

till

ers"

w

et g

rain

tha

t sh

ould

be

incl

uded

in

the

rati

on

Multiple-criteria decision-making techniques and livestock ration formulation 27

This ration costs £1.782/day and Table 2 compares its composition with the specified requirements. The nutrient requirements are binding restraints only for metabolisable energy, undegradable protein and copper. Several nutrients are in surplus supply--particularly magnesium, sodium and cobalt--which may be an undesirable nutritional balance in the diet. This situation could be redeemed by setting upper, as well as lower, limits for each nutrient. This approach, however, produces over- constrained problems with empty feasible sets in many cases. For instance, if, in this example, the supplies of sodium, copper and cobalt are bound to upper limits of 50 g, 400 mg and 6 mg, respectively, there is no solution that satisfies all the restrictions.

TABLE 2 Nutrient Requirements and Ration Composition using the Least-

Cost Criterion Only

Ingredient Requirements Ration

Metabolisable energy (M J) 213.3 213.3 Rumen-degradable protein (g) 1 663.9 2 044.6 Undegradable protein (g) 620.5 620.5 Calcium (g) 73-6 91.6 Phosphorus (g) 67.8 89.8 Magnesium (g) 24.0 36.5 Sodium (g) 26.9 55.1 Copper (mg) 180.0 180 Cobalt (mg) 2.0 8.9 Dry matter (kg) 19-0 18.30

The reduction of surplus supplies of various nutrients in least-cost diets ought to be one of the goals in formulating rations using mathematical techniques. Anderson & Earle (1983) have argued this point and used minimisation of nutritional imbalance as the objective function in formulating human diets using goal-programming techniques. However, their particular purpose could have been equally well served by using a traditional LP approach, as demonstrated by Romero & Rehman (1984b). Nonetheless, the emphasis on cost as the single criterion within the framework of LP does nothing to reduce the chances of producing nutritionally undesirable rations.

Besides cost and nutritional imbalances, another important con- sideration in ration formulation is how bulky the diet is. Obviously, rations


which are bulky, either because of low physical density or because of high moisture content, imply considering extra costs of storage and handling, and the limitations imposed by storage capacity. Therefore, reducing the bulk of the ration becomes a desirable goal. With this in mind, Crabtreds example can now be reformulated as a problem with three different objectives: (a) minimise the cost of the ration; (b) minimise the imbalanced supply of copper, sodium and cobalt and (c) minimise the bulk (i.e. the wet weight) of the diet.

Obviously, such a formulation of the diet selection problem leads to a situation where the achievement of one objective is in conflict with the attainment of another. But such a problem cannot be solved by optimising each objective function independently. However, problems with multiple goals and objectives can be solved using goal-programming and multi-objective programming techniques. This is what we set out to do in the next sections.

RATION FORMULATION AS A WEIGHTED GOAL- PROGRAMMING PROBLEM

The theory behind weighted goal-programming techniques (WGP) was first introduced to the management science literature by Charnes & Cooper (1961). Any problem involving multiple goals or objectives is solved in such a way that the solution ensures their satisfaction. This procedure is in contrast to LP where only one objective can be optimised while the rest has to be treated as restraints. However, the concept of a goal is fundamental to these techniques which is obtained on combining an objective with a target. The targets are desired levels of achievement for any one of a decision-maker's objectives (Romero & Rehman, 1983). In the context of ration formulation minimising the bulk of a ration is an objective, while the specific target of obtaining it at a level less than 40 kg is a goal reflecting the aspirations of the ration-formulator.

WGP minimises the deviations among the desired levels of goals and the actual results. This is included in the model by converting inequalities into equalities through the addition of positive and negative deviation variables that permit either under- or over-achievement of each goal. WGP considers all goals simultaneously in a composite objective function. This function minimises the sum of all the deviations among goals and their aspiration levels. The deviations are weighted according to


the relative importance of each goal. For a detailed explanation of this technique in the context of farm planning see Romero & Rehman (1984a).

Now the three objectives proposed in the previous section can be converted into goals as follows.

Goal gl

The ration should not have a bulk larger than 40 kg. Using the dry matter content of various ingredients from Table 3, the expression for goal g~ is given by:

3.704x 1 + 1.219x 2 + 3-876x 3 + 9.259x 4

+ 1"200x5 + 1"137x6 +nx - P l = 40 (1)

where the deviational variable n~ measures the under-achievement of goal g~ while Pl plays the opposite r61e. As the desired level of ration bulk should not be greater than 40kg, the deviational variable pl must be minimised.

Goal g2

The supply of sodium in the diet should not exceed 100 % of its specified requirements. This imbalance goal, I~v a, is obtained from constraint (7) in Table 1:

INa = 3"8xl + l ' l x 2 + 0"9X a d- 2 ' 6 X 4 -t- 0"2x 5 + 5"4x 6 - 26.9 (2)

Considering eqn. (2) as percentages instead of absolute values, we have:

(3.8x I + l - lx 2 + 0.9x 3 + 2.6x 4 + 0.2x 5 + 5.4x 6 - 26-9). 100 26.9 (3)

Therefore the expression for g2 is:

14-13x 1 + 4.09x 2 + 3.35x 3 + 9.67x 4

+ 0"74x5 + 20"07x6 + n2 --P2 = 200 (4)

To achieve the desired level of this goal P2 must be minimised.

Goals g3 and g4

Using the same procedure as for sodium, the goals for copper and cobalt are derived from constraints (8) and (9) in Table 1, so that their supplies,


TABLE 3 Dry Matter Content of Ingredients (g/kg)

Ingredient Silage Straw Distillers' Swedes Barley Dairy wet grains compound

Dry matter content 270 820 258 108 833 880

Source: Crabtree (1982, p. 299).

too, should not exceed 100 % of the specified requirements. This is stated by eqns (5) and (6).

3.17x 1 + 1.78x 2 + 5.55x 3 + 2.11x 4

+ 2"67xs + 10"55x6 + n3 --P3 = 200 (5)

4"5X 1 + 2X 2 + 10X 3 + 2x 4 + 2x s + 75x 6 + n# - P 4 = 200 (6)

For achieving g3 and g4, P3 and P4 must be minimised.

Goal g5

Lastly, the objective of minimising the cost can be converted into a goal by setting a target of £1.782 as cost, which corresponds to the minimum cost associated with the ration recommended by the ordinary LP approach. Therefore, the expression for g5 using the unit costs of ingredients from Table 1, is:

0.0482x~ + 0.0268x z + 0.0698x 3 + 0.0926x 4

+O'1056xs+O'1625x6+ns-ps= 1"782 (7)

To achieve gs, P5 is minimised. The variables of the objective function must represent percentage

deviations from the targets. Minimisation of absolute deviations does not make sense when each goal is measured in different units. Hence, the elements of the objective function are standardised for the WPG model to give:

Pl 100 P2 100 P3 100 W 1 ~ 6 ~ - + W 2 - - - - + W3

200 1 200 1

P4 100 Ps 100 + W4 200 1 - - + W5 1.78~ 1 (8)

TA

BL

E 4

W

eigh

ted

Goa

l-P

rogr

amm

ing

Mod

el f

or R

atio

n F

orm

ulat

ion

Obj

ecti

ve f

unct

ion

Min

imis

e: 2

.5 W

lp I

+ O

. 50

W2

p 2

+ O

. 50

W3

p 3

+ O

" 50

W4

p 4

qt.

56"

12 W

sp 5

Su

bje

ct t

o:

3.7

04

x 1

+

1.2

19

x 2

+

3.8

76

x s

+

9-2

59

x 4

+

1.2

00

x 5

+

1.1

37

x 6

+

n I

- P

l =

4

0 (

bu

lk)

14

.13

x~

+

4.0

90

x 2

-b

3'3

50

x 3

+

9"6

70

x 4

+

0-7

40

x s

+

20

"07

0x

6 +

n

2 -P

2

=

20

0 (

imba

lanc

e in

sod

ium

)

3.1

7x

, +

1

.78

x 2

+

5.5

5x

3 +

2.

1 Ix

4 +

2

.67

x 5

+

10

.55

x 6

+

n 3

-P3

=

2

00

(im

bala

nce

in c

op

pe

r)

4"5

x I

+

2x

2 +

1

0x

3 -

-F 2

x 4

+

2x

5 +

7

5x

6 J

r-r/

4 -P

4

=

20

0 (

imba

lanc

e in

co

ba

lt)

0"0

48

2x

I +

0

-02

6 8

x2

+

0.0

69

8x

3 +

0

.09

2 6

x 4

+

0.1

05

6x

5 +

0

.16

2 5

x 6

+

n 5

- P

5 =

1

.78

2 (

cost

)

and

Ax

< b

(te

ch

nic

al

cons

trai

nts

from

Ta

ble

1)

x>

0

n>__

0 p

>0

2"

tm

g~

x is

a v

ecto

r of

rat

ion

ingr

edie

nts.

In

this

cas

e x

has

six

elem

ents

or

poss

ible

ing

redi

ents

. n

is t

he v

ecto

r of

dev

iati

onal

var

iabl

es r

epre

sent

ing

unde

r-ac

hiev

emen

t.

p is

the

vec

tor

of d

evia

tion

al v

aria

bles

rep

rese

ntin

g ov

er-a

chie

vem

ent.

A

s th

ere

are

five

goa

ls b

eing

con

side

red

in t

he m

odel

; th

eref

ore,

bot

h n

and

p ha

ve f

ive

elem

ents

eac

h.

E-


O r "

2.5Wlp 1 + 0.50Wzp z + 0-50 Wap s + 0.50W4P 4 + 56.12 Wsp 5 (9)

Where W I , . . . , W s are the weights attached to the deviational variables representing the respective importance given to the achievement of various goals. The structure of this W G P model is given in Table 4. Mathematically, it is a conventional LP problem and can be solved by the Simplex algorithm. Different solutions can be obtained by attaching different values to the W parameters. For instance, if W~ = W 2 . . . . W 5 = 1, the Simplex method provides the following opt imum solution:

x 1 (silage) = 3-511 kg x 3 (distillers' wet grains) = 3.80 kg x 5 (barley) = 6.298 kg

X 2 (straw) = 0"675 kg x4 (swedes) = 0 x 6 (dairy compound) = 4.716 kg

While the opt imum values of the deviational variables are:

n x = 0 Px = 1.477kg n 2 = 35"59~o P2 = 0 n 3 = 100% P3 = 0 n4 = 0 P4 = 221.41% n 5 = 0 P5 =£0"10

This solution permits complete achievement of the goals g2 (sodium

T A B L E 5 Sets o f Weights Used in the Sensit ivi ty Analysis o f the W G P Solu t ion

Run W l (bulk) W z (imbalance W 3 (imbalance W 4 (imbalance W s (cost) in sodium) in copper) in cobalt)

U) (2) (3) (4) (5) (6)

1 1 1 1 1 1 2 1 2 2 2 1 3 2 1 1 1 1 4 1 1 1 ! 2 5 1 3 3 3 i 6 3 1 1 1 1 7 1 1 1 1 3 8 1 1 1 1 4 9 "1 2 2 2 4

10 2 1 1 1 4 11 1 1 1 1 5

TA

BL

E 6

Se

nsit

ivit

y A

naly

sis

of t

he W

GP

Sol

utio

n

4 4.

Run

x~

x 2

x 3

%

x

s x 6

F

resh

wet

ght

(kg/

day)

Su

rplu

s o

f N

a as

Su

rplu

s o]

Cu

as

Surp

lus

o] C

o as

C

ost

(£)

(~da

ge)

(str

aa')

(d

lstd

lers

' (s

wed

es)

(bat

h,y)

(d

airy

pe

rcen

tofr

eqm

rem

ent

perc

ento

freq

uire

men

t pe

reen

toJr

equt

rem

ent

(kg)

(k

g)

wet

gra

ms)

(k

g)

(kg)

co

mpo

und)

A

ctua

l D

e~,i

atio

a A

ctua

l D

et,t

ati

on

A

ctua

l D

el't

at~

on

A

ctua

l D

et't

att

oa

A

ctua

l D

erta

tton

(k

g)

(kg)

a

chw

rem

ent

Jrom

the

ac

hwre

men

t fr

om

the

ac

hwre

men

t fr

om

the

ac

hwt'e

raen

t fr

om

the

ae

hwre

men

t [r

om

the

o] t

he g

oal

targ

et

of

the

goal

ta

rget

o]

the

goa

l ta

rget

o

f th

e go

al

targ

et

of

the

goal

ta

rget

I 3

511

0-67

5 3-

800

0 6

298

4-71

6 41

477

I

477

64.4

10

0 0

0 32

1.41

0 22

1 41

0 1

884

0'10

2 2

4.50

6 0

3800

0

6"11

8 4.

576

43-9

63

3963

72

-771

0

0 0

3137

41

213.

741

1-87

2 0.

090

3 2-

923

0917

3.

800

0 65

19

48

42

40

0

60

24

9

0 0

47

2

0 32

9 15

5 22

9-15

5 I

906

0 12

4 4

3 51

1 06

75

3-80

0 0

62

98

4

716

41.4

77

I 47

7 64

.410

0

0 0

321

410

221'

410

1.88

4 0'

102

5 5

144

0 3,

800

0 5,

520

4-53

6 45

564

5.

564

80.5

40

0 0

0 31

2-38

4 21

2 38

4 I

883

0-05

1

6 2

923

0 91

7 3

800

0 6-

519

4.84

2 40

0

60-2

49

0 0

0 31

3 74

1 21

3'74

1 1.

872

0.1"

24

7 4-

911

0 3

652

0 4

843

4.85

5 43

-678

3.

678

82 6

45

0 0

0 33

2.40

7 23

2.40

7 I

792

0,00

1 8

4-91

1 0

3 65

2 0

4-84

3 4.

855

43.6

78

3.67

8 82

645

0

0 0

332

407

232.

407

I 79

2 0.

001

9 5,

144

0 3.

800

0 5,

520

4-53

6 45

.564

5

564

80.5

40

0 0

0 31

2-38

4 21

2 38

4 1

833

0"05

1 I0

4

275

0 26

3 3

652

0 5.

147

4 92

4 42

-087

2

087

76.3

59

0 0

0 33

5 90

9 23

5-90

9 1-

812

0 02

3 11

4-

911

0 3

652

0 4.

843

4-85

5 43

-678

3

678

82.6

45

0 0

0 33

2 40

7 23

2-40

7 I

792

0 00

1

4. e~

;a

:..

k~

L~


supply imbalance) and g3 (copper supply imbalance) by virtue of the fact that Pl =P2 = 0. The value of 1-477kg for Pl implies that goal gl has exceeded its target by 1.477 kg or the bulk of the ration is 41.477 kg. Similarly, P4 = 221.41~ means that the imbalance in cobalt has surpassed its target by that proportion; that is, the supply of this nutrient is 321.41 ~ of the specified requirements. Finally, P5 = 0-102 means that the cost of the ration is £0.102 over the set target of £1.782/day.

The sensitivity analysis of a WGP solution can provide very useful information for ration formulation. To demonstrate the generation and use of this information we have used eleven sets of weights, as shown in Table 5 and corresponding solutions are presented in Table 6. From this information it is possible to obtain something like a measure of the trade- offs between goals. Thus, if, for instance, the decision-maker is undecided between solutions 1 and 2, the former will be preferred if the over- achievement of bulk by 2.486 kg can be traded off with a reduction in cost of £0.012 and for a drop in cobalt imbalance of 10.15 ~o. Such an array of information giving trade-offs among various goals should enable the decision-maker to choose the ration that best suits his aims. It must, however, be pointed out that if the weight attached to the cost of the ration is given a value higher than 10, then the LP solution given above is obtained again.

RATION FORMULATION AS A LEXICOGRAPHIC GOAL PROGRAMMING (LGP) PROBLEM

The LGP approach was introduced by Charnes & Cooper (1961) and later developed by Ijiri (1965), Lee (1972) and Ignizio (1976). The LGP approach assumes that it is possible to divide goals into priorities. Further pre-emptive weights can be attached to sets of goals situated in different priorities. In other words, the fulfilment of the goals in a given set, Qi, is immeasurably preferable to the achievement of any other set situated in a lower priority, Qj. In LGP higher priority goals are satisfied first, and only then are lower priorities considered. For a non-technical exposition of LGP, see Romero & Rehman (1984a).

To illustrate the use of LGP in ration formulation, we have to establish the priority structure of goals in this problem. Let us assume, for example, the goals of obtaining a ration costing less than £2 and weighing below 50 kg are situated in the first priority, Q1 ; that is, they are to be satisfied in


a pre-emptive way. These targets can be considered as minimum levels of achievement for the cost and bulk of the ration. Using eqns (7) and (1) and on modifying the subscripts of the deviational variables, the goals in priority Q~ are:

Cost: 0-0482x I + 0.0268x 2 + 0.0698x 3 + 0"0926x 4

+O.1056xs+O.1625x6+nl-p1=2 (10)

Bulk: 3.704x 1 + 1.219x 2 + 3.876x 3 + 9.259x 4

+ 1.20x 5 + 1.137x 6 + n 2 - P 2 = 50 (11)

As regards weights attached to the achievement of goals, assume that formulating a ration costing less than £2 is twice as important as the at tainment of required bulk. Now using the same procedure as for deriving the objective function of W G P in eqn. (10), the first component to be minimised in the lexicographic process is:

2. 100 100 2 Pl + ~-P2 = 100pl + 2p2

The next priority QE, is made up of goals g2, g3 and g4 described above; that is, the supply-requirement imbalance for sodium, copper and cobalt should not be greater than 100 ~o. This can be derived from eqns (4), (7) and (8), with the appropriate subscripts for the deviational variables; thus the goals in priority Qz are:

Sodium: 14.13x I + 4.09x z + 3.35x 3 + 9-67x4

+0"74x5 + 20"07x6 + n3 - P 3 = 200 (12)

Copper: 3.17x 1 + 1.78x 2 + 5.55x 3 + 2"11x4

+ 2"67x5 + 10"55x6 + n4 - P 4 = 200 (13)

Cobalt: 4.5x I + 2X 2 + 10X 3 -4- 2X 4 + 2X 5

+ 75X6 + n5 - P5 = 200 (14)

If the three goals in Q2 are of equal importance, and their targets and measurements are homogeneous, then the second component to be minimised in the lexicographic process is given by Pa +P4 +Ps .

Finally, the last priority, Q3, is made up of goals of finding a ration costing less than £1.782with minimum bulk (which is 22.34kg, see the following section). It is interesting to note the similarity that appears to exist between goals in priorities Q1 and Q3. In fact, in Q1 goals represent


something like minimum levels of achievement while Q3 represents desired levels of achievement. When goals are specified at both a minimum and a desired level in a pre-emptive way, some authors call this approach two-stage LGP (Keown & Martin, 1977).

The algebraic expression for the goals in Q3 is derived from eqns (1) and (7) as given below:

Cost: 0.0482x I + 0-0268x 2 + 0-0698x 3 + 0.0926x 4

+ 0"1056x5 + 0"1625x6 + r/6 - P 6 = 1-782 (15)

Bulk: 3"704x 1 + 1.219x 2 + 3.876x 3 + 9.259x 4

+ 1"20x5+ 1"137x6 + n7 - P 7 = 22.34 (16)

Once again, assuming that the cost goal is twice as important as the ration bulk, the last component of the lexicographic minimisation process is given by:

(2) 100 100 1-782 P6 + ~-f-.-.-~P7 = 112"2p6 + 4.47P7

The whole lexicographic minimisation process would be given by the following vector:

M i n a = [(100pl + 2pz) , ( p 3 + P 4 +Ps),(112"2p~ + 4"47pz)] (17)

The vector a is called the achievement function where each component represents the deviational variables that must be minimised to make sure that the goals ranked in this priority come closest to their targets. The structure of our LGP model is given in Table 7.

This problem cannot be solved by a straightforward application of the Simplex method, because we are now seeking a minimum value ordered vector rather than minimising the scalar product of two vectors. There are several algorithmic approaches that can be adopted to solve LGP problems. Perhaps the most efficient algorithms for LGP problems are the Sequential Linear Method (Dauer & Krueger, 1977; Ignizio & Perlis, 1979) and the Partitioning Algorithm (Arthur & Ravindran, 1978, 1980). On using the Sequential Linear Method we obtain the following optimum solution:

x 1 (silage) = 5.145 kg x z (straw) = 0 x 3 (distillers' wet grains)= 3.80 kg x 4 (swedes)= 0 x 5 (barley)= 5.519kg x 6 (dairy compound)= 4.536kg

TA

BL

E 7

L

exic

og

rap

hic

Go

al-P

rog

ram

min

g M

od

el f

or R

atio

n F

orm

ula

tio

n

Ach

iev

emen

t fu

nct

ion

:

Min

a =

[(1

00p~

+ 2

p2),

(P

3 +

P4

+ P

s),

(112

, 2p

6 +

4.47

p7)]

Sub

ject

to:

f 0.

048

2x 1

+ 0

.026

8x

2 +

0.06

9 8x

3 +

0.0

92 6

x 4

+ 0.

105

6x s

+ 0-

162

5x 6

+ n

I -P

l =

2 (c

ost)

Q

I:

t 3.

704x

1 +

1.

219x

2 +

3.9

76x

3 +

9.25

9x 4

+

1.20

0x 5

+

1.13

7x 6

+ n2

-p

2

= 50

(bu

lk)

~1

4"1

3x

l +

4"09

x 2

+ 3"

35x

3 +

9.6

7x

, +

0-74

x 5

+ 20

.07x

6 +

n 3

-P

a =

200

(im

bal

ance

in

sodi

um)

Q2:

~3"

17X

l +

1"78

x2 +

5"5

5x3

+ 2.

1 Ix

4 +

2.67

x 5

+ 10

.55x

6 q

- n 4

-P4

=

200

(im

bal

ance

in

copp

er)

t'4

'5x

l +

2x2

+ 10

x3 +

3x4

+ 2

x5 +

75x

6 +

n5 -

Ps

= 20

0 (i

mb

alan

ce i

n co

bal

t)

Q

f0.0

48

2x~

+ 0

.02

68

x 2

+ 0

.069

8x

3 +

0.0

92

6x

4 +

0.1

05

6x

5 +

0.1

62 5

x 6

+/%

-p

6

= 1.

782

i.e.

cost

3:

i, 3

.704

x I

+ 1.

219x

2 +

3.8

76x

3 +

9.25

9x 4

+ 1.

200x

5 +

1.

137x

6 +

n7 -

p7

--

- 22.

34 i

.e.

bu

lk

and

:

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< b

(tec

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al c

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ints

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m T

able

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3"


While the opt imum values of the deviational variables are:

n 1 = £0.167 Pl = 0 n z = 4.434 kg P2 = 0 n 3 = 19"458 ~ P3 = 0 n 4 = 100 ~ P4 = 0 n 5 = 0 P5 = 212"384~ n 6 = 0 P6 = £0-051 n 7 = 0 P7 = 23-227 kg

This solution permits complete achievement of the goals in Q1; that is, the minimum levels set for cost and bulk are achieved. For Q2 the targets for imbalances in sodium and copper supplies are also achieved but not for cobalt which has a positive deviation of 212.384 ~ which means that more than three times the requirement is supplied. Finally, in Q3 the least cost and minimum bulk goals have positive deviations of £0.051 and 23-277kg, respectively; that is, their actual values are £1.833 and 45.517kg.

Both sensitivity and parametric analyses of the LGP solutions can produce useful information, particularly if the analyst wishes to study the effects of rearranging priorities, changing targets and attaching different weights to the goals in a given priority. These analyses can be implemented easily using the optimal tableau of the LGP solution. (For details see Ignizio, 1976, Chapter 4 and Ignizio, 1982, Chapter 9).

RATION F O R M U L A T I O N AS A MULTI-OBJECTIVE P R O G R A M M I N G (MOP) PROBLEM

When several objectives, which are subject to a set of restraints (usually linear), are to be optimised simultaneously, then such problems can be solved using MO P (vector optimisation) techniques. Thus, while GP works with several goals, M O P deals with multiple objectives. However, when several objectives are involved simultaneously, it is impossible to define an opt imum solution; therefore, MOP seeks to find the set of ef f icient solutions (also called n o n - d o m i n a t e d or P a r e t o optimal solutions).

The elements of an efficient set must be a feasible solution. Further, they have to be such that there are no other feasible solutions that will yield an improvement in any one of the objectives without reducing the


achievement of at least another one. To illustrate this definition, assume that in our ration formulation problem there are two objectives, cost and bulk, with the following three feasible solutions:

a = [2 40] Efficient b = [2.5 40] Inefficient c = [3 20] Efficient

The first component of each vector is cost (£) and the second is bulk (kg). The solution in b is inefficient or inferior and should never be chosen by a rational decision-maker as it is dominated by the solution in a. The bulk of the ration is the same in both solutions, but solution a is cheaper than b. Whereas solution c is efficient and is not dominated by a, because, although more expensive, it is less bulky. The purpose of M O P , therefore, is to generate the efficient set of solutions. The problem is formulated in the following general framework:

subject to:

Eft Z (x) = [ZI(x), Z2(x) , . . . , Zq(X)] 08)

x ~ F

Where Eft means to search for the efficient solutions and F represents the feasible set of solutions.

To demonstrate the use of M O P in ration formulation, assume that in Crabtree's problem the three objectives to be identified are: (a) minimise the cost; (b) minimise the bulk and (c) minimise the aggregated over- supply in sodium, copper and cobalt. The formal structure of the multiobjective model is thus given by:

Eft Z (x) = [Z,(x), Zz(x), Z3(x)] (19)

Where

Z~(x) = 0.048x x + 0.0268x 2 + 0-0698x 3 + 0.0926x 4

+ 0"1056x 5 + 0-1625x 6 cost (£)

Z2(x) = 3.704Xl + 1-219x 2 + Y876x 3 + 0-259x 4

+ l '20x 5 + 1"137x 6 bulk (kg)

Za(x ) = 21.80x 1 + 7.87x 2 + 18.09x 3 + 13.78x 4 + 5-41x 5

+ 105"62x 6 - 300 imbalance in sodium, copper and cobalt


Subject to:

and

A x < b (technical constraints from Table 1)

x>_O

The expression for Za(x) has been derived from the addition of the imbalances of the three nutrients as defined above.

There are basically three approaches to generate, or at least to approximate, the etficient set: (a) weighting method; (b) constraint method and (c) multi-criterion Simplex method. A detailed explanation of these approaches can be seen in Cohon (1978) Chapter 6, and Goicochea et al. (1982) Chapter 3.

For our problem we have used the weighting method. The basic idea of this method is to combine all the objectives into a single one. A weight is attached to each objective function before they are all added into a single one. Subsequently, the efficient set is generated through a parameteric variation of the weights. Therefore, the MOP problem defined by eqn. (18) becomes the following parametric programming problem:

q

WiZ,(x) (20) i = 1

Maximise or Minimise

subject to:

x~F

This method was first proposed by Zadeh (1963) and Geoffrion (1968) provided the formal proof that if the weights are strictly greater than zero (W i > 0), the the solutions generated from eqn. (20) are efficient. These points are efficient only in a mathematical sense and can result in technically unacceptable solutions, particularly when the restraint set has not been properly defined. However, one practical drawback of this method is the possibility that some efficient points may be skipped. To avoid this possibility it is necessary to reduce, considerably, the scale of weights which can result in a large number of calculations. The weighting method is useful only as a tool to approximate the efficient set.

As our problem in eqn. (19) has only three objective functions, the weighting method generates all the efficient points. The six efficient rations of our problem are shown in Table 8. It is interesting to note that solution 1 corresponds to the minimum cost ration, solution 3 to the

Multiple-criteria decision-making techniques and livestock ration.[brmulation

TABLE 8 Multiobjective-Programming Model for Ration Formulation (Efficient Points)

41

EJ[~cient Silage Straw Distillers' Swedes Barley Dairy Cost Bulk Ot'er- solution x I x 2 wet grains x 4 x s compound (£) (kg) supply

(kg) (kg) x 3 (kg) (kg) (kg) )¢6 (kg) (%)

1 4"981 0 3'436 2 2"778 5"069 1"782 59'385 451"499 2 5'007 0 3"555 0 4"815 4.884 1.792 43'655 418.265 3 3-569 0'386 3.800 0 6'566 4.679 1.901 41"618 382-399 4 0.395 3'260 0 0 4.502 10-118 2-226 22.344 827-306 5 3.306 0.494 3-800 0 6-665 4.736 1.911 40.959 384"006 6 2-855 0-945 3"800 0 6.544 4'856 1.908 39"830 389'823

minimum imbalance ration and solution 4 to the minimum bulk ration. These three solutions establish what in MOP literature is called the ideal point or utopian point; that is, the point where all the objectives achieve the optimum value. The ideal or utopian point of our ration formulation problem is: [£1.782; 382.399~; 22.344kg].

The optimum ration is one that is chosen by the decision-maker from the set of efficient solutions. But, of course, the choice depends on the preferences of the decision-maker, depending upon his subjective values attached to the trade-offs among cost, imbalance and bulk. For instance, if solution 1 (least cost ration) is preferred to solution 2, in that situation a reduction of 15.370kg in bulk and 33.234~ in imbalance does not compensate for an increase of £0-010 in cost.

Basically there are two general approaches to choosing the optimal solution from the efficient ones: compromise programming and interactive techniques. The first approach defines as optimum the efficient solution that is closest to the ideal point. Depending on the particular measure of distance used, a set of compromise solutions can be established (Zeleny, 1973, 1974b, 1982; Chapters 5, 6 and 10). The interactive techniques rely on the progressive definition of the decision- maker's preferences; that is, there is an interaction between him and the model. The interaction proceeds by asking the decision-maker his preferences or trade-offs that he is prepared to accept and then solving the model in response to each question. The interactive techniques proposed by Masud & Hwang (1981) and Zionts & Wallenius (1976) are perhaps the most promising. However, the explanation of either interactive or compromise techniques is beyond the scope of this paper.


To conclude this section it should be mentioned that theoretically the efficient points calculated here are actually extreme points or corner solutions; therefore, the efficient points generated by MOP represent only a subset of the whole set of efficient points. In fact, there are two types of efficient points: (i) extreme points (generated by the methods used in this section); and (ii) interior points (points connecting efficient extreme points). The feasible set of solutions for the MOP model is derived from a convex and continuous region and, as such, it contains an infinite number of efficient interior points. But, from a practical point of view, only the subset of efficient extreme points is of interest; so, to find the whole efficient set is unnecessary. There are, however, some algorithms which search for efficient faces; that is, the convex combinations of efficient extreme points (Zeleny 1974b; Yu & Zeleny, 1975; Iserman 1977; Ecker et al., 1980). Thus, the whole efficient set is defined as the union of efficient faces. On applying the Yu and Zeleny algorithm to our example we observe that all the faces are efficient. This result is very common when there is a restraint that is binding for all the efficient extreme points, as is the case for undegradable protein constraint in our example.

PENALTY FUNCTIONS IN RATION FORMULATION

Some nutritionists argue that in formulating least-cost diets for livestock the ratios between some key ingredients such as forage and concentrate, calcium and phosphorus, etc., must be kept within nutritionally desirable limits. If these ratios are violated then a penalty system should operate within the diet formulation model to automatically penalise the cost of the ration. In contrast to the ordinary LP approach to diet formulation, such a penalty system can be built relatively easily into the GP model, which, in fact, provides a new perspective for dealing with the problem of choosing livestock rations.

Stranks (1983) suggests that in Crabtree's example the optimum forage/concentrate ratio should be set at 40 ~o or more forage, using the following penalties:

40-35 ~ - - incur a 1 unit penalty per 1 ~ fall 35-30 ~ - - incur a 2 unit penalty per 1 ~o fall 30-25 ~o--incur a 10 unit penalty per 1 ~ fall

Below 25 ~ - - incur an infinite penalty

Multiple-criteria decision-making techniques and livestock ration formulation 4 3

This suggestion implies that a penalty function similar to the one in Fig. 1 is hooked to our goal programming model by introducing one constraint and three additional goals as follows:

X 1 -t- X 2 - - 0 " 2 5 X 3 + X 4 - - 0"25x 5 - 0 " 2 5 x 6 ~> 0 (21)

X 1 + X 2 7 !- Xxl. + nl - P x = 30 (22)

X 3 --1- X 5 + X 6

x 1 + X z + x 4 + nl + n2 - P 2 = 35 (23)

x 3 + x 5 + x6

x~ + x 2 + x 4 + nl + n2 + n3 --P3 = 40 (24)

x 3 + x 5 + x 6

The contribution to the objective function of the W G P model or to the achievement function of the LGP model will be giyen by 10n I + 2n 2 + n 3 if we work with absolute deviations.

The constraint (21) means that rations with a forage/concentrate ratio below 0.25 are not feasible (i.e. incur an infinite penalty). The negative deviation nx of eqn. (22) measures the percentage points by which forage/concentrate ratio is below 30~o. In the same way the negative deviational variables n~ + r/2 and n~ + n 2 + 113 measure the percentage points by which forage/concentrate ratio is below 35~o and 4 0 ~ , respectively.

It must be pointed out that if variable n 1 in eqn. (23) and variables n~ + n 2 in eqn. (24) are omitted the total penalty is overestimated. For example, if the forage/concentrate opt imum ratio is 0.32 and variables n~ and n I + n 2 are omitted from eqns (23) and (24) then the total penalty (absolute deviations) will be 14, when the actual total penalty is only 11. This drawback is associated with some goal programming formulations using penalty functions (see Kvanli, 1980 and later modifications by Romero, 1984).

The equalities (22) to (24) are not linear, as in each expression cross multiplication by the denominator produces a non-linear constraint. Thus, the GP algorithms devised only for linear structures cannot be used. But for problems with such fractional goals the algorithmic approaches suggested by Kornbluth (1973) (pp. 201-3) and Kornbluth & Steuer (1981) can be used.

A natural progression of this penalty system approach to respect the nutritionally desirable qualities of a ration is to treat most nutrient


P e n a l t y ( u n i t s )

65

15

S l o p e =

= 10

S l o p e = 2 /

f / S l o p e = 1

Fig. 1.

25 30 35 40

F o r a g e / C o n c e n t r a t e r a t i o

Penalty function for forage/concentrate ratio.


Penalty (unite)

12

Slope = ~

0

Fig. 2.

I

Slope

Slope = I ~ i

1.5 4 6

Cobalt (mE)

Penalty function for cobalt.


requirements in a similar way. We demonstrate this by dealing with the particular case of cobalt requirements in the ration; but the approach outlined is a general one and could be used for all nutrients. For Crabtree's matrix assume that the penalty function for cobalt is characterised by:

(a) The intake in the ration is permitted to fall below 2 mg but never lower than 1.5 m g - - t h a t is, there is an infinite penalty for it to go beyond that point.

(b) If the intake is between 1.5 and 4mg no penalty is incurred. (c) If the intake is more than 4 mg and less than 6 mg, then there is a

penalty of one unit per milligram of excess supply. (d) If the intake is more than 6 mg and less than 8 mg, the penalty is

five units per milligram of excess supply. (e) The intake must never be more than 8 mg- - tha t is, there is an

infinite penalty for this situation to occur.

Figure 2 shows the above penalty function while the equations below specify its structure within the G P model:

0.09x I + 0 " 0 4 x 2 q- 0"2X 3 + 0"04X 4 + 0"04X 5

0"09x 1 + 0"04x 2 + 0"2X 3 + 0"04X 4 + 0"04X 5

0.09x~ + 0-04x 2 + 0"2X 3 -k- 0 " 0 4 X 4 q- 0"04X 5

+ 1.5x 6 > 1.5 (25)

-k- l ' 5 X 6 -+- r/1 - - P l - - P 2 = 4 (26)

-k- l ' 5 X 6 ~- n 2 - - P 2 = 6 (27)

0.09x 1 + 0-04x z + 0"2X 3 -k- 0"04X 4 + 0"04X 5 + 1 "5X 6 ~ 8 (28)

The contribution to the objective function of the W G P model or to the achievement function of the LGP model will be given by lpl + 5p 2 in terms of absolute deviations.

SOME C O N C L U D I N G R E M A R K S

The main methodological objections to the use of linear programming for generating livestock rations are its inability to tackle more than one 'ingredient choice' criterion and the rigidity with which it treats the nutrient requirements specified for a diet. This approach could generate either unimplementable rations or a nutritionally undesirable mix of ingredients. But these problems can be overcome if the ration formulation is postulated within the multiple-criteria decision-making framework by using techniques such as lexicographic goal-programming, weighted


goal-programming and multiple-objective programming. Not only that, the information generated by weighted goal-programming permits an exploration of the trade-offs that exist among different criteria such as cost, ration bulk and nutrient over-supply and how they influence the quality of the ration.

If the aims of a diet formulation exercise can be arranged into a priority structure, then the lexicographic approach explores the effects of rearranging these priorities, setting different nutrient requirement targets and attaching different weights to nutritional goals before the most desirable ration can be chosen. The multiple-objective programming presents the nutritionist with an array of nutritionally efficient solutions from which the most suitable one can be picked and implemented. Similarly, the ability to incorporate penalty functions for over- or under- achieving the nutritional specifications underlines the superiority of the goal-programming approach over simple least-cost diet formulation. If the nutritionist and/or his client can 'interact' with the model on a computer terminal within an advisory environment such as the one reported by Crabtree (1982) then the advantages that multiple-criteria decision techniques offer can be actually realised.

We have been mainly concerned to demonstrate the applicability of multiple-criteria techniques to livestock ration formulation problems using a simple example, but their use by nutritionists either in place or in tandem with linear programming requires research. Hopefully this paper provides the conceptual transition that is needed and can trigger work to test the application of multiple-criteria techniques in developing ration formulation models for more realistic situations.

REFERENCES

Anderson, A. M. & Earle, M. D. (1983). Diet planning in the third world by linear and goal programming. J. Opl. Res. Soc., 34, 9-16.

Arthur, J. L. & Ravindran, A. (1978). An efficient goal programming algorithm using constraint partitioning and variable elimination. Mgmt. Sci., 24, 867-8.

Arthur, J. L. & Lawrence, K. D. (1980). A multiple goal blending problem. Comput. & Ops. Res., 7, 215-24.

Arthur, J. L. & Ravindran, A. (1980). PAGP a partitioning algorithm for (linear) goal programming problems. ACM Trans. Math. SoJtware, 6, 378-86.

Chappell, A. E. (1974). Linear programming cuts cost in production of animal feeds. Opl. Res. Q., 25, 19-26.


Charnes, A. & Cooper, W. W. (1961). Management models and industrial applications of linear programming, Vol. 1, John Wiley and Sons, New York.

Chen, J. T. (1973). Quadratic programming for least cost feed formulations under probabilistic protein constraints. Amer. J. Agri. Econ., 55, 175-83.

Cohon, J. L. (1978). Multiobjective programming and planning. Academic Press, New York.

Crabtree, J. R. (1982). Interactive formulation system for cattle diets. Agri. Systems, 8, 291-308.

Dauer, J. P. & Krueger, R. J. (1977). An iterative approach to goal programming. Opl. Res. Q., 28, 671-81.

Dent, J. B. & Casey, H. (1967). Linear programming and animal nutrition. Crosby Lockwood, London.

Ecker, J. G., Hegner, N. S. & Kouada, I. A. (1980). Generating e.ll maximal efficient faces for multiple objective linear programs. J. Optimization Theory and Applications, 30, 353-81.

Geoffrion, A. M. (1968). Proper efficiency and the theory of vector maximization. J. Optimization Theory and Applications, 22, 618-30.

Goichochea, A., Hansen, D. R. & Duckstein, L. (1982). Multiobjectit, e decision analysis with engineering and business applications. John Wiley and Sons, New York.

Ignizio, J. P. (1976). Goal programming and extensions. Lexington Books2 Massachusetts.

Ignizio, J. P. (1982). Linear programming in single and multiple-objecti~'e systems. Prentice-Hall, New York.

Ignizio, J. P. & Perlis, J. H. (1979). Sequential linear goal programming: Implementation via MPSX. Comput & Ops. Res., 6, 141-5.

Ijiri, Y. (1965). Management goals and accounting Jor control. North Holland Publishing Company, Amsterdam.

Iserman, H. (1977). The enumeration of the set of all efficient solutions for a linear multiobjective program. Opl. Res. Q., 28, 711-25.

Keown, A. J. & Martin, J. D. (1977). A chance-constrained goal programming model for working capital management. Engineering Economist, 22, 152-174.

Kornbluth, J. S. H. (1973). A survey of goal programming. Omega, 1,193-205. Kornbluth, J. S. H. & Steuer, R. E. (1981). Goal programming with linear

fractional criteria. Eur. J. Opl. Res., 4, 421-5. Kvanli, A. H. (1980). Financial planning using goal programming. Omega, 8,

207-18. Lee, S. M. (1972). Goalprogrammingfor decision analysis. Auerbach Publishers,

Philadelphia. Masud, A. S. & Hwang, C. L. (1981). Interactive Sequential Goal Programming.

J. Opl. Res. Soc., 32, 391400. Mohr, G. M. (1972). The bulk constraint and computer formulations of least

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Multiple-criteria decision-making techniques and lit,estock ration Jormulation 49

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Multiple-Criteria Decision-Making Techniques and Their ... · dgricultural Systems 15 (1984) 23-49 Multiple-Criteria Decision-Making Techniques and Their Role in Livestock Ration