Chapter nine Livestock ration formulation and multiple criteria decision-making techniques

Chapter nine Livestock ration formulation and multiple criteria

decision-making techniques

One of the most successful areas of application for the conventional linear programming (LP) paradigm has been in the search for the least cost combination of foods that will meet a specified level of nutritional requirements for livestock. This procedure depends on the following fundamental assumptions: 1 there is a single objective (usually the minimisation of the cost of the diet) which is a mathematical function of the decision variables; 2 the decision variables of the problem are the amounts of the available ingredients that will constitute the diet; 3 the nutritional requirements are convertible to mathematical functions of the decision variables and they form the constraint set of the problem; and 4 the optimum diet is the one that minimises the single specified objective without any violation of the constraints imposed. Although LP has been used widely in practice with noticeable success, the above assumptions remain its weaknesses. In formulating rations, the decision maker is interested in an economically optimal ration that achieves a compromise between several conflicting objectives such as minimisation of cost, imbalance of nutrient supplies, food bulk, and so on. 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 really impose nutritional constraints that cannot be violated under any circumstances. On the contrary, a certain relaxation of these constraints imposed would not seriously affect an animal's physical and economic performance. We believe that these methodological weaknesses of LP can be overcome substantially, and its application to diet formulation can be improved considerably, if the problem is analysed within the MCDM framework. This chapter ~transfers the ration formulation problem from a context where the DM wishes to optimise a single objective to one where several objectives are to be optimised simultaneously. The goal programming (GP) and the multiobjective programming (MOP) techniques explained earlier are used. The problem of relaxing the rigidity of the nutritional requirements is dealt with in the next chapter.

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Livestock ration formulation and multiple criteria decision-making techniques

137

A livestock ration formulation example This section adapts a diet formulation problem (Crabtree 1982), which is used as a basis for discussion in the rest of the chapter. Crabtree's example has been chosen because it is well thought out from the animal nutritional standpoint and also because it seems to have many of the problems whose solution we wish to illustrate by using the MCDM techniques. Crabtree's LP model analyses the ration formulation of a 600 kg dairy cow with a milk yield of 30 kg/day, without any change in the liveweight of the animal. The ingredients that are available for making up the diet, along with their compositions, and nutritional requirements of the animal are given in Table 9.1. However, for our purposes we have relaxed the upper limits on total dry matter intake and the amount of swedes and distillers' wet grain in the ration somewhat, as well as the proportion of dry matter derived from silage and straw. Most of Table 9.1 is self-explanatory but the constraint (14) means that at least 20% 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% of the total dry matter. The least-cost diet for this problem is comprised of the following: Silage (x,) = 4.981 kg Distillers' wet grains (x3) = 3.416 kg Barley (Xs) = 2.778 kg

Straw (x2) = 0.0 kg Swedes (x4) = 2.0 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 in units of kg dry matter per day. Table 9.2 Nutrient requirements and ration composition using the least-cost criterion only Ingredients

Requirements

Ration

Metabolisable energy (MJ)

213.3

213.3

Rumen-degradable protein (g)

1663.9

2045

Undegradable protein (g)

620.5

620.5

Calcium (g)

73.6

91.6

Phosphorus (g)

67.8

89.8

Magnesium (g)

24

36.5

Sodium (g)

26.9

55.1

Copper (mg)

180

180

Cobalt (mg)

2

8.9

Dry matter (kg)

19

18.3

138

Multiple criteria analysis for agricultural decisions

This ration costs s and Table 9.2 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 s u p p l y - particularly magnesium, sodium and cobalt, which may be an undesirable nutritional imbalance in the diet. The situation could be improved by setting upper, as well as lower, limits for the level of supply of each of the nutrients. 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. The reduction of surplus supplies of various nutrients in least-cost diets ought to be one of the goals in formulating rations using mathematical programming techniques. Besides cost and nutritional imbalances, another important consideration 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 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, Crabtree's example can now be reformulated as a problem with three different objectives: (1) minimise the cost of the ration; (2) minimise the imbalanced supply of copper, sodium and cobalt, and (3) minimise the bulk (i.e. the net weight) of the diet. Obviously there is a situation of conflict in trying to achieve these objectives. The minimisation of cost is not compatible with minimising either the nutrient imbalances or the physical bulk of the ration. As conflicting objectives are being considered the problem cannot be solved through minimising each objective function subject just to the nutritional requirements. Crabtree's example with multiple goals will be solved in what follows via GP and MOP techniques to illustrate how these techniques can be used fruitfully in ration formulation problems.

Ration formulation as a WGP problem Let us now analyse the ration formulation problem with three objectives discussed in the preceding section and build it as a weighted goal programming (WGP) model. To do that we convert the three objectives into goals: that is, by fixing a target value for each objective and introducing corresponding negative and positive deviational variables and by using the dry matter content of the various ingredients from Table 9.3. Table 9.3 Dry matter content of ingredients (g/kg) Ingredient

Silage

Straw

Dry matter

270

820

Source: Crabtree (1982,p. 299)

Distillers' Swedes wet grains 258

108

Barley

Dairy compound

833

800

Livestock ration formulation and multiple criteria decision-making techniques

139

Thus, the three goals are:

Goalgl The ration should not have a bulk larger than 40 kg of fresh weight and the expression for goal g~ is given by: 3.704Xl + 1"219x2 + 3.876x3 + 9-259X4 + 1.200x5 + 1.250x6 + nl - Pl -- 40

(9.1)

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

Goalg2 The supply of sodium in the diet should not exceed 100% of its specified requirements. This imbalance goal, INa,is obtained from constraint (7) in Table 9.1. (9.2)

INa = 3.8x 1 + 1.1X2 + 0.9x3 + 2.6x 4 + 0.2x 5 + 5.4x 6 - 26.9 Treating equation (9.2) as percentages instead of absolute values, we have:

(9.3)

(3.8x~ + 1.1x2 + 0 . 9 x 3 + 2.6x 4 + 0.2x 5 + 5.4x 6 - 26.9).100 26.9 Therefore the expression for g2 is: 14.13x I + 4.09x 2 + 3.35x 3 + 9.67x 4 + 0.74x 5 + 20.07x 6 + n 2 - -

t9 2 --

200

(9.4)

To achieve the desired level of this goal P 2 must be minimised.

Goals g3 anti g4 Using the same procedure as for sodium, the goals for copper and cobalt are derived from constraints (8) and (9) in Table 9.1, so that their supplies, too, should not exceed 100% of the specified requirements. This is stated by the equations (9.5) and (9.6) below. 3.17x~ + 1.78x 2 + 5.55x 3 + 2.11x 4 + 2.67x 5 + 10.55x 6 + n 3 - - / ) 3

4.5x~ + 2 x 2 + 10x3 + 2x4 + 2x 5 + 75x 6 + n 4 - p 4 = 200 For achieving g3 and g4, P3 and P4 must be minimised.

=

200

(9.5)

(9.6)

140

Multiple criteria analysis for agricultural decisions

Goalg5 Lastly, the objective of minimising the cost can be converted into a goal by setting a target of s

as cost, which c o r r e s p o n d s to the m i n i m u m cost associated w i t h the ration

r e c o m m e n d e d by the ordinary LP approach. Therefore the expression for g5 using the unit costs of ingredients from Table 9.1, is: 0.0482x~ + 0.0268x 2 + 0.0698x 3 + 0.0926x4 + 0.1056x5 + 0.1625x6 + n 5 - P5 = 1.782

(9.7)

To achieve gs, P5 is minimised. The variables in the objective function have to be given in percentage deviations from the targets as minimisation of absolute deviations does not make sense w h e n each goal is measured in different units. Hence, the elements of the objective function are standardised for the W P G model to give: p, 100 192 100 w, 40 1 + W22 ~ ~ 1- ~ -

103

W3 200

100 P4 100 P5 100 1 -I- W, 200 1 "~- W5 1.782 1

(9.8)

or

2.5w~p~ + 0.50w2P 2 +

0 . 5 0 w 3 p 3 + 0.50w4p 4 -I 56.12wsp s

(9.9)

Where w~. . . . . w5 are the weights attached to the deviational variables representing the respective importance given to the achievement of the various goals. The structure of this WGP model is given in Table 9.4. Different solutions can be obtained by attaching different values to the w parameters. For instance, if w~ = w 2 = ... = w5 = 1, the simplex m e t h o d provides the following o p t i m u m solution: x~ (silage) = 3.511 kg

x 2 (straw) = 0.675 kg

x 3 (distillers' wet grain) = 3.80 kg

x 4 (swedes) = 0 x 6 (dairy c o m p o u n d ) = 4.716 kg

x s (barley) = 6.298 kg

While the o p t i m u m values of the deviational variables are: nI = 0

Pl = 1.477kg

n 2 = 35.59%

P2 = 0

n 3 = 100% I/4 " - 0

P3 = 0 /94 = 221.41%

n5 = 0

P5 = 0.102

Livestock ration formulation and multiple criteria decision-making techniques

141

Table 9.4 Weighted goal programming model for ration formulation Objective function Minimise: 2.5wlP 1 + 0.50w2P 2 + 0.50 w3P3+ 0.50 w4P4+ 56.12wsP 5 subject to 3.704x~ + 1.219x 2 + 3.876x 3 + 9.2594x 4 + 1.200x 5 + 1.250x 6 + n 1 -/91 -- 40 (bulk) 14"13Xl + 4"09x2 + 3"35x3 + 9"67x4 + 0-74x5 + 20"07x6 + n2 - / 9 2 - 200 (imbalance in sodium) 3.17x I + 1.78x 2 + 5.55x3 + 2.11x 4 + 2.67x 5 + 10.55x 6 + n 3 - / 9 3 = 200 (imbalance in copper) 4.5x I -I- 2x 2 + 10x 3 + 2x 4 + 2x 5 + 75x 6 + n4 - / 9 4 = 200 (imbalance in cobalt) 0"0482Xl + 0"0268x2 + 0"0698x3 + 0-09-26x4 + 0"1056x5 + 0"1625x6 + n5 - / 9 5 = 1.782 (cost), a n d x e F [technical constraints from Table 9.1 excluding sodium, copper and cobalt r e q u i r e m e n t restraints]

Table 9.5 Sets of weights used in the sensitivity analysis of the WGP solution Run

Weight w 1

Weight w 2

Weight w 3

Weight w4

for gl (bulk)

for g2

for g3

for g4

Weight w s for gs

(imbalance

(imbalance

(imbalance

(cost)

in sodium)

in sodium)

in s o d i u m )

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1

1

1

1

1

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2

2

2

1

3

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1

1

1

1

4

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1

1

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9

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This solution permits complete achievement of the goals g2 (sodium supply imbalance) and g3 (copper supply imbalance) by virtue ofthe fact that p~ = P2 = 0. The value of 1.477 kg for p~ implies that goal g~ 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 of 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 s over the set target of s 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 9.5 and corresponding solutions are presented in Table 9.6. From this information it is possible to obtain something like a measure of the trade-offs

142

Multiple criteria analysis for agricultural decisions

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 offwith a reduction in cost of s and for a drop in cobalt imbalance of 10.15%. 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 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 an LGP problem To illustrate the use of lexicographic goal programming (LGP) in ration formulation, we have to establish the priority structure of goals in our example. Let us assume, for example, the goals of obtaining a ration costing less than s and weighing below 50 kg are situated in the first priority Q1. These targets can be considered as minimum levels of achievement; that is, for the decision maker it is completely and absolutely necessary to use rations cheaper than s in order to obtain profits and with a bulk below 50 kg in order not to exceed the storage capacity. Using equations (9.7) and (9.1) and on modifying the subscripts of the deviational variables, the goals in priority Q! are: Cost: 0.0482x~ + 0.0268x2 + 0.0698x3 + 0.0926x4 + 0.1056xs + 0.1625x6 + n~ - p~ = 2 (9.10) Bulk: 3.704x~ + 1.219x2 + 3.876x3 + 9.259x4 + 1.200~ + 1.250x6 + n 2 -

P2

--

50 (9.11)

As regards weights attached to the achievement of the various goals, assume that formulating a ration costing less than s is twice as important as the attainment of required bulk. Now using the same procedure as for deriving the objective function of WGP the first component to be minimised in the lexicographic process is: 2 • 100 100 2 P' + ~ P2 = 100p, + 2p2 The next priority Q2, is made up of goals g2, g3 and g, described above: that is, the supplyrequirement imbalance for sodium, copper and cobalt should not be greater than 100%. This can be derived from equations (9.4), (9.7) and (9.8), with the appropriate subscripts for the deviational variables: thus the goals in priority Q2 are:

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Livestock ration formulation and multiple criteria decision-making techniques

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Multiple criteria analysis for agricultural decisions

Sodium: 14.13x~ + 4.09x 2 + 3.35x 3 + 9.67x 4 + 0.74x 5 + 20.07x 6 + rt3 - - / 9 3

=

200

(9.12)

Copper: 3.17x~ + 1.78x 2 + 5.55x 3 + 2.11x 4 + 2.67x 5 + 10.55x 6 + n 4 - P4 = 200

(9.13)

Cobalt: 4"5Xl

(9.14)

q- 2ah + 10X3 + 2X4 + 2X5 + 75X6 + n 5 - Ps = 200

If the three goals in Q2 are of equal importance, and as their targets and measurements are homogenous, then the second components to be minimised in the lexicographic process is given by P3 + P, + PsFinally, the last priority, 03, is made up of goals of finding a ration costing less than s with m i n i m u m bulk (which is 22.34 kg, see next section). It is interesting to note the similarity that appears to exist between goals in priorities Q~ and 03. In fact, in Q~ goals represent something like m i n i m u m levels of achievement while 03 represents desired levels of achievement. W h e n goals are specified at both a m i n i m u m and desired level in a preemptive way, some authors call this approach two-stage LGP (Keown and Martin 1977). The algebraic expression for the goals in Q3 is derived from equations (9.1) and (9.7) as given below: Cost: 0.0482x~ + 0.0268x 2 + 0.0698x 3 + 0.0926x 4 + 0.1056x s + 0.1625x 6 +

rt 6 - - / 9 6 ---- 1.782

(9.15) Bulk: 3.704x~ + 1.219x 2 + 3.876x 3 + 9.259x 4 + 1.200x 5 + 1.250x 6 + n 7 - - P 7 -

22.34 (9.16)

Once again, assuming that the cost goal is twice as important as the ration bulk, the last c o m p o n e n t of the lexicographic minimisation process is given by: 2 • 100 1.782

P6 +

100 22.3----~ t)7

~

112.2p6 + 4.47p7

Livestock ration formulation and multiple criteria decision-making techniques

145

The w h o l e lexicographic m i n i m i s a t i o n process w o u l d be given by the following achievement function: Minimise_a = [(100p, + 2P2), (P3 + P4 + P5), (112.2p, + 4.47P2)]

(9.17)

The structure of our LGP m o d e l is given in Table 9.7. Table 9.7 Lexicographic goal programming model for ration formulation Achievement function: Minimise a = [(100Pl + 2P2 ), (P3 + P4 + P5), (112-2pl + 4"47/'2)] subject to: QI" 0-0482xl + 0-0268x2 + 0.0698x3 + 0-0926x4 + 0.1056x5 + 0-1625x6 + nl - P~ = 2 (cost) 3.704x~ + 1.219x 2 + 3.876x 3 + 9.2594x 4 + 1.200x 5 + 1.250x 6 + n 2 - P2 = 50 (bulk) Q2: 14.13x~ + 4.09x 2 + 3.35x 3 + 9.67x 4 + 0.74x 5 + 20.07x 6 + n 3 - P3 = 200 (imbalance in sodium) 3.17x~ + 1.78x 2 + 5.55x 3 + 2.11x 4 + 2.67x 5 + 10.55x 6 + n 4 - / ' 4 = 200 (imbalance in copper)

4.5Xl + 2x2 + 10x3 + 2x4 + 2x5 + 75x6 + n 5 - P5 = 200 (imbalance in cobalt) Q3: 0"0482Xl + 0-0268x2 + + 0.0698x3 + 0"0926x4 + 0-1056x5 + 0-1625x6 + n 6 - t96-- 1.782 (cost) i.e. cost 3"704Xl + 1-219x2 + 3-876x3 + 9"2594x4 + 1-200x5 + 1-250x6 + n7 - P7 = 22.34 i.e. b u l k a n d x e F [technical constraints from Table 9.1 excluding sodium, copper and cobalt requirement restraints]

On using the sequential linear m e t h o d (see Chapter 3), we obtain the following o p t i m u m solution: x 1 (silage) = 5.145 kg x 3 (distillers' wet grain) = 3.80 kg x5 (barley) = 5.519 kg

x 2 (straw) -- 0 x 4 (swedes) - 0 x 6 (dairy c o m p o u n d ) = 4.536 kg

While the o p t i m u m values of the deviational variables are: n1= s n 2 = 4.434 kg n 3 = 19.458% n4 =

100%

ns=O n6=O nv=O

pl--0

/,2=0 /,3=0 /,4=0 /'5 = 212.384% P6 = 0.051 /'7 = 23.227 kg

146

Multiple criteria analysis for agricultural decisions

This solution permits complete achievement of the goals in Q,; that is, the m i n i m u m 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 m i n i m u m bulk goals have positive deviations of s and 23.277 kg respectively; that is, their actual values are s and 45.517 kg.

Ration formulation as a MOP problem To demonstrate the use of multiobjective p r o g r a m m i n g (MOP) in ration formulation, assume that in Crabtree's problem the three objectives to be considered are: (1) minimise the cost; (2) minimise the bulk and (3) minimise the aggregated over-supply in sodium, copper and cobalt. The formal structure of the multiobjective model is thus given by:

g f f g (x) = [z, (x), z~ (x), z~ (x)]

(9.18)

where z I (__x)= 0.0482x1 + 0.0268x 2 + 0.0698x 3 + 0.0926x 4 + 0.1056x s + 0.1625x 6 (cost (s z2 (x) = 3.704x, + 1.219x 2 + 3.876x 3 + 9.259x 4 + 1.200x s + 1.250x 6 (bulk (kg)) z3 (__x) = 21.80x~ + 7.87x z + 18.09x 3 + 13.78x 4 + 5.41x 5 + 105.62x 6 - 300 (imbalances in sodium, copper and cobalt) subject to x e F [technical constraints from Table 9.1 excluding sodium, copper and cobalt requirement restraints] The expression for Z3(_x) has been derived from the addition of the imbalances of the three nutrients as defined above. For generating the efficient set we resort to the weighting method presented in Chapter 4. Applying this method six extreme efficient rations were generated. These rations are shown in Table 9.8. It is interesting to note that solution 1 corresponds to the m i n i m u m cost ration, solution 3 to m i n i m u m imbalance and solution 4 to the m i n i m u m bulk ration. In other words, these three solutions establish the ideal or utopian point, which for our problem is: [s 382.399%; 22.334 kg].

Livestock ration formulation and multiple criteria decision-making techniques

147

Table 9.8 Multiobjective programming for ration formulations (efficient points) Efficient solution

Silage x1 (kg)

Straw x2 (kg)

Distillers' wet grain x 3 (kg)

Swedes x4 (kg)

Barley x5 (kg)

Dairy Cost (s Compound X6 (kg)

Bulk (kg)

Oversupply

1

4.981

0

3.343

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

3.8

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

3.8

0

6.665

4.736

1.911

40.959

384.006

6

2.855

0.945

3.8

0

6.544

4.856

1.908

39.830

389.823

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 between cost, nutrient supply-requirement imbalance and bulk. For instance, if solution 1 (least cost ration) is preferred to solution 2, in that situation a reduction of 15.370 kg in bulk and 33.234% in imbalance does not compensate for an increase of s in cost. Alternatively, if the DM is not clear about the actual trade-offs then he could follow either the CP methods of Chapter 5 or the interactive approach as discussed in Chapter 6. This chapter has demonstrated that the problem of diet formulation can be represented by GP and MOP far more realistically than is the case when ordinary LP is used. However, the problem that the specified nutrient requirements for a given animal ought to be met in a rigid and inflexible way still remains. However, we believe that a variation of GP incorporating penalty functions provides a better framework of analysis, as shown in the next chapter.

Note 1 This chapter is a slightly modified version of a paper by Rehman and Romero 1984.