A feasible solution x ∈ S is an efficient solution if and only if it does not exist any other feasible solution x ¯ ∈ S such that 1) z i ( x ¯ ) ≤ z i ( x ) for i = 1 , 2 , and 2) z j ( x ¯ ) < z j ( x ) for at least one j ∈ { 1,2 } . The set of all efficient solutions is called the efficiency set noted E , also called Pareto set. The corresponding set in the criterion space is the set E c = C E .

Under the assumption that the two cost vectors c 1 and c 2 are linearly independant, Using weighted-sums, we can replace the bicriteria linear programming problem by a single criterion linear programming problem. We consider λ ∈ [ 0,1 ] and the weighted-sum function is

z ( x ; λ ) = ( 1 − λ ) z 1 ( x ) + λ z 2 ( x ) = [ ( 1 − λ ) c 1 + λ c 2 ] x ,

and we consider the single criteria problem for λ ∈ [ 0,1 ]

( P ( λ ) )   { m i n   z ( x ; λ ) = ( 1 − λ ) z 1 ( x ) + λ z 2 ( x ) = [ ( 1 − λ ) c 1 + λ c 2 ] x   subject   to   x ∈ S .

The value function φ ( λ ) of ( P ( λ ) ) is defined by

φ ( λ ) = m i n { z ( x ; λ ) | x ∈ S } .

From [11] we have

E = ∪ λ ∈ ( 0,1 ) arg min x ∈ S z ( x ; λ ) .

Hence the efficiency set E in the decision space is a connected set and is the union of faces, edges and vertices of S . This set may be quite complex due to the high dimension of the decision space. On the other side E c , which is the image in ℝ 2 of E by a linear transform, is a much simpler set.

Since we have assumed that both criteria are lower bounded on S , it follows that E c is a simple compact polygonal line. Indeed in that case E c is the union of a finite number L of segments [ Q l − 1 , Q l ]

E c = ∪ l = 1 L [ Q l − 1 , Q l ]

where

[ Q l − 1 , Q l ] = { Q ∈ ℝ 2 | Q = ( 1 − σ ) Q l − 1 + σ Q l     for     σ ∈ [ 0,1 ] } ,

and such that

( Q l − 1 , Q l ) ∩ ( Q l ˜ − 1 , Q l ˜ ) = ∅     if     l ≠ l ˜ ,

with

( Q l − 1 , Q l ) = { Q ∈ ℝ 2 | Q = ( 1 − σ ) Q l − 1 + σ Q l     for     σ ∈ ( 0 , 1 ) } .

To each segment is associated a weight λ l − 1, l such that the vector ( 1 − λ l − 1, l , λ l − 1, l ) t is orthogonal to the segment [ Q l − 1 , Q l ] in ℝ 2 . To each point Q of E c is associated an interval Λ ( Q ) defined by

Λ ( Q ) = { [ λ _ l , λ ¯ l ]                   if   Q = Q l     ( l = 0 , ⋯ , L ) , [ λ l − 1 , l , λ l − 1 , l ]       if   Q ∈ ( Q l − 1 , Q l )     ( l = 1 , ⋯ , L ) ,

where

{ λ _ 0 = 0 , λ ¯ l − 1 = λ _ l = λ l − 1 , l     for     l = 1 , ⋯ , L , λ ¯ L = 1 ,  

with λ ¯ l − λ _ l > 0 for l = 0 , ⋯ , L .

We will call weak efficiency set, or weak Pareto set, the set defined by

E f = ∪ λ ∈ [ 0,1 ] arg min x ∈ S z ( x ; λ ) .

Obviously E ⊆ E f . In the criteria space we will have E c f = C E f . Geometrically in the criterion space ℝ 2 , this means we add to E c possibly a vertical segment or a ray from Q 0 in the positive direction of z 2 , D 0 = ( 0 , 1 ) ,

R ( Q 0 ; D 0 ) = { Q 0 + η D 0 | η ∈ ( 0 , η 0 ] } ⊂ S c ,

and/or a horizontal segment or a ray from Q L in the positive direction of z 1 , D L = ( 1 , 0 ) ,

R ( Q L ; D L ) = { Q L + η D L | η ∈ ( 0 , η L ] } ⊂ S c ,

where η 0 and η L are nonnegative finite or infinite values. They are the maximal values of η such that R ( Q 0 ; D 0 ) and R ( Q L ; D L ) are both subsets of S c . To these points on E c we set

Λ ( Q ) = { [ 0,0 ]       if   Q ∈ R ( Q 0 ; D 0 ) , [ 1,1 ]           if   Q ∈ R ( Q L ; D L ) .

The parametric analysis is based on the weighted-sum given by

z ˜ ( x ; μ ) = z 1 ( x ) + μ z 2 (x)

for μ ∈ [ 0, + ∞ ) , and the value function in this case is defined by

φ ˜ ( μ ) = m i n { z ˜ ( x ; μ ) | x ∈ S } .

Instead of ( P ( λ ) ) , we could consider the single criteria problem for μ ≥ 0

( P ( μ ) )   { min z ˜ ( x ; μ ) = z 1 ( x ) + μ z 2 ( x ) = ( c 1 + μ c 2 ) x   subject   to   x ∈ S .

Since λ and μ are related by the formulae

λ = μ 1 + μ     and     μ = λ 1 − λ ,

to the efficient extreme points { Q l } l = 0 L on the efficiency set E c correspond also the following intervals for the parameter μ

Λ ˜ ( Q ) = { [ μ _ l , μ ¯ l ]                     if   Q = Q l     ( l = 0 , ⋯ , L ) , [ μ l − 1 , l , μ l − 1 , l ]           if   Q ∈ ( Q l − 1 , Q l )     ( l = 1 , ⋯ , L ) ,

where

{ μ _ 0 = 0 , μ ¯ l − 1 = μ _ l = μ l − 1 , l       for   l = 1 , ⋯ , L ,   μ ¯ L = + ∞ .

In many applications, the parameter μ is in fact a tax over the the second criteria (for a minimization problem). Interesting enough is to observe that the behavior change (extreme point) only for the critical values μ l − 1, l of the parameter μ . Indeed when μ increases and its value passes through μ l − 1, l , the optimal point, extreme point, move from Q l − 1 to Q l . Thus, any level of taxes μ strictly between the values μ l − 1 , l = μ _ l and μ l , l + 1 = μ ¯ l causes the same behavior described by Q l .

Let us associate to any Q = ( z 1 , z 2 ) ∈ S c the weighted-sum function given by

φ Q ( λ ) = ( 1 − λ ) z 1 + λ z 2 .

Then the value function φ ( λ ) associated to ( P ( λ ) ) is such that

φ ( λ ) = min { φ Q ( λ ) | Q ∈ S c } = min { φ Q ( λ ) | Q ∈ E c } = min { φ Q l ( λ ) | l = 0 , ⋯ , L } .

Hence we have the following results.

Theorem 4.1. [12] Let Q ∈ E c , we have φ ( λ ) = φ Q ( λ ) if and only if λ ∈ Λ ( Q ) .

Theorem 4.2. [12] Let Q ∈ E c and 0 ≤ λ 1 < λ 2 ≤ 1 . Then λ 1 and λ 2 ∈ Λ ( Q ) if and only if [ λ 1 , λ 2 ] ⊆ Λ ( Q ) . It follows that Q is one of the Q l ( l ∈ 0, ⋯ , L ).

Theorem 4.3. [17] The function φ ( λ ) is continuous, piecewise linear and concave. The abscissae of slope changes are the increasing values λ l − 1, l for l = 1 , ⋯ , L .

Let us observe that the slope associated to φ Q ( λ ) strictly decreases for Q going from Q 0 to Q L on E c , since z 1 increases and z 2 decreases steadily. We deduce the next results.

Theorem 4.4. [12] Let Q ′ i and Q ′ j be two distinct points on E c . For λ ∈ [ 0,1 ] , ( 1 − λ , λ ) t is orthogonal to the segment [ Q ′ i , Q ′ j ] if and only if φ Q ′ i ( λ ) = φ Q ′ j ( λ ) .

Theorem 4.5. Let Q ′ i and Q ′ j be two distinct points on E c and λ ∈ [ 0,1 ] , such that ( 1 − λ , λ ) t is orthogonal to the segment [ Q ′ i , Q ′ j ] . For a fixed λ , the function φ Q ( λ ) is constant as a function of Q on the segment [ Q ′ i , Q ′ j ] . Let us note this constant value by φ Q ′ i Q ′ j . Moreover

1) if φ ( λ ) = φ Q ′ i Q ′ j then [ Q ′ i , Q ′ j ] ⊂ E c , λ ∈ Λ ( Q ′ i ) and λ ∈ Λ ( Q ′ j ) ;

2) if φ ( λ ) > φ Q ′ i Q ′ j then ( Q ′ i , Q ′ j ) ∩ E c = ∅ .

Theorem 4.6. Let Q ′ i and Q ′ j be two distinct points on E c . If λ ∈ Λ ( Q ′ i ) is such that φ ( λ ) = φ Q ′ j ( λ ) then λ ∈ Λ ( Q ′ j ) . Moreover there exists l ∈ { 0 , ⋯ , L } such that λ = λ l − 1 , l and [ Q ′ i , Q ′ j ] ⊆ [ Q l − 1 , Q l ] ⊆ E c .

Theorem 4.7. Let Q ′ i and Q ′ j be two distinct points on E c f . Let λ ′ i ∈ Λ ( Q ′ i ) and λ ′ j ∈ Λ ( Q ′ j ) , and consider the following two lines

L i ( λ ′ i ) = { Q ∈ ℝ 2 | φ Q ( λ ′ i ) = φ ( λ ′ i ) }

and

L j ( λ ′ j ) = { Q ∈ ℝ 2 | φ Q ( λ ′ j ) = φ ( λ ′ j ) } .

(A) If λ ′ i ≠ λ ′ j , the point of intersection of L i ( λ ′ i ) and L j ( λ ′ j ) is Q ˜ ( λ ′ i , λ ′ j ) = ( ψ ( 0 ; λ ′ i , λ ′ j ) , ψ ( 1 ; λ ′ i , λ ′ j ) ) where

ψ ( λ ; λ ′ i , λ ′ j ) = λ ′ j − λ λ ′ j − λ ′ i φ ( λ ′ i ) + λ − λ ′ i λ ′ j − λ ′ i φ ( λ ′ j ) ,

so

ψ ( 0 ; λ ′ i , λ ′ j ) = λ ′ j φ ( λ ′ i ) λ ′ j − λ ′ i − λ ′ i φ ( λ ′ j ) λ ′ j − λ ′ i

and

ψ ( 1 ; λ ′ i , λ ′ j ) = λ ′ j − 1 λ j − λ ′ i φ ( λ ′ i ) + 1 − λ ′ i λ ′ j − λ ′ i φ ( λ ′ j ) .

(B) If λ ′ i = λ ′ j , then L i ( λ ′ i ) = L j ( λ ′ j ) which contains the segment [ Q ′ i , Q ′ j ] .

In this section we consider both criteria upper bounded on S . In the forthcoming algorithm we initialize the process with the two points Q 0 and Q L on E c . Then we gradually obtain a sequence of points { Q ′ i } i = 0 I on E c , and

a sequence of intervals associated to these points { Λ ′ ( Q ′ i ) = [ λ _ ′ i , λ ¯ ′ i ] } i = 0 I such that Λ ′ ( Q ′ i ) ⊆ Λ ( Q ′ i ) and

φ ( λ ) = φ Q ′ i ( λ )       for   all     λ ∈ Λ ′ ( Q ′ i ) .

At the end of the process I = L and we have Q ′ l = Q l with

[ λ _ ′ l , λ ¯ ′ l ] = Λ ′ ( Q ′ l ) = Λ ( Q l ) = [ λ _ l , λ ¯ l ]

for.

Algorithm (Pareto bicriteria)

STEP 0. Initialization.

(A) Enter the data of the problem.

(B) Determine for and set. For and set. We get the initial point which as the same first coordinate as, and which as the same second coordinate as. Those two points might not be on, but are on.

(C) Set and;

(D) Set and;

(E) Set.

STEP 1. As long that there exists an index i such that, select one such index i and do:

(A) Find such that, hence such that is orthogonal to the segment (see Theorem 4.4);

(B) Solve, compute with;

(C) Update the list of points and their intervals :

I) Modification of the intervals. If then all the segment is in (see Theorem 4.5), and is defined on by (see Theorems 4.1 and 4.5)

We modify as follow:

a) for:;

b) for:;

In the sequel no more point will be generated on.

II) Point insertion and interval modification. If then, insert the point and modify intervals as follows (see Theorem 4.6):

a) Insert between and in the list with ;

b) Set;

c) If, then and any is in, hence we modify by setting;

d) If, then and any is in, hence we modify by setting.

STEP 2. For any i such that, remove from the list and set.

STEP 3. End of the process (and). The output is the list.

Let us observe that this process use only optimal solutions of, optimal values of the decision variables, which is easily obtained from any elementary linear program solver.

Remark 4.8. This algorithm produces at each iteration an inner and an outer approximation. The inner approximation is the polygonal line joining the for. The outer approximation is the polygonal line joining the points, , , , , , , , , , , as long as the’s are well determined (see Theorem 4.7). At the end of the algorithm the two approximations agree.

In this section we are going to determine the maximum number of calls to a linear program solver to completely determine the Pareto set, or equivalently its efficient extreme points. The result is given in the last theorem of this section and says that it takes at most calls to a linear program solver to generates the extreme points.

We will use the following ordering on. For any two distinct points and on, we will say that precedes on, or equivalently that follows on, if moving from on in the direction from to we move from to. We will note or equivalently.

Theorem 4.9. The algorithm generates at most 3 points on on and two of these points are and.

Proof. Let us remark that the algorithm will eventually find a point in for any. Let be the first point generated by the algorithm in. This first point can be generated at STEP 0, an initial point, if for or for. Otherwise, it is generated through STEP 1-C-II, with and. Then this point is included in the list, and there are three cases to study:

1) for a and we have with. We will have, or if the lower bound is modified through STEP 1-C-II-c (if and).

2) for a and we have with. We will have, or if the upper bound is modified through STEP 1-C-II-d (if and).

3) for and we have.

Let be the second point generated by the algorithm in. must be one of the two points used to generate, and hence. This point is generated through STEP 1-C-II, and it is included in the list. There are two cases to study:

1) and, we will have and . Consequently and

with modified as in the preceding case. Moreover if we will modify the upper bound to get.

2) and, we will have and

. Consequently and with modified as in the preceding case. Moreover if we will modify the lower bound to get.

Two points of are now in the list. We can have a point with or an extreme point, with, or, with. Otherwise the two points are the extreme points and. In that case, if and, it can happen that and we will have terminated with the interval. Otherwise let us note the third point generated in. There are two cases to study:

1) We have only one extreme point, or, of the segment in the list and. As in the preceding paragraph, we will introduce it in the list, and depending of the case, by passing through STEP 1-C-II, if and, or if and. Moreover, we will have respectively or .

2) We already have two extreme points and in the list. In that case and and we will have and. We pass through STEP 1-C-I and we modify the intervals to get et. Since , is not added to the list.

In the sequel, the algorithm generate no more point on because if we have two points and we have, or else, if we have three points, and and we have and.

Theorem 4.10. If, respectively, then, respectively, is eventually removed of the list without any supplementary call to the linear program solver.

Proof. When is introduced in the list, there is no supplementary call for. Similarly for the interval when is introduced in the list. The points and are removed from the list at STEP 2 since and.

Theorem 4.11. The algorithm generates the extreme points of the Pareto set in at most calls to a linear program solver.

Proof. The initialization STEP 0 requires 2 calls. For STEP 1, as we generate the for and possibly one supplementary call for each segment for, there is at most calls. Hence the algorithm requires at most calls.

The least cost diet problem, introduced in [18] , is a classical linear programming problem [19] [20] [21] . A decision variable is assigned to each ingredient and represents the amount (in kg) of the j^th ingredient per unit weight (1 kg) of the feed. Together, they form the decision vector in our model. The model's objective function is the diet cost. A vector of unit costs is used, where each represents the unit cost of the j^th ingredient (euro/kg or $/kg). Thus the total cost of a unit of weight (1 kg) of diet is which must be minimized over the set of feasible diets denoted by. The classic least cost animal diet formulation model is:

The constraints impose some bounds on the quantity of the different ingredients in the diet. For example a unit of feed is produced (a 1 kg mix), expressed by the constraint. Some ingredients, or combinations of ingredients, can be imposed on the diet. These restrictions give rise to equality constraints (=) or inequality constraints (≥ or ≤). More specifically, to satisfy protein requirements, the following constraints are introduced for the L groups of amino acids contained in the ingredients. We set

where represents the amount of digestible amino acid l contained in a unit of ingredient j and is the minimum amount of digestible amino acid l required. Finally, the diet must satisfy the digestible phosphorus requirements given by

where is the amount of digestible phosphorus contained in a unit of ingredient j.

Nitrogen and phosphorus excretions are directly related to the excess of amounts of protein (amino acids) and phosphorus in the diet. Hence, we have to establish the protein and the phosphorus contents of the diet and take into account the parts that are actually assimilated.

The protein content of a diet is, where is the amount of protein per unit of ingredient j. The total excretion of protein is then given by the amount in protein of the diet from which we remove the amount of protein effectively digested given by, then

Hence decreasing the total excretion is equivalent to decrease the protein content of the diet while maintained fixed the needs in protein.

As for the nitrogen, the amount of phosphorus of a unit weight diet is, where is the amount of phosphorus per unit of ingredient j. The amount is the the amount of phosphorus which is actually digested. In this way the phosphorus excretion is given by the phosphorus content of the diet from which we remove the amount of phosphorus which is actually digested

Hence, decreasing the phosphorus excretion is equivalent to decreasing the phosphorus content of the diet while maintained fixed the needs in phosphorus.

The ingredients and their corresponding variables are described in Table 1. Table 2 contains the entire model together with the values of the technical coefficients of the model.

The algorithm was programmed in MATLAB, which includes in its standard library the linear program solver called Linprog. This software can use the simplex method or an interior point method.

At first we analyse the relation between the cost of the diet and the two different excretions (nitrogen and phosphorus). As a curiosity, we also consider the interactions between the two kind of excretions: nitrogen and phosphorus.