Methods of Least cost dairy feed formulation

Least cost feed formulation for dairy cows

Obtaining animal products at desired quality and amounts from farm animals is only possible with a balanced and sufficient diet. The costs of feed and feeding vary from 50 to 80% in the operating costs depending in the operating area in animal production (beef, dairy, egg). It can be thus understood that decreasing the costs of feed and feeding is the most important factor in achieving quality as well as economical production.

A least cost ration incorporates all the available feedstuffs having good nutritive value and being available at a reasonable low cost. It can also be defined as an economic ration for animal production (dairy, beef, sheep, goats, poultry etc.) that provides nutrients in balanced proportion with lowest possible cost per kg or 100 kg.

It is the ration containing all essential nutrients that are needed to meet the requirements of the animal (growth, maintenance, production, reproduction, work etc.) without affecting quality and with least cost.

Least cost ration formulation using linear programming (LP) is an ideal approach to optimize cost of feed without compromising on the quality of feed using local feeds.

Methods of Least cost feed formulation
Methods of Least cost feed formulation

Advancements in computer technology, mathematical modeling and animal nutrition have resulted in vast improvements in the last 20-25 years. Computers have provided affordable feed formulation solutions to the feed industry. Many entrepreneurs have come out with different feed formulation software for commercial use often limited by high costs. In this paper we have presented a method for formulating least cost ration using Solver in MS-Excel. MS-Excel comes bundled with MS-Office and hence no extra cost is involved by the use of this method. It is also user friendly as minimum computer skills are required.


  • List all the available feeds, fodders and other available ingredients.
  • Enlist the components of each ingredient
  • Feed the computer with the cost of all available feed ingredients.
  • Give instructions to the computer for the type of ration desired depending upon the requirements of animal (growth, maintenance, production, reproduction, work or starter, grower, layer etc; high energy, high protein, low energy, low protein etc.)
  • Give instructions to the computer regarding the amount of feed ingredients (for example say DM of 20 kg or DCP of 1.5 kg). Similarly, amount of certain feed ingredients in the ration can be fixed like fish meal (say 10 %) and mineral mixture (say 2 %).
  • Now, the computer will take the least cost feed ingredients for formulating least cost ration.
  • It is a linear program based model that includes the following stepwise approach: i e. Proximate values (DM and nutritive value i.e. CP / DCP and TDN / ME).

Therefore, preparation of economical rations that address the needs of animals is important to achieve sustainable and profitable animal production, and it will be useful for the technical staff and formulators of farms and feed factories to get assistance from specialists and attend trainings to prepare rations and combo feeds with the lowest costs.

A balanced and sufficient diet can only be achieved by providing the animals with the whole nutrients in the feed that can be voluntarily consumed by the animals.

Building on this, we need to define and understand certain terms used in preparation of rations.

Feed: These are materials that are orally consumed by animals, which contain at least one of the nutrients needed by the animals and which do not cause any harmful effects when given at certain limits.

Ration: It refers to all feeds given to animals within one day (24 hours).

Balanced ration: A balanced ration is achieved when the feeds consumed by animals during the day meet all needs of the animals.

Unbalanced ration: It refers to intake of less or more than the required amount of nutrients in the feed consumed by animals during the day.

Voluntary feed consumption: It refers to the amount of feed that the animals can consume in one day (24 hours).

When preparing a ration, one should be familiar with the daily feed consumption and required nutrients for the relevant farm animals, accurate and precise analyses of the nutrients used in feed raw materials in the ration, the restricting specifications of the raw materials in the ration for the relevant farm animal, if any, and the raw material costs.

It is related to a healthy determination of the amount of feed that can be consumed by the animal group which will be rationed in a day. The basic starting point of preparing a ration is the feed consumption.

For example, assume that two different people are preparing ration; and one has estimated that the dairy cattle group to be rationed has a daily dry matter consumption of 22 kg/day, and the other has estimated it as 18 kg/day.

Also assume that according to the ambient conditions, the milk yield, lactation period, age, condition, milk composition, the ME requirement for the group is 50 Mcal/day, and raw protein requirement is 3400 g/day.

Thus, the person who estimated a need of 22 kg DM should ensure;
(50 Mcal ME/day) / (22 kg KM/day) = 2.27 Mcal/kg DM,
(3400 g HP/day) / (22 kg KM/day) = 154.5 g HP/kg DM,

The person who estimated a need of 18 kg DM/day should ensure;
(50 Mcal ME/day) / (18 kg KM/day) = 2.78 Mcal/kg DM,
(3400 g HP/day) / (18 kg KM/day) = 189 g HP/kg DM,

When these rations are given, the feed consumption of the group will vary. The animal will try to consume as much as possible of the feed with low content to the limits of its stomach capacity, but it will consume less of the feed with higher content or will be exposed to various risks such as acidosis.

Assuming that the actual average consumption is 20 kg DM/day, the person who prepares a ration on the basis of consumption of 22 kg DM will provide 45.5 Mcal ME/day to the animals, while the person who prepares a ration on the basis of consumption of 18 kg DM will provide 55.5 Mcal ME/day to the animals.

Similarly, the procurement of daily raw proteins for these two rations will be 3091 g HP/day and 3778 g HP/day, respectively. Here one can understand the importance of determination and estimation of actual DM consumption.

Nutritional models (e.g. NRC, INRA, CNCP) estimate the consumption of feed (dry matter) for animal groups which are to be rationed. However, feed consumption is shaped by various factors.

These factors include the quality, content and form of the raw materials used in the ration, nutrient contents and particle size of the ration and other factors related to the feed, herd management and environment, as well as the health condition, yield and physiological condition of the animals.

Besides feed (DM) consumption, the needs of the animal group to be rationed for daily nutrients should be accurately determined. There are different nutritional models to do this.

The most important models include NRC, INRA, CNCP, NORDIC, and CSIRO. These models strive to develop estimations that comply with actual needs with updates in developments to identify needs in the international literature.

For example, NRC published and updated the estimation of needs of dairy cattle in 1989 and 2001. Again NRC published and updated the needs of beef cattle for nutrients in 1984, 1996, 2000 and 2016. Again INRA published and updated the models estimating the needs of ruminants for nutrients in 1989 and 2007.

In poultry, NRC 1994 is still applicable. The estimates of these models can be taken as a starting point. In general the approach of these models to the needs are quite similar.

There are not any major differences. For example, there are certain differences in modeling of rumen fermentation for ruminants.

This results from a different treatment of carbohydrate and protein fractions in feeds and different views of efficiency in their use.

The nutrient content of the feeds should be clearly determined. Particularly the dry matter content of rough feed vary significantly, and should be closely followed.

This plays the most important role in preparing a balanced and/or unbalanced ration, ensuring a healthy intake of DM and provision of nutrients for animals.

Besides, the content for other basic nutrients of feeds should be determined in an up-to-date and healthy way.

This allows a clearer knowledge of the daily nutrient amounts given to animals and minimizes the potential imbalances.

The restricting specifications of raw materials to be used in rations and combo feeds for the animal to be rationed should be known, and these specifications should be taken into account when preparing a ration.

For example, urea cannot be used in rations for poultry, and if sufficient amount of easily destructible carbohydrates (starch, sugar) is used in rations for ruminants, it may be used in combo feed at a rate of 1.5% or not more than 25% of total proteins.

Again, husk grains should be used carefully in the rations for poultry, and when used, use of enzymes should be taken into consideration. In other words, these issues should be considered as restrictions when preparing a formulation.

The current prices of feed raw materials should be taken into account to create economical rations. The cost is an important factor in selection of feeds to be included in formulations of rations with lowest costs.

Furthermore, although ration can be handled manually, the low-cost raw materials should be used with due consideration of the overall ration balance, and the yield, physiological condition of the animal and the general characteristics of the feed.

The ration preparation methods in practice of ration preparation include trial and error, Pearson square method, algebraic method, and linear and non-linear programming methods.

With the basic information above, rations can be prepared using different approaches.

Trial and Error Method
The feed raw materials to be included in the ration can be balanced with the nutrient amounts needed by the animal or the nutrient concentrations in the ration using the trial and error method.

The success here depends on the experience of the person preparing the ration.

Furthermore, the higher the number of factors (restrictions) to be considered (costs, energy, protein, RYP, Ca, P, Vitamin A, vitamin D, Cu, Zn, etc.), the more difficult and time-consuming the preparation of ration.

Cost efficiency of the prepared ration will always be questionable. Furthermore, if solution is done manually, the trial and error method will often result in calculation errors since the process involves numerous mathematical calculations.

Excel or other software may be used to avoid this. In such applications, the calculations are set once, and then the calculations are made automatically by changing the amounts of feed to avoid mistakes in calculation.

Pearson Square Method

It is a simple method that has been developed to obtain mixtures with desired content by mixing two materials with known contents.

With this method, two feed raw materials with different nutrient contents can be mixed or desired nutrient content can be achieved for two mixtures.

In the Pearson square method, the desired nutrient level is placed in the middle of the square, and the nutrient content of the first feed raw materials is placed in upper left, and the nutrient content of the second raw material is placed in lower left.

The difference between the nutrient content of the feed raw materials and the desired nutrient level in the mixture is determined, and it is placed in the diagonal corner as positive, regardless of its mark.

The sum of differences is calculated and the rate of each difference in total difference is identified to determine the rate of mixture. Pearson square method can be applied if the desired level of nutrients in the mixture is between the same nutrient content of two raw materials.

Example 1: We want to prepare an mixture containing 20% RP, using PTK containing 36% RP and corn containing 9% RP. The illustration and solution in Pearson square is as follows:

Balanced mixtures with multiple nutrients can be prepared using the Double Pearson Square. Here, for example, if the aim is to balance the metabolic energy (ME) as well as the raw protein (RP), firstly the RP levels of two mixtures should be equated; however, their ME contents should be as distinct as possible.

Then the two mixtures are solved in the Pearson square so as to equal them to a certain energy level, thus evening out the RP and ME level. However, balancing more contents may not be possible with this method, since it will be more difficult to deal with confusing calculations.

Algebraic Method

Basically, Pearson square method has been derived from this approach. In the existing method, the equations to be used as restrictions in the solution are written as equations with variables and solved with proper mathematical methods.

These equations can be written down as a matrix, and solved with matrix operations. However, there may be many issues in solution and it may require complex calculations.

The simplest way of this is creating an equation with two variables. When the number of variables and restrictions is more than two, mathematical operations become complex.

We can prepare the same mixture in the Pearson square example using this method as follows:
Corn + PTK = 100 kg (restriction of amount)
0.09 Corn + 0.36 PTK = 20 kg (restriction of RP)

If we multiply both sides of the first equation by 0.09, and change their mark by multiplying them with -1, the equations and sums will be as follows:
-0.09 Corn – 0.09 PTK = -9
0.09 Corn + 0.36 PTK = 20
Total = 0.27 PTK = 11
PTK = 11 / 0.27 = 40.74 kg
Corn = 100 – 44 = 59.26 kg
As can be seen in the example, the mixture rates are the same as in the Pearson square method.

Preparing a Ration Using the Linear

Programming Technique

Linear programming is an optimization tool used in efficient use of rare sources. In linear programming the purpose and the conditions that can be realized with this purpose should be expressed as linear equation or inequation functions in a measurable way.

Ration preparation is an exact linear programming problem due to its purpose and its restrictions (restriction of raw materials or ration content). Basically, there are two elements in linear programming.

The first is linearity, and the second is the restriction. The task of ration preparation perfectly complies with these assumptions.

The developments in desktop and laptop computer technology and the transfer of techniques used in solution of linear programming problems into computers has allowed common use of linear programming in all areas related to economical use of limited sources.


Least-Cost Ration Formulation
The most important issue to be highlighted in ration preparation using computers with the least-costs is the accurate definition of the problem as linear equations or inequations.

In the least-cost production, the unit product cost is minimized against existing restrictions.

After creating the models for the least cost ration solution, these models should be properly entered into an appropriate linear programming package.

The most important issue in computed ration preparation is the knowledge of the person(s) preparing the ration in feeds and animal nutrition.

This is because computers mathematically solves a sum of mathematical operations as we introduce them to computers.

In this technique, the solution is obtained through mutual communication and interaction between the computer and the person preparing the ration.

The breeder, feeder or student who is preparing a ration should achieve the least-cost solution by controlling the data on raw materials, cost factors during the interactive operations with the computer using and interpreting his/her knowledge of feeds and animal breeding, and, if necessary, reformulate the whole ration.

There are 3 main elements in linear programming models. The first of these elements is the function, the second is the decision variables, and the third is the restrictions.

The purpose in least-cost feed formulation is to ensure an equation that includes the unit cost of function feed. Decision variables are the feed raw materials to be included in the ration.

The restrictions are the levels of nutrients to be included in the feed and restrictions of the feed raw materials.

Purpose Function: MIN ∑i 1 FiYi

Yi = amount of feed i (are also decision variables),

Fi = price of feed i.

The value of the purpose function is found according to the value of ∑i 1 Yi. If the feed is prepared in kg, the result will be in TL/kg; and if it is prepared in tons, the result will be in TL/ton.

Restrictions: ∑i 1 BMij * Yi ≥ veya ≤ Bj

BMj*Yi: Refers to the sum of multiplications of the amount of feed raw materials “i” and nutrient material “j” of feed raw material “i”; and “Bj” refers to restriction value for the right-hand side element of the norm of nutrient “j”, that is the nutrient “j” norm of the ration. Restrictions may be defined as equations or inequations.

Ths section includes an example on identifying the needs for dairy cattle and preparing the rations that address these needs.

Example 2 involves ration preparation applied in a computer, and in this example the ration is prepared according to the total mixed ration system (TMR).

Example 2: We want to prepare a ration to address the needs using the feed raw materials below for a cow, weighing 600 kg and producing 30 kg 3.5% fat milk.

Step 1: The needs of the animal are identified using the NRC (1989) models or the values.  According to NRC (1989) the daily needs of the animal are 50.76 Mcal ME, 2922 g raw protein, 113.4 g Ca and 72 g P. Maximum dry matter consumption of the animal is 21 kg, assuming that it is 3.5% of the live weight.

Step 2: Besides the restrictions related to needs, the other restrictions to be addressed in the ration should be identified.

In the current solution, it has been decided to keep the rough feed level in the ration at 40-60% for dry matter and at 50-100 g/day for salt. This is the total mixed ration approach (feeding system that freely offers rough and concentrate feed).

Total ration is prepared, rough and concentrate feeds are considered together.

The restrictions related to the rough feed level of the ration are calculated as follows.

(0.91alfalfa dry hay + 0.33corn silage)
————————————————————————————————————>=0.40 and <=0.60
(0.91alfalfa dry hay + 0.33corn silage + 0.88corn + 0.88barley + 0.89wheat bran + 0.89sfk + 0.92cottonseed + 1lime stone + 1dcp + 1salt)

The linear expression of the lower restriction for the ration rough feed ratio as specified above is demonstrated in row 3 of restrictions under step 3, and the upper limit is demonstrated in row 4.

Step 3: For the optimization approach, linear equations or inequations should be created as follows, the problem should be defined mathematically.

Here the most important issue is the amount of total ration. In the existing approach, the daily dry matter consumption capacity of the dairy cow is considered as the total feed.

As is well known, the daily nutrients needed by an animal should be given to the animal in the feed that it can consume in a day.

The equations or inequations of the planned optimization model are given below:

1) MIN 0.50 alfalfa dry hay + 0.30corn silage + 0.90corn + 0.85barley + 0.55wheat bran + 1.75sfk + 1.20cottonseed + 0.03lime stone + 20.00dcp + 0.10salt
Subjected To;

2) 0.91alfalfa dry hay + 0.33corn silage + 0.88corn + 0.88barley + 0.89wheat bran + 0.89sfk +0.92cottonseed + 1lime stone + 1dcp + 1salt<=21 !(kg/day)dry matter consumption capacity

3) 0.546alfalfa dry hay + 0.198corn silage – 0.352corn – 0.352barley – 0.356bwheat bran – 0.356sfk – 0.368cottonseed – 0.4lime stone – 0.4dcp – 0.4salt>=0 !(lower limit for rough feed ratio (not less than 40%)

4) 0.364alfalfa dry hay + 0.132corn silage – 0.528 corn – 0.528barley – 0.534wheat bran – 0.534sfk – 0.552cottonseed – 0.6lime stone – 0.6dcp – 0.6salt<=0 !(upper limit for rough feed ratio (not more than 60%)

5) 2alfalfa dry hay + 0.88corn silage + 2.75corn + 2.89barley + 2.3wheat bran + 2.9sfk + 3.52cottonseed + 0lime stone + 0dcp + 0salt=50.76 !(Mcal/day) ME restriction

6) 130alfalfa dry hay + 26.7corn silage + 90corn + 119barley + 150wheat bran + 440sfk + 219cottonseed + 0lime stone + 0dcp + 0salt=2922 !(g/day) raw protein restriction

7) 10.3alfalfa dry hay + 0.8corn silage + 0.2corn + 0.5barley + 1.2wheat bran + 2.9sfk + 1.5cottonseed + 360lime stone + 237dcp + 0salt=113.4 !(g/day) Ca restriction

8) 1.64alfalfa dry hay + 0.7corn silage + 3corn + 3.4barley + 12.3wheat bran + 6.3sfk + 6.9cottonseed + 0.2lime stone + 188dcp + 0salt=72 !(g/day) P restriction

9) salt>0.05 !(kg/day) lower limit for salt

10) salt<0.1 !(kg/day)upper limit for salt

Step 4: If restrictions are required for feed raw materials, these should be defined. Restrictions resulting from the raw material or the animal physiology or the stock availability may make these definitions mandatory.

For example, in the approach adopted for the second solution, the maximum use level for barley is defined at 5 kg (barley <=5).

In the third solution, the lower level for use of corn silage is 20 kg, and the amount of cottonseed is defined at not more than 1 kg, and these restrictions are added to the solutions for the existing model.

Step 5: Ration formulation consisting of linear equations or inequations should be properly entered into an appropriate linear programming package software.

There are many package software prepared for ration solutions which are either public or intended for private use.

In special package software, the formulations above are automatically prepared by the computer, considering the restrictions defined by us for the ration.

Step 6: The ration formulation which is properly entered and solved should be checked at the final step. Here, what is important is the sufficiency of the basic information of the person(s) preparing the ration on feeds and animal nutrition.

The applicability of the obtained solution can only be checked with a knowledge of feeds and animal nutrition. In the light of this information, the controlled ration should be reformulated, if necessary, and new and better solutions are obtained.

Here the basic criteria is whether the needs of the animal is satisfied in a well-balanced way. In conclusion, the decision should be made by the person preparing the ration.

Linear programming or computed ration solution application basically allows preparing balanced and cost-efficient rations in a rapid way.

When ration 1 is checked, it is understood that a high amount of barley is used in the ration. As is well known, grains with high destructibility may increase the risk of acidosis in dairy cattle rations.

Thus it may have to be restricted (barley<=5 kg), and the relevant restriction should be entered in the model.

When the ration is re-run, the solution no. 2 is obtained. When solution no. 2 is controlled, it is understood that 4 kg cottonseed is used in the ration. As is known, oil seeds can be fed to animals up to 2 kg per day.

However, the upper limit here is 1 kg (cottonseed<=1). Furthermore, it is seen that corn silage is not used at all in this solution. Assuming that corn silage is available, its use may be included.

Thus, using not less than 20 kg corn silage may be included in the model as a restriction (corn silage>=20 kg), and the last solution can be obtained (ration 3).

As can be understood from an examination of the solutions, every intervention of the person preparing the ration in the formulation has increased the cost of feed.

However, the main issue is to address the needs of the animals in a healthy and cost-efficient way. Preparing rations on computers using linear programming entails the following advantages.

In Pearson square and other simple algebraic methods, the number of variables and restrictions is limited.

Solutions can only be made with one or two restrictions. However, in computed ration preparation, there is not a major limitation on the number of variables and restrictions.

Restrictions are fixed in simple methods. We can assign only one value for a restriction; however, when preparing rations on a computer, we can define the top and bottom values.

In other words, ranges (inequation) and ratios for top and bottom can be defined for restrictions and variables.

In simple methods, economics depends on observation and personal skills in trial and error, considering the prices of raw materials.

When the computed ration preparation method is used, well balanced rations can be obtained in a cost efficient way.

Computed ration preparation allows achieving more complex solutions in a shorter period of time.

Besides, computer application with linear programming and optimization approach allows making certain plans and have an insight of ration preparation by offering information on the impact of changes in raw material prices and amounts and the nutrient content of rations on the solution and the cost of the ration.

There are numerous commercial ration programs which turn linear programming into computer applications as well as scientific package software that solve equation systems with multiple variables.

Furthermore, Excel and other office applications feature add-ins (such as solver add-in) which can handle optimization.

When the same feed and contents, the same requirements and the same restrictions are given, the results will be the same in the most professional software and in a simple Excel solver.

Here, professional software offers decision-making tools such as database management, multiple solution, storage of solved rations, stock control and other operation management tasks.

In this regard, the costs of professional software may vary from 1.000 to 50.000 USD.


Ration formulation has a share to play a significant role when the objective is economical and sustainable animal production in terms of costs of feed.

In this regard, the technical staff at the relevant department for feed production at feed factories and farms will understand the linear programming logic and have a better command of the ration formulation, either with package programs or with free packages for general purpose.

Besides, a well-balanced ration may not be achieved even when the ration is formulated with due consideration of the issues above.

That is because when this ration prepared on paper is given to the formulator, the differences in dosaging result in a different ration than the one on paper.

When the ration prepared in a mixer is served to animals, and the animals select the feed due to particle distribution, taste, etc., the animal will receive a different ration than planned.

Again, the nutrients absorbed in the intestines from the feed consumed by animals in a day and the metabolized part of absorbed nutrients may also be considered as rations.

In that case, we can talk about at least 4 different rations. When managing feeding, the purpose should be to minimize the difference among these rations.

On the other hand, it was stated that a well-balanced ration can only be achieved by providing the nutrients (energy, protein, vitamins, minerals, etc.) required daily in the feed that is consumed.

Therefore, the animals should be allowed to feed freely. It should also be noted that the proper distance of the feedbox according to the animal population and the cleanliness of feedboxes are important for nutrition and feed management and for achieving balanced ration.

Restricted feeding results in constant fluctuation in feed consumption among animals due to the hierarchy in the herds, and therefore causes metabolic problems among animals such as acidosis or swelling as well as fluctuations in their performances.

Limitations of computer based model

  • Certain constraints need to be imposed on ingredients (maximum and minimum levels) or otherwise, it may take all low cost ingredients with poor nutritive value. Such a ration would not result in high milk production at least cost and hence, milk production may get adversely affected.
  • Computer can not encounter the toxic material in the ingredients.
  • Computer will not count the additive effect of feeds.
  • Computer can not judge the digestibility and palatability of ingredients. It may be a least cost ration, but with poor palatability.
  • Needs skill and good programming.

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