100 LINEAR PROGRAMMING FOR MULTIPLE FEED FORMULATION R.A. Clements* Summary This paper first briefly reviews the classical feed mix problem formulated as an LP and discusses some of the problems encountered in practice while using such models. It then extends the formulation to a multiple-formulae problem and explores some of the uses of post-optimal analysis for ingredient buying strategy. 1. Introduction Table 1 shows the approximate tonnages of prepared feed manufactured in New Zealand. Poultry predominate. Birds are highly efficient converters of protein from cheap sources into high-grade protein in eggs as well as cheap plant food into meat. Larger animals are much less efficient, using a large proportion of their food intake for maintenance. Poultry keeping is intensive - units range from 5000 to 20,000 birds. Food is bought and consequently the poultry keeper is sharply aware of the cost of feed. This is not so for other animals. Pigs are traditionally fed from by-products of milk and in most grazing-type farming, the quantity and quality of feed consumed is not measured. Food costs are thus difficult to ascertain. Feed-lot operation has not yet started in New Zealand. For these reasons, the discussion in this paper is confined to poultry only. * Paper presented at the 1973 Conference of the Operational Research Society of New Zealand, October 26th, 1973 in Wellington. The author wishes to acknowledge with thanks the extensive editorial suggestions made by Dr. H.G. Daellenbach
- 101 - TABLE 1: Formula Feeds in New Zealand Annual Tons of Feed Egg layers (4m) 200,000 Growers 40,000 Broilers (10m) 50,000 Other poultry 10,000 All other animals 8,000 308,000 2. The Diet Problem The feed mix problem is a variation of the well known diet problem of mixing ingredients to obtain a feed that satisfies certain nutrient qualities so as to minimize the cost of the feed, usually formulated as a linear programme. An ingredient is a material that can be purchased commercially for use in the manufacture of feed. A nutrient is a chemical substance which is useful for the growth and maintenance of an animal species. Generally, an ingredient contains a number of nutrients plus foreign matter of no recognised value. Some ingredients consist of a single nutrient. Let x- denote the amount of the j*'*1 ingredient used in a J f e e d, b^ denote the minimum requirement of nutrient i, a^j denote the quantity of nutrient i contained in one kilogram of ingredient j, and Cj denote the cost of ingredient j. Then the feed mix problem is: Find values for x ^, j=l,2,..., n which (1) MINIMIZE Ec.x. j 3 3 subject to Ea..x. _> b., i = 1,2,..., m j 1 1 and Xj ^ 0, j = l, 2,..., n.
102 3. Input Problems Table 2 lists the main ingredients available in New Zealand. The major items are grains and meat-meal which are relatively cheap. Some of the minor items are however very important and also expensive, but may only be added in minute proportions. Therefore, the quantities of ingredients needed to make up a 1000 kg batch of feed may range from 500 kg down to a few grams, while prices vary inversely from 4 cents/kg to up to $25/kg. TABLE 2: Available Ingredients in New Zealand Wheat Barley Maize Oats Bran Pollard Brewers grain Maltings De-hy lucerne Mollasses Tallow Meat-meal Liver meal Dried blood Milk powder Peas Fish meal Bone flour Lime Salt Vitamins Minerals Anti-biotics In the analysis of ingredients we are interested only in the metabolizable content. Since animals differ in their ability to digest, the analysis of the same ingredient will vary with the animal for which it is used. The a ^ coefficients represent thus the metabolizable content for a particular animal species. Chemical analysis is usually not complete and covers only nutrients of importance. Ingredients also vary highly in quality and therefore the nutrient content in an ingredient is an average. For any given animal, the nutritionist will specify the amounts of nutrient present in each ingredient, lower and upper bounds on the amounts of each ingredient desired in a feed formula, the quantities and the prices of each ingredient available for mixing. Specification of nutrient requirements may also be in the form of composition ratios, giving rise to expressions of the form
103 (2) Ea..x./Ea,.x. = r. / r 1 j ij J k k] j i k where r./r, is the ratio for nutrients l and k and Ea..x. is i k. ij i 3 the amount of nutrients i present in the feed. Rearranging, we get the following constraint equation (3) r, Ea..x. - r -Ea..x. = 0 k 13 3 1. kj 1 J J Certain nutrients can be substituted for others, but not necessarily vice-versa. Say nutrient i can be substituted for nutrient k but not vice-versa. Then the LHS coefficients in constraint k would be equal to the sums (aj^+a^j) for all j, whereas in constraint i only the a.'s would be shown. For practical reasons, a limited number of predetermined vitamin-mineral premixes containing 30 to 40 ingredients are used in whole bag lots of 1 kg per 1000 kg of feed to provide necessary additional nutrients. Their choice is made arbitrarily by the nutritionist and is only reflected by adjusting nutrient specifications accordingly if necessary. Bulk or density must sometimes be included for practical or mechanical reasons, such as bagging or pelletting, giving rise to the following type of constraints: (4) Ev-x. > v j J J - where Vj is the volume per kilogram of ingredient j and v is the minimum volume of the mix. One of the ingredients may essentially serve as bulk source. The mechanics of handling and mixing may restrict the number of ingredients or the type, quality or quantity which can be used. Restrictions on the number of ingredients gives rise to non-linearities that can only be handled by integer programming. With linear programming such restrictions are ignored in the formulation and the optimal solution is checked
104 for possible violations. If violations occur the problem is resolved after excluding some ingredients. Prices of certain ingredients may not be known exactly in advance to purchases and vary over the seasons. For others, such as meat-meal,prices actually paid vary depending on the source due to transportation cost differentials. The quantities available from each source may not be known in advance. Consequently average prices may have to be used. 4. Multiple Feed Formulation A feed manufacturer usually prepares a number of different formulae for the same or different animal species. As long as ingredients are available in unlimited quantities each formula would be optimized individually However, if some ingredients are only available in restricted quantities and their output is not responsive to price increases, such as for by-products, ingredient allocation to each formula cannot be done independently of the allocations to other formulae. Consequently the optimization has to cover all formulae in one LP problem. We shall formulate this problem such that ingredient composition is directly expressed in percentages. Let Xjt denote the percentage of formula t made up from ingredient j, y. denote the total amount of ingredient j used in all p formulae, b. denote the minimum or maximum percentage of the i constraint for nutrient specifications (which may include ratio and bulk restrictions) a-denote the metabolizable nutrient content for ijt ingredient j in formula t in the it^1 constraint for nutrient specifications, sj denote the sales budget in kilograms for formula j dj denote the availability of ingredient j, Cj denote the cost of ingredient j.
TABLE 3: Coefficient Matrix Structure of the Multiple Formulae Problem 105
106 Then the problem is: Find values for y^, xjt > j = 1»2, n, t = l,2,...,p which MINIMIZE Zc.y. 3 3 3 subject to looyj = 0 all j (allocation of ingredients to formulae) (5) all i,t (nutrient specifications) all j all j,t. (ingredient availability) Table 3 shows the coefficient matrix of the problem, which has the rather striking structure of the decomposition problem. S_. Post-Optimal Analysis Cost ranging for ingredients which are in short supply can be highly valuable to indicate maximum prices that could be offered for additional quantities of these ingredients. However, to cover the range of prices over which such ingredients are likely to vary it is usually necessary to have recourse to parametric programming. Unfortunately, prices of various ingredients may not fluctuate in the same direction and this analysis therefore has to be done individually for each ingredient that is subject to large price fluctuations. The value of parametric programming lies mainly in the discovery that some ingredients attain some sort of plateau, i.e., the volume used remains constant over a wide range of prices of other ingredients. Such ingredients can be bought forward with relative confidence. Others, which respond sharply have to be treated more cautiously and bought in small lots.
107 6. Conclusion Ind ivi dual formula optimization generates a cost savi ngs of about $2.00 per ton of feed over an non-optimi zed formula of the same nut ritional value. Multiple formulae opt imi zat ion doubles this fi gure, an d, besides guarantees that the var i ous fo rmulae can be made to plan from the ingredients ava ilable as of a given point in time. There are two extensions that could be pursued. The first is to go to a multi-period model where, given future prices of commodities, ingredients may be purchased forward or stored in inventory for later use. The difficulty here is that future prices and availabilities are difficult to predict. The second extension is to optimize the nutritional content of a feed. Higher nutrient specifications can produce more meat per ton of feed, but will also increase the cost of the feed. The optimum balance can be determined provided sufficient data on nutrient/performance correlations are available. This is presently done by American beef producers but is not yet used in New Zealand.