Tails-up is an established pet food company that manufactures both wet and dry food and treats for dogs and cats. One of the products produced by Tails-up is a low-kilojoule dry dog food for overweight dogs. This product is made from beef products and grain. Each kilogram of beef costs R18.00 and each kilogram of grain costs R12.00. A kilogram of the dog food must contain at least 9 units of Mineral 1 and 8 units of Mineral 2. A kilogram of beef contains 10 units of Mineral 1 and 12 units of Mineral 2. A kilogram of grain contains 5 units of Mineral 1 and 9 units of Mineral 2. A quality standard set by one of the largest breeder's associations stipulates that at least 25% of the content of dry dog food should be meat. As a result, the management of Tails-up has committed to meet these minimum criteria in order obtain the endorsement of this association. At the last management meeting, the decision was taken to also venture into producing two types specialized wet cat foods, one for Siamese cats and the other for Persian cats. As part of this one of the factories has been instructed to use their spare capacity in order to initiate these new lines. A total of 1750 minutes of mincing time, 2200 minutes of mixing time, 1600 minutes of canning time, and 90 minutes of labeling time are available every month. It takes slightly longer to mince and mix the Siamese cat food when compared to Persian cat food, but the Persian cat food require longer canning time to complete the process. It was decided to manufacture and sell these specialized cat foods in packaging sizes of 12 cans. A batch of 12 cans of Siamese cat food requires 30 minutes of mincing time, 40 minutes of mixing time, 20 minutes of canning time and 1 minute of labeling time, while a batch of 12 cans of Persian cat food requires 20 minutes of mincing time, fifteen minutes of mixing time, 30 minutes of canning time and 1 minute of labeling time. The expected market price of a can of Siamese cat food is R8.00, while that of a can of Persian cat food is R14.00. As a result of an upcoming Siamese world championship cat show, an increase in the demand for Siamese cat food is expected and therefore more than 384 cans of Siamese cat food will have to be produced each month. The head animal dietician at Tails-up has formulated minimum requirements for a new dry cat food for kittens. The dry kitten food is mixed by combining different quantities of three ingredients. It has to be done in such a way that the end result contains adequate energy levels, at the same time keeping the cost of the mixture to a minimum and maintaining taste. Each of the three ingredients contains certain amounts of five important substances. The following table shows the minimum substance requirement levels per kilogram of dry kitten food, as well as the cost of each of the mixtures

In addition to this it is important to note that based on nutritional industry standards, the mixture of ingredients in the final product may not exceed a total of 30 units.

Ingredients available for mixing

Subs Req Grain (Units/ kg) Meat (Units per kg) Milk powder (Units per kg) Min Req(Units) Iron 6 8 1 13

Fat 3 3 3 4

Protein 7 10 8 16 Carbohydrates 13 2 4 18

Fiber 8 1 3 12

Cost per kg 930c 770c 1450c

Tails-up has also identified the need for a low-cost dry dog food that still has sufficient nutritional value to maintain basic animal health, but is sufficiently cheap to exploit a rapidly growing market of low-income consumers. This dog food will be marketed under the brand Toptails and manufactured from excess raw materials available on some of the other production lines. The current excess raw materials have been estimated at 190kg animal protein derivative and 1,800kg of grain. Two packaging formulations are considered, one for large dog breeds and the other for small dog breeds. It is estimated that the profit on the large breed dog food will be R27 per bag and the profit on the small breed dog food will be R17 per bag. The assistance of a veterinary scientist has been obtained to ensure that the product will at least have sufficient nutritional value to maintain basic animal health. Based on this it was determined that a bag of dog food for a large breed will require 500g of animal protein derivative and 6.5kg of grain, while a bag of dog food for a small breed will require 800g of animal protein derivative and 6.2kg of grain. Because a whole new production line has to be set up, management has decided that at least 150 bags of dog food for small breeds and 300 bags of dog food for large breeds have to be produced to make this production line worthwhile from a management perspective.

3.1 Formulate the LP Model required to minimize the cost of low-kilojoule dry dog food for overweight dogs.

3.2 Determine how many kilograms of beef and grain should be included in each kilogram of low-kilojoule dry dog food for overweight dogs.

3.3 Determine what the cost and mineral content of the final product would be. 3.4Formulate and solve the LP Model required to maximize the profit on low-cost dry dog food, Toptails, for the rapidly growing market of low-income consumers