# Formulating a nonlinear program for profit maximization

A bakery produces muffins and doughnuts. Let x1 be the number of doughnuts produced and x2 be the number of muffins produced. The profit function for the bakery is expressed by the following equation: profit = 4x1 + 2x2 + 0.003x12 + 0.004x22. The bakery has the capacity to produce 800 units of muffins and doughnuts combined and it takes 30 minutes to produce 100 muffins and 20 minutes to produce 100 doughnuts. There is a total of 4 hours available for baking time. There must be at least 200 units of muffins and at least 200 units of doughnuts produced. Formulate a nonlinear program representing the profit maximization problem for the bakery. What is the equation associated with the constraint regarding the amount of time available for baking?

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#### Solution Preview

Let x1 be the number of doughnuts produced andx2 be the number of muffins produced.

We want to maximize profit=4x1 + 2x2 + 0.003x1^2 + 0.004x2^2

Subject to ...

#### Solution Summary

The solution gives detailed steps on setting up a nonlinear programming model. Each equation is written based on each given statement.

Linear Programming Formulation Problems

Linear Programming Formulation Problem Set (Show all works.)

1. A farmer can purchase 3 kinds of feed for his stock, with various percentages of each of 4 nutrients, called A, B, C, and D. A mixture of feeds gives proportional amounts of nutrients. The following table gives the minimum daily requirements (lb), cost (¢ / lb), and nutritional elements per pound of feed.

Nutrient Feed 1 Feed 2 Feed 3 Minimum Daily Requirements

A 2 3 7 950

B 1 1 0 250

C 5 3 0 900

D 0.5 0.5 1 230

Cost 4.0 3.5 9.6

(a) Formulate this as a Linear Program.

(b) Show that it is feasible to use 475lb of feed 1 and none of the other feeds, at a total cost of $19.

(c) Suppose the farmer wants to use equal amounts of feeds 1, 2 and 3. How much must he use, and what does it cost?

2. A local manufacturing firm produces 4 different metal products, each of which must be machined, polished, and assembled. The specific time requirements (in hours) for each product are as follows.

Machining, h Polishing, h Assembling, h

Product I 3 1 2

Product II 2 1 1

Product III 2 2 2

Product IV 4 3 1

The firm has available to it on a weekly basis 480h of machine time, 400h of polishing time, and 400h of assembly time The unit profits on the products are $6, $4, $6, and $8, respectively. The firm has a contract with a distributor to provide 50 units of product I and 100 units of any combination of product II and III each week. Through other customers, the firm can sell each week as many units of products I, II, and III as it can produce, but only a maximum of 25 units of product IV. How many units of each product should the firm manufacture each week to meet all contractual obligations and maximize its total profit? Assume that any unfinished pieces can be completed the following week.

3. a caterer must prepare from 5 fruit drinks in stock 500 gal of a punch containing at least 20% orange juice, 10% grapefruit juice, and 5% cranberry juice. If inventory data are as shown below, how much of each fruit drink should the caterer use to obtain the required composition at minimum total cost?

Orange Juice, % Grapefruit Juice, % Cranberry Juice, % Supply,

gal Cost,

$ / gal

Drink A 40 40 0 200 1.50

Drink B 5 10 20 400 0.75

Drink C 100 0 0 100 2.00

Drink D 0 100 0 50 1.75

Drink E 0 0 0 800 0.25

4. A town has budgeted $250,000 for the development of new rubbish disposal areas. 7 sites are available, whose projected capacities and development costs are given below. Which sites should the town develop?

Site A B C D E F G

Capacity, tons/wk 20 17 15 15 10 8 5

Cost, $1000 145 92 70 70 84 14 47

5. Recreational Motors manufactures golf carts and snowmobiles at its 3 plants. Plant A produces 40 golf carts and 35 snowmobiles daily; plant B produces 65 golf carts daily, but no snowmobiles; plant C produces 53 snowmobiles daily, but no golf carts. The costs of operating plants A, B, and C are respectively $210000, $190000, and $182000 per day. How many days (including Sundays and holidays) should each plant operate during September to fulfill a production schedule of 1500 golf carts and 1100 snowmobiles at minimum cost? Assume that labor contracts require that once a plant is opened, workers must be paid for the entire day.

6. Formulate the following nonlinear program as a Linear Program.

min : Ax = b.

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