i$)(26$ pts$)$ Regular linear Programming problems:

$1.(7$ pts$)$ The WorldLight Company produces two light fixtures $($products $1$ and $2)$ that require both metal frame parts and electrical components. Management wants to determine how many units of each product to produce so as to maximize profit. For each unit of product $1,$ one unit of frame parts and two units of electrical components are required. For each unit of product $2,$ three units of frame parts and two units of electrical components are required. The company has $200$ units of frame parts and $300$ units of electrical components. Each unit of product $1$ gives a profit of $$1,$ and each unit of product $2,$ up to $60$ units, gives a profit of $$2.$ Any excess over $60$ units of a product $2$ brings no profit, so such an excess has been ruled out.

$2.(8$ pts$)$ Fred Jonson manages a family$-$owned farm. To supplement several food products grown on the farm, Fred also raises pigs for market. He now wishes to determine the quantities of the available types of feed $($corn$,$ tankage$,$ and alfalfa$)$ that should be given to each pig. Since pigs will eat any mix of these feed types, the objective is to determine which mix will meet certain nutritional requirements at a minimum cost. The number of units of each type of basic nutritional ingredient contained within a kilogram of each type of feed is given in the following table, along with the daily nutritional requirements and feed costs:

Nutritional Ingredient Kilogram of Corn Kilogram of Tankage Kilogram of Alfalfa Minimum Daily Requirement

Carbohydrates $902040200$

Protein $308060180$

Vitamins $102060160$

Cost $($cents$)847260$

$3.(11$ pts$)$ Joyce and Marvin run a daycare for preschoolers. They are trying to decide what to feed the children for lunch. They would like to keep their costs down, but they also need to meet the nutritional requirements of the children. They have already decided to go with peanut butter and jelly sandwiches, and some combination of graham crackers, milk, and orange juice. The nutritional content of each food choice and its cost are given in the table below:

Food Item Calories from Fat Total Calories Vitamin C $($mg$)$ Protein

$($g$)$ Cost

$($cents$)$

Bread $(1$ slice$)1070035$

Peanut butter $(1$ tbsp$.)75100044$

Strawberry jelly $(1$ tbsp$.)050307$

Graham cracker $(1$ cracker$)2060018$

Milk $(1$ cup$)701502815$

Juice $(1$ cup$)0100120135$

The nutritional requirements are as follows. Each child should receive between $400$ and $600$ calories. No more than $30$ percent of the total calories should come from fat. Each child should consume at least $55$ milligrams $($mg$)$ of Vitamin C and $12$ grams $($g$)$ of protein. Furthermore, for practical reasons, each child needs exactly $2$ slices of bread $($to make the sandwich$),$ at least twice as much peanut butter as strawberry jelly, and at least $1$ cup of liquid $($milk and$/$or juice$).$

Joyce and Marvin would like to select the food choices for each child that minimizes cost while meeting the above requirements.

ii$).(24$ pts$)$ Reliability Problems:

$4)(5$ pts$)$ What is the reliability of the following production process? R$1=0\mathrm{.}95,$ R$2=0\mathrm{.}90,$

R$3=0\mathrm{.}98.$

$5)(6$ pts$)$ What is the reliability that bank loans will be processed accurately if each of the five clerks shown in the chart has the reliability shown?

$6)(8$ pts$)$ A production line has three machines A$,$ B$,$ C$,$ with reliabilities of $\mathrm{.}99,\mathrm{.}96,$ and $\mathrm{.}93$ respectively. The machines are arranges so that if one breaks down, the others must shut down. Engineers are weighing two alternative designs for increasing the line$$s reliability. Plan $1$ involves adding an identical backup line, and plan $2$ involves providing a backup for each machine. In either case, three machines $($A$,$ B$,$ and C$)$ would be used with reliabilities equal to the original three.

Plan$1$:

Plan $2$:

Which plan will provide the higher reliability?

$7)(5$ pts$)$ A Web server has five major components that must all function in order for it to operate as intended. Assuming that each component of the system has the same reliability, what is the minimum reliability each one must have in order for the overall system to have a reliability of $\mathrm{.}98?$