Problem #$1$ Lorie Nutritional Diet Planning $($This problem includes $14$ questions $$ Questions $1$ to $14)$

Lorie, a diet nutritionist, is working to create a diet plan for a healthy human consumption. She believes

that a healthy diet should be made to consist of three possible foods, namely apples, bread and corn.

She is planning the diet to provide calories, vitamin X and mineral Y$.$ Lorie gathers the information for

the diet problem and summarized it in the table seen below.

Cost per pound

$($in cents$)$

Availability Calories $($per lb$.)$ Vitamin X $($per lb$.)$ Mineral Y $($per lb$.)$

Apples $25$ Unlimited $3202040$

Bread $501$ lb$./$person$/$day $1,20015020$

Corn $152$ lb$./$person$/$day $100030$

Minimum Daily

requirements

$1,10015060$

Lorie also determined that a person needs to have at least $0\mathrm{.}5$ pounds $($lb$.)$ of apples in the daily diet.

Lorie needs you to help her design a minimum cost diet plan which meets the minimum daily nutritional

needs.

You are asked to set up a proper management decision analysis model to solve Lorie$$s diet problem.

Upon solving the model your computer output generates the following result tables for further analysis

and discussions with Lorie.

Table A$.$ Assume that upon solving Lorie problem and obtain the following sensitivity report.

Variable Cells

Final Reduced Objective Allowable Allowable Lower Upper

Variable Name Value Cost Coefficient Increase Decrease Limit Limit

$1$

Decision values

Apples $1\mathrm{.}0714285710250\mathrm{.}33333333318\mathrm{.}333333336\mathrm{.}6725\mathrm{.}333$

$2$ Decision values Bread $0\mathrm{.}857142857050137\mathrm{.}52\mathrm{.}547\mathrm{.}5187\mathrm{.}5$

$3$ Decision values Corn $00\mathrm{.}267857143151$E$+300\mathrm{.}26785714314\mathrm{.}721$E$+30$

Constraints

Final Shadow Constraint Allowable Allowable Lower Upper

Const # Name Value Price R$.$H$.$ Side Increase Decrease Limit Limit

$1$ Bread Equation $0\mathrm{.}857142857011$E$+300\mathrm{.}1428571430\mathrm{.}861$E$+30$

$2$ Corn Equation $0021$E$+30201$E$+30$

$3$ Calories Equation $1371\mathrm{.}42857101100271\mathrm{.}42857141$E$+301\mathrm{.}00$E$-301371\mathrm{.}43$

$4$ Vitamin X Equation $1500\mathrm{.}2678571431502036\mathrm{.}53846154113\mathrm{.}46170$

$5$ Vitamin Y Equation $600\mathrm{.}4910714296024021\mathrm{.}3333333338\mathrm{.}67300$

$6$ Apples Equation $1\mathrm{.}07142857100\mathrm{.}50\mathrm{.}5714285711$E$+301\mathrm{.}00$E$-301\mathrm{.}07$

All of questions $1$ to $15$ referred to Lorie Diet Problem. Use the sensitivity analysis results seen in Table A

seen above to perform the necessary calculations to answer the following $15$ questions in the multiple$-$

choice space of the final exam.

Question #$1$: the optimal total minimum cost of Lorie nutritional plan is:

Question #$2$: Which of the nutritional constraints are fully utilized $($or requirements are exactly met$)?$

Question #$3$: The daily cost of food for a group of $1,000-$person institution $($in dollars$)$ is:

Questions #$4$: How many calories will Lorie be recommending to supply daily to each person in the

nutritional program?

Question #$5$: Which food is not recommended for inclusion in the program?

Question #$6$: If a certain food is not in the optimal diet, what should the price of this food be to make it

attractive enough to be included in the solution $($in cents, approximately$)?$

Question #$7$: What is the utilization level of bread $($in percentage, approximately$)?$

Question #$8$: Suppose we increase the requirement of Vitamin X from $150$ to $155$; the cost of the diet,

in such a case, will be $($in cents, per person, per day, approximately$)$ what amount?

Question #$9$: What is the necessary condition $($or conditions$)$ which can motivate Lorie to require the

inclusion of more apples in the nutritional program in a cost$-$effective way?

Question #$10$: Why is dual price of the Apples requirement $($i$.$e$.,$ constraint $6)$ equal to zero?

Question #$11$: In the right$-$hand side $($RHS$)$ sensitivity analysis $($in constraint #$2),$ there is no upper limit$.$

This is because:

Question #$12$: How much money can be saved, per person, per day, in cents, if the mineral requirement

is changed from $60$ t$055($approximately$)?$

Question #$13$: Assume that the price of apples increases $10$ percent; the diet composition in such a case

will:

Question #$14$: Is it possible that the optimal solution will include corn only, if the price of corn is low

enough?

Problem #$2$: Your Castle Construction Project $($This problem includes $8$ questions $$ Questions # $15-22)$

You are working to build your private castle. You have determined and listed the following $10$ major

activities that need to take place to complete your project. The table below $($Table B$)$ gives the detailed

input you need for your project tasks and relationships.

Table B$.$

Task # Task Name

Immediate

Predecessor

Task #

Expected

time

$($days$)$

$1$ Walls & Ceiling $25$

$2$ Foundation None $3$

$3$ Roof Timbers $12$

$4$ Roof Sheathing $33$

$5$

Electrical

Wiring $14$

$6$ Roof Shingles $48$

$7$ Exterior Siding $85$

$8$ Windows $12$

$9$ Paint $6,7,82$

$10$

Inside wall

Board $8,53$

Table C$.$

Task # Task Name

Activity

Slack

Times

Answer A

Activity

Slack

Times

Answer

B

Activity

Slack

Times

Answer C

Activity

Slack

Times

Answer

D

$2$ Foundation $0000$

$1$

Walls and

Ceiling $0010$

$3$ Roof Tim