Problem $1$: $(30)$ Problem $2$: $(30)$

The water engineer of a city's municipal department plans to build a water tank to address the increasing

water demand of the city's fast$-$growing population. The available space for the tank $($excluding access$),$ is

$600f{t}^2.$ The planned inside depth of the tank is $10ft.,$ and it is desired that its length is at least twice its

width $($Length and width are outside dimensions$).$ Space limitations dictate that the width cannot exceed

$20ft.$ The tank will be built with reinforced concrete of $1ft.$ thickness $($walls$,$ base, and cover$).$ The city

seeks to reduce the total cost of the concrete construction. The unit cost of concrete construction, which

includes not only concrete material costs but also formwork and other related costs, is

$\{$:$35\mathrm{.}315f{t}^3)$

Find:

$($a$)$ Formulate the problem indicating decision variables, objective function, and constraints. $(15)$

$($b$)$ Solve the problem using Excel Solver and provide optimal tank dimension and cost. Round off tank

dimensions to the nearest $3^{\text{'}\text{'}}.($Hint: Use GRG Nonlinear solving method and initial non$-$zero

decision variable values$)(15)$ Problem $3$: $(30)$

A project manager is faced with assigning three separate projects amongst four $(4)$ different profit

centers. The weekly profit$/$person $($p$)$ between the profit center and the projects varies because of the

distance and efficiency of each profit center. The available$/$required staffing for each profit center and

project are shown below. The project manager wishes to maximize total weekly profits.

Find:

a$)$ Formulate the above problem as linear program optimization model with Network Diagram,

Objectives, Decision Variables, Constraints. Determine optimal assignments of persons $($p$)$ from

each profit center to each project and maximum weekly profits using Excel Solver. $($Note: General

Mathematical Expressions of the Linear Programming Model are acceptable$)(15)$

b$)$ Conduct a post$-$optimization sensitivity analysis using Excel Solver and answer the following

questions. $(15)$

i$.$ Are all profit centers fully utilized? If not, which one is underutilized?

ii$.$ Which profit center could contribute the most increase in weekly profit by increasing

available staff?

iii. Which assignments are sensitive to any decrease in profits?

A contractor has found natural reserves of sand and gravel at Bloomingdale and Valley Springs where he

may purchase such material. The net profit $($markup$),$ including delivery from Bloomingdale and Valley

Springs, is $$2$ and $$3$ per $m^3,$ respectively. After the material is brought to the site, it is mixed thoroughly

and uniformly, and the contract specifications state that the mix should contain a minimum of $30\%$ sand.

A maximum total volume of $100,000m^3$ of mixed material is estimated for the project. The Bloomingdale

Pit contains $25\%$ sand, and the Valley Springs Pit contains $50\%$ sand. Total volume of material taken from

the Valley Spring pit must not exceed $60\%$ of the total volume from Bloomingdale. As the new

construction engineer on the project, you are asked to determine how much material should be taken

from each pit to maximize profit.

Find: Using graphical methods only

a$)$ Formulate the problem with mathematical expressions for decision variables, objective function,

and constraints. $(10)$

b$)$ Solve the problem and determine the optimal volume of material transported from each pit as

well as the maximum expected profit. $(10)$

c$)$ Perform post$-$optimization sensitivity analysis and provide range of optimality of objective

function coefficients and shadow price $($Dual Value$)$ for each constraint limit$.(10)$

Solutions to the following problems must include screen shots of formulated problems in Excel Solver.