Page 3 of 3 BUS 4110 Project 4 Problem 1 Rutland Corporation has a one - year contract to supply motors for all refrigerators produced

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BUS $\mathrm{4110}$ Project $4$

Problem $1$

Rutland Corporation has a one$-$year contract to supply motors for all refrigerators produced by

the Ice Age Corporation. Ice Age manufactures the refrigerators at four locations around the

country: Boston, Dallas, Los Angeles, and St$.$ Paul. Plans call for these numbers $($in thousands$)$

of refrigerators to be produced at the four locations.

Boston $\mathrm{60}$

Dallas $\mathrm{80}$

Los Angeles $\mathrm{70}$

St$.$ Paul $\mathrm{90}$

Rutland has three plants that can produce the motors. The plants and their production capacities

$($in thousands$)$ follow:

Denver $\mathrm{100}$

Atlanta $\mathrm{75}$

Chicago $\mathrm{125}$

Because of varying production and transportation costs, the profit Rutland earns on each lot of

$1,\mathrm{000}$ units depends on which plant produced it and to which destination it was shipped. The

accounting department estimates of the profit per unit $($shipments are made in lots of $1,\mathrm{000}$ units$)$

are as follows:

Per Unit Shipping Costs

From$\backslash$To Boston Dallas Los Angeles St$.$ Paul

Denver $ $\mathrm{7.00}$ $ $\mathrm{11.00}$ $ $\mathrm{8.00}$ $ $\mathrm{13.00}$

Atlanta $ $\mathrm{20.00}$ $ $\mathrm{17.00}$ $ $\mathrm{12.00}$ $ $\mathrm{10.00}$

Chicago $ $\mathrm{8.00}$ $ $\mathrm{18.00}$ $ $\mathrm{13.00}$ $ $\mathrm{16.00}$

Given profit maximization as a criterion, Rutland would like to determine how many motors

should be produced at each plant and how many should be shipped from each plant to each

destination.

Perform the following:

$\mathrm{�}$ Put problem in standard form.

$\mathrm{�}$ Calculate the initial feasible solution.

$\mathrm{�}$ Check for optimality.

$\mathrm{�}$ If not optimal, find the entering and leaving variables.

$\mathrm{�}$ Calculate the improved solution.

Answer the following questions based on your calculations $($next page$)$:

BUS $\mathrm{4110}$ Project $4$

$\mathrm{1.}$ In standard form, where do the per unit shipping costs go$?$

a$.$ Small box

b$.$ Large box

c$.$ Right side of table

d$.$ Below the table

$\mathrm{2.}$ For the initial feasible solution, what is the quantity of units shipped from Chicago to St$.$

Paul?

$\mathrm{3.}$ What is the shipping cost for the initial feasible solution?

$\mathrm{4.}$ How do we know if we have an optimal solution?

a$.$ All results in the evaluation column are $>=0$

b$.$ All results in the evaluation column are $<=0$

c$.$ If any of the numbers in the evaluation column are $>=0$

d$.$ In case of a tie, flip a coin

$\mathrm{5.}$ What is the entering variable $($which cell in the table$)?$

$\mathrm{6.}$ How do we determine the leaving variable?

a$.$ The smaller of two occupied cells

b$.$ The larger of the two occupied cells

c$.$ The smaller of the two per unit shipping costs

d$.$ The larger of the two per unit shipping costs

$\mathrm{7.}$ What is the leaving variable $($which cell in the table$)?$

$\mathrm{8.}$ What is the total shipping cost for the improved solution?

$\mathrm{9.}$ How many units will be shipped from Chicago to Boston?

$\mathrm{10.}$ What is the value of the U$1$ indices?

BUS $\mathrm{4110}$ Project $4$

Problem $2$

The distribution system for the Gwinn Corporation consists of three plants $($A$,$ B$,$ and C$),$ two

warehouses $($D and E$),$ and four customers $($W$,$ X$,$ Y$,$ and Z$).$

Customer W

Plant A

Warehouse D Customer X

Plant B

Warehouse E Customer Y

Plant C

Customer Z

The $3$ manufacturing plants ship to the $2$ warehouses.

The $2$ warehouses ship to the $4$ customers.

The supply, demand, and unit shipping cost information are given in the table. Set up and solve

the transportation problem to minimize total shipping costs.

Plant Supply Store Demand Warehouse Capacity

A $\mathrm{500}$ W $\mathrm{450}$ D $\mathrm{725}$

B $\mathrm{550}$ X $\mathrm{300}$ E $\mathrm{725}$

C $\mathrm{400}$ Y $\mathrm{300}$

Z $\mathrm{400}$

Per unit shipping costs Per unit shipping costs

D E W X Y Z

A $47$ D $6584$

B $85$ E $3677$

C $66$

$\mathrm{1.}$ How many units should be shipped from Plant A to Distribution Center D$?$

$\mathrm{2.}$ How many units should be shipped from Plant C to Distribution Center E$?$

$\mathrm{3.}$ What is the evaluation $($E$=$C$-$U$-$V$)$ for cell Plant B to Distribution Center D$?$

$\mathrm{4.}$ Is the Initial Feasible solution for shipping from plants to distribution centers optimal?

a$.$ Yes

b$.$ No

$\mathrm{5.}$ What is the total shipping cost from the three plants to the two distribution centers?

$\mathrm{6.}$ How many units should be shipped from Distribution Center E to Customer Y$?$

$\mathrm{7.}$ What is the evaluation for cell Distribution Center E to Customer Z$?$

$\mathrm{8.}$ Is the Initial Feasible solution for shipping from distribution centers to customers

optimal?

a$.$ Yes

b$.$ No

$\mathrm{9.}$ What is the total shipping costs from the distributions centers to customers?

$\mathrm{10.}$ What is the total shipping costs for the entire system, from the manufacturing plants to

the end customers?