Exercise $9\mathrm{.}1-3$ Hillier & Lieberman

The Versatech Corporation has decided to produce three new products. Five branch plants now have excess

product capacity. The unit manufacturing cost of the first product would be $$31,$$$29,$$$32,$$$28,$ and $$29$ in

Plants $1,2,3,4,$ and $5,$ respectively. The unit manufacturing cost of the second product would be $$45,$$$41,$

$$46,$$$42,$ and $$43$ in Plants $1,2,3,4,$ and $5,$ respectively. The unit manufacturing cost of the third product

would be $$38,$$$35,$ and $$40$ in Plants $1,2,$ and $3,$ respectively, whereas Plants $4$ and $5$ do not have the capability

for producing this product. Sales forecasts indicate that $600,1,000,$ and $800$ units of products $1,2,$ and $3,$

respectively, should be produced per day. Plants $1,2,3,4,$ and $5$ have the capacity to produce $400,600,400,$

$600,$ and $1,000$ units daily, respectively, regardless of the product or combination of products involved. Assume

that any plant having the capability and capacity to produce them can produce any combination of the products

in any quantity.

Management wishes to know how to allocate the new products to the plants to minimize total manufacturing

cost$.$

Formulate this problem as a transportation problem by constructing the appropriate parameter table

Obtain an optimal solutionExercise #$2$

Leading an army of stormtroopers can be highly demanding. The Chief Operations Officer of the Galactic

Empire has learned that Operations Research may come in handy for planning the weekly schedules of their

army. Three types of daily shifts are usually put in place: AM shifts from $8$AM to $4$PM; PM shifts from $4$PM

to midnight; and Night shifts between midnight and $8$AM$.$ A stormtrooper in a week will perform one type of

shift only: i$)$ A stormtrooper doing AM shifts will work five consecutive days starting Monday, Wednesday or

Thursday; ii$)$ A stormtrooper doing PM shifts only will work four consecutive days starting either Tuesday or

Friday; iii$)$ A stormtrooper doing Night shifts only will work three consecutive days, starting Tuesday, Thursday

or Saturday. The needs of stormtroopers in a new small outpost in the Outer Rim can be summarized as follows:

Describe the decision variables of this scheduling problem.

If the hourly wages are of $10GC($Galactic Credits$)$ for the AM and PM shifts, and of $15GC$ for the Night

shifts, write the objective function of minimizing the total expenditure.

Write the demand constraints associated with the following three periods: $1)$ Tuesday AM; $2)$ Thursday

PM; $3)$ Friday Night.

The Chief Operations Officer has learned that the stormtroopers have unionized, and are asking to add

the following constraint to the model: the total number of stormtroopers doing night shifts should not

exceed a $50\%$ of the total number of stormtroopers doing either AM or PM shifts. Write this constraint.