Venus Robotics has two employee training programs $-$ Teamwork and Creative Thinking. The

training programs are delivered to $20$ employees at a time. Teamwork training is a $3$ day program

and costs $$10,000$ per offering; Creative Thinking lasts $2$ days and costs $$8,000.$ Venus Robotics'

management has decided that at least $15$ training programs should be offered total, with at least $6$

of these on Teamwork and at least $6$ on Creative Thinking. However, they do not wish their

employees to use more than $1000$ days of training. $($One Teamwork training program occupies $20$

employees for $3$ days, so requires $60$ days of training time.$)$

Formulate a linear programming model for minimizing cost of training programs. Define

your decision variables and constraints.

Solve the model using graphical solution and find the optimum solution and values of the

basic variables.

Solve the model by Simplex method. Define your binding and non$-$binding constraints

and show how to find shadow prices using optimal tableau.

Solve the model using Excel Solver, show your Excel spreadsheet and report your

solutions here.

Create the optimal tableau manually by using basic variables in the initial tableau.

Manually calculate the range of values for the cost of Teamwork training for which the

current basis remains optimal. Then, show it in Excel Solver sensitivity report.

Manually calculate the range of values for the cost of Creative Thinking training for which

the current basis remains optimal. Then, show it in Excel Solver sensitivity report.

Manually calculate the range of values for the number of training programs for which the

current basis remains optimal. Then, show it in Excel Solver sensitivity report.

Manually calculate the range of values for the number of training days for which the current

basis remains optimal. Then, show it in Excel Solver sensitivity report.

Manually calculate the range of values for the number of Teamwork training programs for

which the current basis remains optimal. Then, show it in Excel Solver sensitivity report.

Manually calculate the range of values for the number of Creative Thinking training

programs for which the current basis remains optimal. Then, show it in Excel Solver

sensitivity report.

If the minimum number of Teamwork training programs were increased to $8,$ would the

current basis remain optimal and what would be the new cost value?

Suppose Venus Robotics is considering another type of $2-$day$-$long training which costs

$$5,000$ per offering. Should Venus Robotics offer this training?

Find the dual of the initial model and define decision variables and constraints in the dual

model.

Solve the dual model with Excel Solver. Comment on optimum solution and on values of

basic variables.

Use complementary slackness to find the dual LP values using the primal LP solution.

Solve the primal model using GAMS, upload your file, show your code and report your

solutions here $($rename the file as HW$1\_$NameSurname$\_$StudentID$\_$Q$1)$

A Touche Young Inc. produces two types of products. For each product, the required hours of

labor and contribution to profit is different and shown in the following table.

The company has a goal of $$48000$ in profits and incurs a $$1$ penalty for each dollar it falls short

of this goal. A total of $60$ hours of labor are available. A $$2000$ penalty is incurred for each hour of

overtime $($labor over $60$ hours$)$ used, and a $$1000$ penalty is incurred for each hour of available

labor that is unused. Marketing considerations require that at least $4$ and $10$ units of product $1$ and

$2$ be produced. For each unit $($of either product$)$ for which production falls short of demand, a

penalty of $$4000$ is assessed.

Formulate an LP that can be used to minimize the penalty incurred by the company.

Define your decision variables and constraints.

Solve the LP using GAMS and upload your code and output files.