Please solve all parts manually not using excel or any other software. Please solve all parts from (a) to (j)

Last Name, Name: 4. Gapco has a daily budget of 100 hours of labor and 200 units of raw material to manufacture two products. If necessary, the company can employ up to 10 hours daily of overtime labor hours at an additional cost of $2/hr. It takes 2 labor hour and 2 units of raw material to produce one unit of product 1, and 2 labor hours and 1 unit of raw material to produce 1 unit of product 2. The profit per unit of product 1 is $10, and that of product 2 is $12. Let x1 and x2 define the daily number of units produced of products 1 and 2, and x3 as the daily hours of overtime used. The LP model and its associated optimal simplex tableau are then given as Maximize z=10x1+12x2-2x3 Subject to 2x1+2x2-x3=0 s2 s3 RHS s1 x1 2 x2 1 2 0 0 1 1 0.5 0 1 0 0 -0.5 1 0 0 1 0 0 a) What is dual problem (i.e., write the formulation)? (2 pts) b)What is the primal solution? (2 pts) c)What is the dual solution? (2 pts) d) Show that the strong duality theorem holds. (2 pts) e) Suppose the unit sales price of product 1 is increased to $11.5. What will be the new optimal solution? (2 pts) x3 0 0 6 4 0.5 -0.5 1 640 55 145 10 f) Find the allowable range of price for product 2 so that the solution does not change. (2 pts) g)Will the optimal solution change if the price of the model 2 drops to $11? Explain. (2 pts) h) Determine the allowable range for labor hours constraint RHS. (2 pts) i) What will be the new optimal solution if the available labor hours is 200 hours? (2 pts) j)Suppose there is a new product which may sell for $20 using 3 hours of labor and 2 units of raw material. Should Gapco manufacture that product? (2 pts)