1. For the optimization problem as presented below assume all coefficients c, a, and b are random and their joint distribution is known. max sin(cixi) + c2x2 Such that: anxi + a12x2 < bi anxi + a22x2 < b2 Xi, X2 ?: Formulate this problem using two-stage stochastic programming for maximizing expected benefits for 2 samples clearly indicating what is a single scenario given their probabilities are Pt and p2 , and any corresponding surplus and deficit variables1 [10 Marks] 2. In a different setting if we assume that only

1. For the optimization problem as presented below assume all coefficients c, a, and b are random and their joint distribution is known. max sin(cixi) + c2x2 Such that:
anxi + a12x2 < bi anxi + a22x2 < b2
Xi, X2 ?: Formulate this problem using two-stage stochastic programming for maximizing expected benefits for 2 samples clearly indicating what is a single scenario given their probabilities are Pt and p2 , and any corresponding surplus and deficit variables1 [10 Marks] 2. In a different setting if we assume that only the RHS of constraints (b) are random. Formulate a chance constrained optimization model. List all your assumptions. [5 Marks] 3. Formulate a probabilistic optimization problem assuming only the objective function coefficients c1 and c2 as random (so no other coefficients are random) in the original problem (1) using the mean-variance objective function (Markovitz type method). Show the expressions for the mean of the objective function and the variance of the objective function for the cases of (i) two random coefficients being independent and (ii) they are joint normally distributed. Show how you would develop a tradeoff function between risk and expected value of the objective function. [10 Marks] 4. Assume only the LHS coefficients in the optimization problem in (1) to be random and the joint distributions coefficients a in each constraints are known. Formulate the chance-constraint for such as problem. [5 Marks] 5. What do we mean by Value of Stochastic Solution (VSS) and Expected Value of Perfect Information (EVPI).Why are these two indicators important. [3 Marks] 6. In the context of Yield optimization , convert the above problem into a three constraint yield optimization problem assuming x1 and x2 as the random design variables. So the objective function becomes another constraint and hence you will have three constraints with each having the x1 and x2 as random variables like the yield optimization problems (at least two) we have seen in the course. Describe how you would get an approximate feasible region, what would be probability distribution model for any general shape of the randomness of each of the variable, and what model would you use to maximize the yield? How would you validate the yield? [12 Marks] 7. BONUS question: Can you use the method in 6 to apply in 4 when ALL four LHS coefficients in constraints have random dependence but x1 and x2 are deterministic . How? [5 Marks]