[SOLVED] SP and RO SQL

SP and RO

Problem 1

A company is considering producing a chemical C which can be manufactured with either process II or process III, both of which use as raw material chemical B.  B can be purchased from another company or manufactured with process I which uses A as a raw material.  Process II and process III are exclusive, i.e. at most one of them can be built. There are five possible outcomes of demand and selling price combination for chemical C. The probability of each scenario, as well as the corresponding demand and selling price are listed below:

	Demand of C (million tons)	Selling Price ($/ton)	Probability
Scenario 1	5	2000	10%
Scenario 2	8	1900	20%
Scenario 3	10	1800	40%
Scenario 4	12	1600	20%
Scenario 5	15	1000	10%

Given the specifications, formulate a two-stage stochastic MILP model to maximize the expected profit and solve it with GAMS/Pyomo to determine:

a)  Which process to build and what would be the corresponding capacity?

b)  How to obtain chemical B and how much product C should be sold in each scenario?

c)  What would be the “value of perfect information” for this problem?

Hints:

a)    Facility  selection  (binary  0-1  decisions)  and  capacity  (continuous  decisions)  are  first-stage  decisions (independent  of  demand/price  realization);  purchase,  production  and  sale  are  second-stage  decisions (dependent on demand/price realization).

b)    The production is each scenario should not exceed the capacity, which is scenario independent.

c)    If production/sale exceeds the demand for a certain scenario, unsold product cannot generate any revenue (but incurs production cost). The sale amount should not exceed the demand and should not exceed the available/production amount.

Data:

Investment Costs

Fixed capital cost (million $)              Variable capital cost ($/ton of product)

Process I	100	250
Process II	150	400
Process III	200	550

Operating Costs

Variable operating/production cost ($/ton of product)

Process I                   100

Process II                   150

Process III                   200

Prices:    A:     $250/ton

B:     $450/ton

Conversions:

Process  I       90% of A to B

Process II        82% of B to C

Process III      95% of B to C

Maximum supply of A:  16 million tons

Problem 2

Consider the newsboy problem: Newspaper unit cost is $0.60/piece, selling price is $1.50/piece; the newsboy needs to order the newspapers tonight, before knowing tomorrow’s demand. Assume there are 100 possible outcomes of demand, ranging from  1 piece to  100 pieces with uniform probability distribution. In other words, there are 100 scenarios for demand and 1% chance for each scenario (i.e. 1% chance for the scenario of 1 piece, 1% chance for the scenario of 2 pieces, 1% chance for the scenario of 3 pieces, … , 1% chance for the scenario of 99 pieces, and 1% chance for the scenario of 100 pieces). The objective is maximizing the expected profit over these 100 demand scenarios while managing the risk.

Formulate the following four risk management models and solve them with GAMS/Pyomo to determine the optimal number of newspapers that should be ordered under each risk management strategy.

a)  Maximizing the mean/expected profit (i.e. risk-neural)

b)  Optimizing mean-variance (using equal weight for expectation and variance, i.e. ρ =-1)

c)  Minimizing probabilistic financial risk for the threshold of 0 profit (i.e. Ω = 0)

d)  Minimizing downside risk for the threshold of 0 profit (i.e. Ω = 0)

e)   Minimizing CVaR for the 20% “loss” quantile (i.e. α = 20% for the “low” profit)

Hints: The example in lecture slides is for cost minimization, while this problem is for profit maximization. Please make sure to revise the model formulations accordingly to account for the “maximization” problem for low profit risk (instead of minimizing “high cost” risk).

Problem 3

Derive the robust counterpart for the following problem:

max  10x1 + 5x2 x1 , x2

s.t.   (6 + u1 )x1 + (2 + u2 )x2  ≤ 80,   ∀ (u1, u2 ) ∈ U

a)   U is a ellipsoidal uncertainty set:  U = {(u1 ,  u2 ) :    ≤1}.

b)   U is a box uncertainty set:  U = {(u1,  u2 ) :  u1   ≤1,  u2   ≤1}.

Problem 4

Consider the following two-stage adaptive robust optimization (ARO) problem:

0.5x1 +100x2  + max u∈U

x1  ≤ 200x2    y1 + y2  ≤ x1 y1  ≤ u1

y2  ≤ u2

(−6y1 − 5y2 )

x1, y1, y2  ≥ 0, x2  ∈ {0,1}

where uncertainty set  U = 

Use linear decision rule to solve the two-stage ARO problem.

Problem 5

Consider the following robust optimization problem with implementation errors.

mn Δx(m)U(x) f (x + Δx)

where f is a nonconvex function defined by,

a)   Find the robust optimal solution x* when  U = {Δx −0.5 ≤ Δx ≤ 0.5 }.

b)  Find the robust optimal solution x* when  U = {Δx −1.5 ≤ Δx ≤1.5 }.