Formulations of two-stage and multistage Stochastic Programming

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1 Formulations of two-stage and multistage Stochastic Programming Yi Fang, Yuping Huang Department of Industrial and Management Systems Engineering West Virginia University Yi Fang, Yuping Huang Intro. to Stochastic Programming 11/01/ / 17

2 Stochastic Programming Contents Two-stage stochastic programming The farmer s problem example General model formulation Multistage stochastic programming Financial planning problem example Node based formulation Scenario based formulation Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

3 The farmer s problem A farmer has 500 acres of land he should decide how many acres to use for growing wheat, corn, and/or sugar beet planting an acre costs $150/acre, $230/acre, and $260/acre respectively at least 200 tons of wheat and 240 tons of corn are needed to feed cattle excess production is sold at $170/t and $150/t if less is produced, it is purchased at $238/t and $210/t sugar beet sells at $36/t up to 6,000t, and $10/t above that quota the average yield of crop are: 2.5t/acre for wheat, 3t/acre for corn, and 20t/acre for beet depending on how good the weather, the yields may decrease or increase by 20% Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

4 Model Decision Variables x W, x C, x B : acres of wheat, corn and sugar beet w Ws, w Cs, w Bs tons of wheat, corn and beet (at favorable price) to be sold in scenario s e Bs tons of beets sold at lower price in scenario s y Ws, y Cs tons of wheat, corn purchased in scenario s Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

5 Formulation The objective function is to maximize the expected profit: max 150x W 230x C 260x B Total acre costs +1/3( 238y W w W 1 210y C w C1 + 36w B1 + 10e B1 ) Scenario 1 +1/3( 238y W w W 2 210y C w C2 + 36w B2 + 10e B2 ) Scenario 2 +1/3( 238y W w W 3 210y C w C3 + 36w B3 + 10e B3 ) Scenario 3 (1) Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

6 Formulation First Stage (Land area limit): x W + x C + x B 500 (2) Second Stage Scenario 1(highest yield) Wheat: 3x W + y W 1 w W 1 = 200 (3) Corn: 3.6x C + y C1 w C1 = 240 (4) Beet: 24x B w B1 e B1 = 0 (5) Scenario 2(neutral yield) Wheat: 2.5x W + y W 2 w W 2 = 200 (6) Corn: 3x C + y C2 w C2 = 240 (7) Beet: 20x B w B2 e B2 = 0 (8) Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

7 Formulation Scenario 3(lowest yield) Wheat: 2x W + y W 3 w W 3 = 200 (9) Corn: 2.4x C + y C3 w C3 = 240 (10) Beet: 16x B w B3 e B3 = 0 (11) Higher-priced Beet Limit: w B1, w B2, w B (12) Variable Restriction: All variables 0 (13) Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

8 General formulation of two-stage stochastic program There is a set of decisions to be taken without full information on some random events. These decisions are called first-stage decisions and are usually represented by a vector x. Later full information is received on the realization of some random vector ξ. Then second-stage or corrective actions y are taken. min c T x + E ξ Q(x, ξ) s.t. Ax = b x 0 where Q(x, ξ) = min{q T y ξ Wy ξ = h ξ T ξ x, y ξ 0}, E ξ denotes expectation with respect to ξ. Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

9 A Three-stage Financial Investment problem We currently have $B for investment. There are three types of investment: stock,bond and saving. After a period, the income will vary depending on the market.,,,, B neutral (,, ),,,, Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

10 A Two-stage Financial Investment problem Variables x ξi : the amount of money to invest i investment when the market situation is ξ y: the income at the end of each period α, β, γ: interest rates based on three types of investment, respectively Example : in the period 1, y 1 = (1 + α 1 )x 1 + (1 + β 1 )x 2 + (1 + γ 1 )x 3 y 1 = x 11 + x 12 + x 13 in the period 2, y 11 = (1 + α 11 )x 11 + (1 + β 11 )x 12 + (1 + γ 11 )x 13 Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

11 A Two-stage Financial Investment problem Path-based Scenario tree neutral v v Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7 ) (x01 S 1 x02 1 x x11 1 x12 1 x13 1 ) (x01 S 2 x02 2 x x11 2 x12 2 x13 2 ) (x01 S 3 x02 3 x x11 3 x12 3 x13 3 Scenario 8 Scenario 9 Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

12 Multistage Stochastic Programming Financial Planning Problem We wish to provide for a child s college education M years from now we currently have $B to invest in any of I investments after M years we would like to have exceed a tuition goal of $G we can change investment every v years, so we have T = M/v investment periods after M years, there will be an income of q% of the excess if exceed the goal after M years, there will be a cost of r% of the amount short Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

13 Financial Planning Problem Variables x it : the amount of money to invest in investment i during period t y: the excess money at the end of horizon w: the shortage in money at the end of the horizon Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

14 Scenario Tree S 3 = 1 Scenario 1 S 2 = 1 S 3 = 2 Scenario 2 S 1 = 1 S 2 = 2 S 3 = 1 Scenario 3 S 3 = 2 Scenario 4 S 3 = 1 Scenario 5 S 1 = 2 S 2 = 1 S 3 = 2 Scenario 6 S 2 = 2 S 3 = 1 Scenario 7 S 3 = 2 Scenario 8 Decision Tree for three stage stochastic programming Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

15 Node based formulation We have a vector of variables for each node in the tree This vector corresponds to what our decision would be given the realization of the random variables we have seen so far Index the nodes l = 1, 2,..., L. We will need to know the parent of any node Let A(l) be the ancestor of node l L in the scenario tree Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

16 Node Based Formulation We need to enforce nonanticipativity Define S t s as the set of scenarios that are equivalent to scenario s at time t max s.t. Prob s (Qy s Rw s ) s S x i1 = B (1) i N x il, l L \ 1 (2) C il x ia(l) = i N i N C ia(s) x ia(s) y s + w s = G, s S (3) i N x il 0, i N, l L (4) y s, w s 0, s S (5) where l is a node within a set of nodes L. A(s) and A(l) represent ancestor nodes. S = (s 1, s 2,..., s t ). Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17

17 Scenario Based Formulation max s.t. Prob s (Qy s Rw s ) s S x it = B, for t = 1 (6) i N x its, for t = 2,..., T 1, s S (7) C its x i,t 1,s = i N i N C its x its y s + w s = G, for t = T (8) i N x its = x its, i N, t T, s S, s S t s (9) x its 0, i N, t T, s S (10) y s, w s 0, s S (11) where s is a scenario within a set of scenarios S. There exists S scenarios at stage T. Yi Fang, Yuping Huang (IMSE@WVU) Intro. to Stochastic Programming 11/01/ / 17