System Planning Lecture 7, F7: Optimization

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1 System Planning 04 Lecture 7, F7: Optimization

2 System Planning 04 Lecture 7, F7: Optimization Course goals Appendi A Content: Generally about optimization Formulate optimization problems Linear Programming (LP) Mied Integer Linear Programming (MILP)

3 Course Goals Short term planning To pass the course, the students should show that they are able to formulate short-term planning problems of hydro-thermal power systems, To receive a higher grade the students should also show that they are able to create specialised models for short-term planning problems, Computation Ability solving short-term planning problems essential Computation, not a goal, not eamined Home assignments give bonus points some are mandatory

4 Optimization Generally (/) Applied mathematics Application: technology, science, economics, etc. Something to be maimized or minimized Profit, cost, energy, speed, losses Given some constraints Physical, technical, economical, legal, etc.

5 Optimization Generally (/) In general form min f( ), subject to g ( ) b h ( ) = c Constraints f objective function optimization variable (multidimensional, e.g. vector) g, h constraints (multidimensional, e.g. vector), variable bounds (like the variable)

6 Optimization Formulating problems (/6). Formulate it verbally:. Think the problem through. Define denotations. Which parameters and variables are needed? 3. Formulate the problem in equation form. That is; define the objective and constraints ( and 3 are interrelated)

7 Optimization Formulating problems (/6) Eample: A corporation owns a number of factories How to deliver items to consumers??

8 Optimization Formulating problems (3/6). Formulate verbally:. How to deliver the goods, while. Minimizing the transport costs? 3. Subject to:.each factory's production capacity limit.fulfil the consumer demand

9 Optimization Formulating problems (4/6). Define denotations:. Indices and parameters:. m factories, factory i has the capacity a i. n customers, customer j demands b j units 3. Transportation costs from factory i to customer j is c ij per unit. Variables. Let ij, i =,..., m, j =,..., n denote the number of units transported from factory i to customer j

10 Optimization Formulating problems (5/6) 3. Formulate the problem mathematically: Objective function Transportation cost: m n i= j= c ij ij Constraints Production capacity: Demand: Variable bounds: n j= m i= ij ai i =... m ij b j j =... n ij 0 i =... m, j =... n

11 Optimization Formulating problems (6/6) The optimization problem: min m i= j= n s.t. a, i =... m j= n m b, j =... n i= ij ij ij ij i j 0, i =... m, j =... n ij c

12 LP Generally (/) Linear problems (LP-problems) A class of optimization problems Linear objective, linear constraints All LPs may be formulated (standard form): T min c subject to A = b 0

13 LP Generally (/) Various types of LP-problems eplained Eamples similar to those in Appendi A We will study: Etreme points Slack variables No feasible solution Inactive constraints Unbounded problem Degenerate problem Flat optimum Dual formulation Solution methods

14 LP Basic Eample (/) The problem: min z = subject to , 0

15 LP Basic Eample (/) Optimum at: =.5 = 0.5 Objective value: z =.5

16 LP Etreme points Corners in feasible space Denoted etreme points Optimum always in etreme point(s) z = 0 z = 30 z =

17 LP Slack variables (/) LP-problem in standard form: min s.t. T c A = b 0 Introduce slack variables All constraints can be equalities

18 LP Slack variables (/) Without slack variables (A b) : 0 0, , 0, 0, = + = + With slack variables (A = b):

19 LP Infeasible problem (/) Feasible space empty Bad formulation min z = s.t , 0 Added constraint

20 LP Infeasible problem (/) 4 (a) + + (c) Feasible space following (c) Feasible following (a) & (b) (b)

21 LP Inactive Constraints (/) Some constraints does not constrain Given the objective min z = subject to , 0 Added constraint

22 LP Inactive Constraints (/) Active constraints z = 40 3 z = 30 z = Inactive constraints (Some algorithms like it!)

23 LP Unbounded Problem (/) If no constraint is active The objective is unbounded z min z = Objective function modified subject to , 0

24 LP Unbounded Problem (/) z = -3 z = -4 z = -5 min z

25 LP Degenerate solution (/) More than one etreme point optimal All linear combination of these optimal min z = subject to , 0 New objective function

26 LP Degenerate solution (/) Line between etreme points optimal

27 LP Flat optimum (/) Etreme points with similar objective values min z = subject to , 0 New objective function

28 LP Flat optimum (/)

29 LP Duality (/5) Any LP problem (primal problem) has a corresponding dual problem Primal problem: T min c subject to A = b 0 Dual problem: T ma b λ λ subject to A T λ c λ 0 λ is said to be a dual variable

30 LP Duality (/5) Theorem (strong duality): If the primal problem has an optimal solution, then also the dual problem has an optimal solution. The objective values of these two problems equals.

31 LP Duality (3/5) min subject to T c A = b 0 One dual variable for each constraint! Question: What dual variables technically describe? Answer: Marginal value of the corresponding constraint i.e. objective value s dependence on RHS

32 LP Duality (4/5) In optimum: λ > 0 (active) λ > 0 (active) λ 3 = 0 (inactive)

33 LP Duality (5/5) Dual variables: λ Small perturbations in the right-hand-side, b Changes in the objective value, z: z = λ T b

34 MILP Generally Mied Integer Linear Programming problems Class of optimization problems Linear objective functions and linear constraints Some variables may be integers min z = such that A = b 0 T c { } { }, i,,..., m i, i m+, m+,..., n i i = n

35 MILP Eample Optimum at: = (.5) = (0.5) z = 40 z = 30 z = 0 Optimal objective: z = 5 (.5)

36 MILP Solution Integer variables are easily implemented Integer problems generally hard to solve Computation time may increase eponentially If possible, avoid integer variables! Special case integer variable: binary variable {0,}

37 MILP Applying Binary Variable (/6) Minimize cost buying certain product Variable, 0: amount of the product, Decreasing marginal cost, e.g.: Economies of scale (outside this course, square root) Volume discount Cost [SEK] b Amount consumed

38 MILP Applying Binary Variable (/6) Split up in two variables = + b Cost [SEK] z = c + c c > c b Amount consumed

39 MILP Applying Binary Variable (3/6) Segment cheaper Prevent being positive as long < b Typically done by using binary variable Cost [SEK] b Amount consumed

40 MILP Applying Binary Variable (4/6) Introduce the binary variable s And the constraints s 0 + Ms 0 b Note: In real-life, M not arbitrary Cost [SEK] b Amount consumed s=0 s=

41 MILP Applying Binary Variable (5/6) Sizing of parameter M Not a part of the course subject to: ( m) Or, simpler M = min m, ( ) min z = + m + 0 m> 0 m ( ) M = min Cost [SEK] b s=0 s= Amount consumed

42 If s=0: 0 0 0, 0 = If s=: M M b b b + = 0 0, Cost [SEK] Amount consumed b s=0 s= Ms s b MILP Applying Binary Variable (6/6)

43 End of lecture 7 Net time short-term planning

Sensitivity Analysis and Duality in LP

Sensitivity Analysis and Duality in LP Xiaoxi Li EMS & IAS, Wuhan University Oct. 13th, 2016 (week vi) Operations Research (Li, X.) Sensitivity Analysis and Duality in LP Oct. 13th, 2016 (week vi) 1 /