Connecting Calculus to Linear Programming

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Miles Hutchinson
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1 Connecting Calculus to Marcel Y., Ph.D. Worcester Polytechnic Institute Dept. of Mathematical Sciences July 27

2 Motivation Goal: To help students make connections between high school math and real world applications of mathematics. programming is based on a simple idea from calculus.

3 Motivation Goal: To help students make connections between high school math and real world applications of mathematics. programming is based on a simple idea from calculus. We can connect calculus to real-world industrial problems.

4 Motivation Goal: To help students make connections between high school math and real world applications of mathematics. programming is based on a simple idea from calculus. We can connect calculus to real-world industrial problems. We ll explore the some basic principles of linear programming and some modern-day applications.

5 y Continuous Functions on Closed Intervals x a b What can we say about a continuous function on a closed interval?

6 y Continuous Functions on Closed Intervals x a b Theorem If f is continuous on [a, b] then f attains both an absolute minimum value and an absolute maximum value on [a, b].

7 y Continuous Functions on Closed Intervals Theorem x a If f is continuous on [a, b] then f attains both an absolute minimum value and an absolute maximum value on [a, b]. Further these occur at critical points of f or endpoints of [a, b]. b

8 y Case Theorem If f is continuous linear on [a, b] then f attains both an absolute minimum value and an absolute maximum value on [a, b], and these occur at endpoints of [a, b] x a b

9 z y.2..2 x What does this mean for a continuous function f (x, y) on a closed and bounded plane region R?

10 z Theorem y If f (x, y) is continuous on closed and bounded plane region R then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur either at critical points of f in the interior of R or on the boundary of R x.4.6.8

11 z y.2..2 x Theorem If f (x, x 2,..., x n ) is continuous on closed and bounded region R R n then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur either at critical points of f in the interior of R or on the boundary of R.

12 z Theorem y.2. If f (x, x 2,..., x n ) is linear on closed and bounded region R R n defined by a system of linear constraints then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur on the boundary of R..2.3 x

13 z y.2..2 x Theorem If f (x, x 2,..., x n ) is linear on closed and bounded region R R n defined by a system of linear constraints then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur on corner points of the boundary of R.

14 z y..2 x Example: A possible feasible region f (x, y, z) = ax + by + cz subjuect to 5 linear constraints. A solution to minimize f over this region will occur at a corner.

15 The was developed by George Dantzig in 947. programming synonymous with optimization. Algorithm to traverse the corner points of the feasible polyhedron for a linear programming problem to find an optimal feasible solution. Standard form for a linear programming problem: min x T c such that Ax b and x. () x, c R n, b R m, A R m n - n variables and m constraints.

16 min x T c such that Ax b and x. For each constraint (row i of A), add a new slack variable y i. [ ] x New system: min x T c such that [A + I ] = b, y x, y. Underdetermined (more variables than equations). A + I The slack now in the system is the key. Each slack variable is associated with a constraint boundary

17 Simplex algorithm: [ ] x Split entries of into 2 subsets: y Basic variables: x B R m (non-zero valued) Non-basic variables: x N R n (zero valued) Represents a corner point of the feasible region. If not optimal, move to an adjacent corner point by swapping one entry in x B with one entry in x N and re-solving the system of equations.

18 z Simplex Geometry y..2 x Figure: Feasible Region Example: min f (x, x 2, x 3 ) = ax + bx 2 + cx 3 subject to 5 linear constraints.

19 z Simplex Geometry y..2 x Example: min f (x, x 2, x 3 ) = ax + bx 2 + cx 3 subject to 5 linear constraints. Choose initial basis to correspond to the origin.

20 z Simplex Geometry y..2 x Example: min f (x, x 2, x 3 ) = ax + bx 2 + cx 3 subject to 5 linear constraints. If the objective function is not optimal, move to a better adjacent corner.

21 z Simplex Geometry y..2 x Example: min f (x, x 2, x 3 ) = ax + bx 2 + cx 3 subject to 5 linear constraints. Repeat until the objective function is minimized (no remaining adjacent corners will reduce it).

22 Application: Shipping Network

23 Shipping Problem K warehouses: S, S 2,..., S K

24 Shipping Problem K warehouses: S, S 2,..., S K L retail locations: D, D 2,..., D L

25 Shipping Problem K warehouses: S, S 2,..., S K L retail locations: D, D 2,..., D L Cost to ship x amount of product from S i to D j is c ij x

26 Shipping Problem K warehouses: S, S 2,..., S K L retail locations: D, D 2,..., D L Cost to ship x amount of product from S i to D j is c ij x Warehouse S i can supply s i amount of product

27 Shipping Problem K warehouses: S, S 2,..., S K L retail locations: D, D 2,..., D L Cost to ship x amount of product from S i to D j is c ij x Warehouse S i can supply s i amount of product Retail location D j demands d j amount of product

28 Shipping Problem K warehouses: S, S 2,..., S K L retail locations: D, D 2,..., D L Cost to ship x amount of product from S i to D j is c ij x Warehouse S i can supply s i amount of product Retail location D j demands d j amount of product Problem: Design a shipping schedule that satisfies demand at all retail locations at a minimal cost.

29 Shipping Problem K warehouses: S, S 2,..., S K L retail locations: D, D 2,..., D L Cost to ship x amount of product from S i to D j is c ij x Warehouse S i can supply s i amount of product Retail location D j demands d j amount of product Problem: Design a shipping schedule that satisfies demand at all retail locations at a minimal cost. Example: As of July 27, Target Corp. has 37 distribution centers and, 82 stores.

30 Shipping Network

31 Shipping Network

32 Shipping Problem Mathematical Model Constraints Objective x ij - amount of product shipped from S i to D j

33 Shipping Problem Mathematical Model Constraints x ij - amount of product shipped from S i to D j Amount leaving S i : x i + x i2 + + x il s i Objective

34 Shipping Problem Mathematical Model Constraints x ij - amount of product shipped from S i to D j Amount leaving S i : x i + x i2 + + x il s i Amount entering D j : x j + x 2j + + x Kj = d j Objective

35 Shipping Problem Mathematical Model Constraints x ij - amount of product shipped from S i to D j Amount leaving S i : x i + x i2 + + x il s i Amount entering D j : x j + x 2j + + x Kj = d j Objective K L Total cost of shipping schedule: c ij x ij i= j=

36 Shipping Problem Mathematical Model Constraints x ij - amount of product shipped from S i to D j Amount leaving S i : x i + x i2 + + x il s i Amount entering D j : x j + x 2j + + x Kj = d j Objective K L Total cost of shipping schedule: c ij x ij i= j= Minimize cost of shipping such that demand is satisfied

37 Shipping Problem Mathematical Model Constraints x ij - amount of product shipped from S i to D j Amount leaving S i : x i + x i2 + + x il s i Amount entering D j : x j + x 2j + + x Kj = d j Objective K L Total cost of shipping schedule: c ij x ij i= j= Minimize cost of shipping such that demand is satisfied Target example: 37, 82 = 66, 674 variables

38 Shipping Problem Mathematical Model Amount leaving S i : x i + x i2 + + x il s i A = I I I K L min c ij x ij such that Ax = i= j= [ s d ], x.

39 Shipping Problem Mathematical Model Amount leaving S i : x i + x i2 + + x il s i Amount entering D j : x j + x 2j + + x Kj = d j A =... I I I K L min c ij x ij such that Ax = i= j= [ s d ], x.

40 TV Project - MA 323

41 TV Project - MA 323

42 TV Project - MA 323

43 References, Marcel., MA 323 Lecture Notes, WPI Department of Mathematical Sciecnes, Griva, Igor, Stephen G. Nash, and Ariela Sofer., and Nonlinear, Second Edition. SIAM, Hillier, Frederick and Gerald Lieberman., Introduction to Operations Research, Tenth Edition. McGraw Hill Education, 25.

Math Models of OR: Some Definitions

Math Models of OR: Some Definitions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Some Definitions 1 / 20 Active constraints Outline 1 Active constraints