EMIS 3360 1

Lecture 3 – pages 39 - 57

Pi Hybrids Model On Page 39

Facilities Sales Regions

1

2

3

4

OKTX

MI

AR

LATN

4x6=24

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Multiple Commodities

h in the set {a,b,c,d,e}

For commodity a we have For commodity b we have

4x6x5 = 120 arcs!

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Subscripts & Sets

f – Facilities f F = {1,2,3,4} h – Corn Type h H = {a,b,c,d,e} r – Sales Region r R = {OK,TX,MI,AR,LA,TN}

Constants

pfh – cost/bag for producing corn type h at facility f (4x5=20)

uf – capacity of facility f (bushels) (4)

ah – bushels of corn that must be processed to produce 1 bag of corn type h (bushels/bag) (5)

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More Constants

dhr – demand for h in region r (5x6=30) (bags)

sfhr – cost to ship one unit of product h from facility f to sales region r (4x5x6=120) ($/bag)

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Variables

xfh – bags of corn type h produced at facility f (4x5=20)

yfhr – bags of corn of type h shipped from facility f to sales region r (4x5x6 = 120)

Note: There are 140 unknowns in this problem.

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Constraints

Capacity Of Facilities (4)

hH ahxfh < uf, for all f F

Demands At Sales Regions (5x6=30)

fF yfhr = dhr, for all h H and r R

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More Constraints

Balance (4x5=20)

rR yfhr = xfh, for all f F, h H

Nonnegativity (140)

xfh > 0, for all f F, h H

yfhr > 0, for all f F, h H, r R

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Objective Function

Minimize fF hH pfhxfh + fF hH rR sfhryfhr

Production Cost Shipping

Cost

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AMPL Model For Pi Problem

# Define Setsset F; set H; set R;

#Define Constantsparam p {F,H}; param u {F}; param a {H};

param d {H,R}; param s {F,H,R};

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AMPL Model Continued

#Define Variablesvar x {F,H} >= 0; var y {F,H,R} >= 0;

#Define Constraints

subject to CoF {f in F}: sum {h in H} a[h]*x[f,h] <= u[f];

subject to DaR {h in H, r in R}:sum {f in F} y[f,h,r] = d[h,r];

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AMPL Model Continued

subject to B {f in F, h in H}:sum {r in R} y[f,h,r] = x[f,h];

#Define Objective Function

minimize cost:sum {f in F, h in H} p[f,h]*x[f,h] +sum {f in F, h in H, r in R} s[f,h,r]*y[f,h,r];

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Section 2.4 Linear & Nonlinear Functions

General Optimization Problem

minimize f(x) subject to gi(x) < bi, for all i

Linear Function Is Simply:

i=1..n aixi = a1x1 + a2x2 + …+ anxn

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Nonlinear Optimization

Everything That Is Not Linear Is NonlinearOne Nonlinear Function Is The Log Function

X Log(X+1)

0 0

10 1.04

20 1.32

30 1.49

40 1.61

50 1.71

100 2.00

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E-Mart Example On Page 50

Subscriptsg – denotes the product type (g=1,2,3,4)c – advertising type (c=1,2,3)

Note: Advertising has decreasing returns (a nonlinear return function involving a log)

Constantspg – denotes the profit percentage for product

g

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More E-Mart

sgc – denotes the increase in sales constant for product g using advertising type c

b – denotes the advertising budget

Variablesxc – denotes the amount of money to

spend on advertising type c

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E-Mart Continued

ConstraintsBudget Restrictionc=1,2,3 xc < b

Nonnegativity xc > 0, for c=1,2,3

Objectivemaximize g=1,2,3,4 pg c=1,2,3 sgc log(xc+1)

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MINOS Can Solve But Not CPLEX

Solution:x1 = 34148.1 x2 = 34542.4 x3 =

31309.6

MINOS can solve linear and nonlinear problems

CPLEX can solve linear, quadratic, and linear integer problems. We use CPLEX in all of our research models.

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Integer Programming Section 2.5

The variables must assume integer values. There are two types of integer variables, binary (0,1) and standard integer.

var X binary; implies that X is either 0 or 1

var Y integer >= 4, <= 10; implies that Y is one of the following values: 4,5,6,7,8,9,10

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Bethlehem Ingot Mold

Subscripts i – mold design number ( i=1,2,3,4) j – product number (j=1,..,6)

SetsMj – set of molds that can be used to produce product j

That is M1 = {1,2,3}, M2 = {2,3,4}, …, M6 = {2,4}

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ConstantsP – max number of molds that can be used

Cji – waste produced when mold i is used to create product j

(j = 1,…,6; i Mj)

Variablesyi = 1, if mold type i is used

= 0, otherwisexji = 1, if mold type i is used to produce product j

= 0, otherwise

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Constraintssum { i in {1..4}} yi < P

(max # molds that can be used)

sum {i in Mj} xji = 1; j = 1,..,6

(each product must be assigned to 1 mold)

xji < yi; j=1,…,6; i Mj

(products can be made from mold i only if mold i is selected for use)

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Objective

Minimize sum { j in {1..6}} sum {i in Mj} cji xji

That is, minimize scrap.