1. EMIS 33601 Lecture 3 – pages 39 - 57 Pi Hybrids Model On Page 39 FacilitiesSales Regions 1 2 3 4 OK TX MI AR LA TN 4x6=24

2. EMIS 33602 Multiple Commodities h in the set {a,b,c,d,e} For commodity a we have For commodity b we have 4x6x5 = 120 arcs!

3. EMIS 33603 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 p fh – cost/bag for producing corn type h at facility f (4x5=20) u f – capacity of facility f (bushels) (4) a h – bushels of corn that must be processed to produce 1 bag of corn type h (bushels/bag) (5)

4. EMIS 33604 More Constants d hr – demand for h in region r (5x6=30) (bags) s fhr – cost to ship one unit of product h from facility f to sales region r (4x5x6=120) ($/bag)

5. EMIS 33605 Variables x fh – bags of corn type h produced at facility f (4x5=20) y fhr – bags of corn of type h shipped from facility f to sales region r (4x5x6 = 120) Note: There are 140 unknowns in this problem.

6. EMIS 33606 Constraints Capacity Of Facilities (4)  h  H a h x fh < u f, for all f  F Demands At Sales Regions (5x6=30)  f  F y fhr = d hr, for all h  H and r  R

7. EMIS 33607 More Constraints Balance (4x5=20)  r  R y fhr = x fh, for all f  F, h  H Nonnegativity (140) x fh > 0, for all f  F, h  H y fhr > 0, for all f  F, h  H, r  R

8. EMIS 33608 Objective Function Minimize  f  F  h  H p fh x fh +  f  F  h  H  r  R s fhr y fhr Production Cost Shipping Cost

9. EMIS 33609 AMPL Model For Pi Problem # Define Sets set F;set H;set R; #Define Constants param p {F,H}; param u {F};param a {H}; param d {H,R};param s {F,H,R};

10. EMIS 336010 AMPL Model Continued #Define Variables var 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];

11. EMIS 336011 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];

12. EMIS 336012 Section 2.4 Linear & Nonlinear Functions General Optimization Problem minimize f(x) subject to g i (x) < b i, for all i Linear Function Is Simply:  i=1..n a i x i = a 1 x 1 + a 2 x 2 + …+ a n x n

13. 13 Nonlinear Optimization Everything That Is Not Linear Is Nonlinear One Nonlinear Function Is The Log Function XLog(X+1) 00 101.04 201.32 301.49 401.61 501.71 1002.00

14. EMIS 336014 E-Mart Example On Page 50 Subscripts g – 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) Constants p g – denotes the profit percentage for product g

15. EMIS 336015 More E-Mart s gc – denotes the increase in sales constant for product g using advertising type c b – denotes the advertising budget Variables x c – denotes the amount of money to spend on advertising type c

16. EMIS 336016 E-Mart Continued Constraints Budget Restriction  c=1,2,3 x c < b Nonnegativity x c > 0, for c=1,2,3 Objective maximize  g=1,2,3,4 p g  c=1,2,3 s gc log(x c +1)

17. EMIS 336017 MINOS Can Solve But Not CPLEX Solution: x 1 = 34148.1x 2 = 34542.4x 3 = 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.

18. EMIS 336018 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

19. EMIS 336019 Bethlehem Ingot Mold Subscripts i – mold design number ( i=1,2,3,4) j – product number (j=1,..,6) Sets M j – set of molds that can be used to produce product j That is M 1 = {1,2,3}, M 2 = {2,3,4}, …, M 6 = {2,4}

20. EMIS 336020 Constants P – max number of molds that can be used C ji – waste produced when mold i is used to create product j (j = 1,…,6; i  M j ) Variables y i = 1, if mold type i is used = 0, otherwise x ji = 1, if mold type i is used to produce product j = 0, otherwise

21. EMIS 336021 Constraints sum { i in {1..4}} y i < P (max # molds that can be used) sum {i in M j } x ji = 1; j = 1,..,6 (each product must be assigned to 1 mold) x ji < y i ; j=1,…,6; i  M j (products can be made from mold i only if mold i is selected for use)

22. EMIS 336022 Objective Minimize sum { j in {1..6}} sum {i in M j } c ji x ji That is, minimize scrap.