AMS341 HW1. Graphically solving this LP we find the optimal solution to be z = $10,000, W = C = 20 acres.

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1 AMS341 HW1 p.63 #6 Let W = acres of wheat planted and C = acres of corn planted. Then the appropriate LP is max = 200W + 300C st W + C 45 (Land) 3W + 2C 100 (Workers) 2W + 4C 120 (Fertilizer) W, C 0 Graphically solving this LP we find the optimal solution to be z = $10,000, W = C = 20 acres. p. 68 #2. AB is 8x 1 + 2x 2 = 16. CD is 5x 1 + 2x 2 = 12. Dotted line is z = 4x 1 + x 2 = 4. Feasible region is bounded by AEDF. Since isoprofit line is parallel to AE, entire line segment AE is optimal. Thus we have alternative or multiple optimal solutions. #3. AB is x 1 - x 2 = 4. AC is x 1 + 2x 2 = 4. Feasible region is bounded by AC and infinite line segment AB. Dotted line is isoprofit line z = 0. To increase z we move parallel to isoprofit line in an upward and `leftward' direction. We will never entirely lose contact with the feasible region, so we have an unbounded LP (Case 4).

2 #7. If an LP's feasible region is bounded, then the LP must have at least one extreme point that is optimal. Thus if an LP has an optimal solution, we can find an optimal solution to the LP by considering only the LP's extreme points. # 8. The LP is infeasible, because no points can satisfy the last two constraints. p. 71 #4. Let X1=Number of days spent in mine 1 and X2=Number of days spent in mine 2. Then appropriate LP is min z=x1+x2 s.t. 2X1+ X2 12 2X1+3X2 18 X1 0 X2 0 Graphically we find the optimal solution to be X1=4.5 days X2=3 days, z = 7.5 days. p.76: #6. Let Xi=Number of policemen who work for 12 hours and begin during Shift i and Yi=Number of policemen who work for 18 hours and begin during shift i. Beginning times for each shift are Shift 1=12AM, Shift 2=6 AM, Shift 3=12 PM, Shift 4=6 PM Then appropriate LP is min z=48(x1+x2+x3+x4)+84(y1+y2+y3+y4) s.t. X1+X4+Y1+Y3+Y4 12 X1+X2+Y1+Y2+Y4 8 X2+X3+Y3+Y1+Y2 6 X3+X4+Y2+Y4+Y3 15 All variables 0

3 AMS341 HW2 p.92: 2. Let x9j = Pounds of Grade 9 oranges used in orange juice. x9b = Pounds of Grade 9 oranges used in bags. x6j = Pounds of Grade 6 oranges used in orange juice. x6b = Pounds of Grade 6 oranges used in bags. max z =.45(x9J + x6j)+.30(x9b + x6b) s.t. x9j + x9b 100,000(Grade 9 Const.) x6j + x6 120,000(Grade 6 Const.) (1) (9x9J + 6x6J)/(x9J + x6j) 8 (OJ at least 8) (2) (9x9B + 6x6B)/(x9B + x6b) 7 (Bags at least 7) All variables 0 (1) Should be replaced by 9x9J + 6x6J 8x9J + 8x6J or x9j - 2x6J 0. (2) Should be replaced by 9x9B + 6x6B 7x9B + 7x6B or 2x9B - x6b Let xi = amount invested in bond i(we assume entire $1,000,000 will be invested since all investments are always profitable) max z =.13x1/1,000, x2/1,000, x3/1,000, x4/1,000,000 st.06x1 +.08x2 +.10x3 +.09x4.08(x1 + x2 + x3 + x4) x1.4(x1 + x2 + x3 + x4) x2.4(x1 + x2 + x3 + x4) x3.4(x1 + x2 + x3 + x4) x4.4(x1 + x2 + x3 + x4 ) x1 + x2 + x3 + x4 = 1,000,000 3x1 + 4x2 + 7x3 + 9x4 6(x1 + x2 + x3 + x4) x1, x2, x3, x4 0 p. 98: #2. Let UT = Unfinished tables produced. FT = Finished tables produced. UC = Unfinished chairs produced. FC = Finished Chairs produced. W = Wood purchased (in board feet). max z = 70UT+140FT+60UC+110FC- W s.t.40(ft+ut)+30(uc+fc) W(Wood Used Wood Purchased) W 40,000(Limitation on Wood Purchased) 2UT+2UC+5FT+4FC 6,000(Labor Used Labor Available) All Variables 0. p. 104: 4. Let XAij = units of product A produced during month i on line j XBij = units of product B produced during month i on line j IAt = units of product A in inventory at end of month t IBt = units of product B in inventory at end of month t Note: i = 1 is March, i = 2 is April

4 Then a correct formulation is min z =.75XA XA XA XA XB XB XB XB IA1 +.20IA2 +.20IB1 +.20IB2 s.t..12xb XA XA XB XA XB XA XB XA12 + IA1 - XA11 = XA21 - XA22 - IA1 +IA2 = XB11 - XB12 + IB1 = XB21 - XB22 - IB1 + IB2 = IA IB All variables 0 p. 109: 8. Let BSt = ending month t Saks balance, BBt = ending month t Bloomingdale s balance, BMt = ending month t Macy s balance. Let PSt = Month t Saks payment, PBt = Month t Bloomingdale s payment, PMt = Month t Macy s payment t 36 MIN = ( PSt + PBt + PMt) t= 1 st PBt + PSt + PMt<=5000 (t= 1,2, 60) BS0 = 20000, BS0 = 50000, BM0 = 40000, BS36=BM36=BS36=0, For t = 1, 2, 35 BSt = BSt- 1- (PSt -.005BSt- 1) BBt = BBt- 1 (PBt -.01BBt- 1) BMt = BMt- 1 (PMt -.015BMt- 1) All variables >=0

5 HW 3 AMS 341 p. 139: 1.From Figure 2 of Chapter 3 we see that the extreme points of the feasible region are Basic Feasible Solution H = (0, 0) s 1 = 100, s 2 = 80, s 3 = 40 x 1 = x 2 = x 3 = 0 E = (40, 0) x 1 = 40, s 1 = 20, s 2 = 40 x 2 = x 3 = s 3 = 0 F = (40, 20) x 1 = 40, x 2 = 20, s 2 = 20 x 3 = s 1 = s 3 = 0 G = (20, 60) x 1 = 20, x 2 = 60, s 3 = 20 x 3 = s 1 = s 2 = 0 D = (0, 80) s 1 = 20, x 2 = 80, s 3 = 40 s 2 = x 1 = x 3 = 0p.149: 3. z x 1 x 2 x 3 s 1 s 2 s 3 RHS Ratio * Enter x 1 in row / None * Enter x 2 in row /2 0 3/2 1/ /2 0 1/2 1/ /2 0-1/2 1/2 5 This is an optimal tableau with optimal solution z = 25, s 1 = 10, x 1 = 15, x 2 = 5, s 2 = s 3 = 0.

6 p.151: #4 Z X1 X2 S1 S2 RHS X1 enters the basis. We arbitrarily choose to enter X1 in row 2. The resulting optimal tableau follows: Z X1 X2 S1 S2 RHS The LP s optimal solution is Z = -9, X1=3, X2=0.

7 AMS 341 HW 4 p. 154:#5 5. Z X1 X2 S1 S2 RHS We now arbitrarily choose to enter X1 into basis. X1 enters basis in Row 1 yielding following optimal tableau. Z X1 X2 S1 S2 RHS This tableau yields the optimal solution Z = 12, X1 = 6, X2=0. Pivoting X2 into the basis yields the alternative optimal solution Z=12, X1=0, X2=6. All optimal solutions are of the form c(1 st optimal solution) + (1-c)(2 nd optimal solution) where 0<=c<=1. This shows all optimal solutions are of form Z=12, X1= 6c, X2=6-6c, 0<=c<=1. p. 158:#3 From the given tableau we find that for x 2>=0, points of the form z = 2x 2, x 1 = 0, x 3 = 3 + x 2, x 4 = 4 are feasible. By letting x 2 grow large, we see that the LP is unbounded. p. 178: #4 Adding excess and artificial variables we obtain min z = 3x 1 + Ma 1 + Ma 2 s.t. 2x 1 + x 2- e 1 + a 1 = 6 3x 1 + 2x 2 + a 2 = 4 Eliminating a 1 and a 2 from z- 3x 1- Ma 1- Ma 2 = 0 yields z + (5M- 3)x 1 + 3Mx 2- Me 1 = 10M. The simplex now yields z x 1 x 2 e 1 a 1 a 2 RHS 1 5M- 3 3M - M M (6- M)/3 - M 0 (3-5M)/3 10M/3 + 4 This optimal tableau has a1 positive 0 0-1/ /3 10/3 and so the original problem is infeasible 0 1 2/ /3 4/3

8 p. 256 (Section 5 Review) 5. Let p A = price of ale and p B = price of beer 5a. Current basis remains optimal if isoprofit line has slope between malt constraint and corn slope. Thus current basis remains optimal if p A/ or 25 p A b. Current basis remains optimal if /p B -.5 or 20 p B 80. 5c. Optimal solution is where corn and malt constraints are binding. If we change amount of available corn we lose feasibility if corn constraint moves below (20, 0) (using 20 bushels of corn) or moves beyond hops- malt intersection (10, 20) (using 50 pounds of corn). Thus current basis remains optimal for 20 Corn availability 50. If 40 + bushels of corn are available, then new solution is where A + 2B = 40 + and 2A + B = 40, or A = 40/3 - /3 and B = 40/3 + 2 /3, and z = Thus corn shadow price is $20. 5d. Current basis remains optimal until hops constraint is moved in beyond corn- malt intersection of (40/3, 40/3). At this point 80/3 pounds of hops are used. Thus current basis remains optimal for Hops availability 80/3. If 30 + pounds of hops are available, decision variables and z- value remain unchanged, so shadow price of hops constraint is $0. p (Chap6 Review) 4. a ( ) x 88, b. $.88 c. $1.20 d. profit increases by 2 * 20; (SKIP part e) p. 301 (Section 6.5) 4. max z = 6x 1 + 8x 2 s.t. x 1 + x 2 4 2x 1 - x 2 2 2x 2 = - 1 x 1 0 x 2 u.r.s.