Question

A photocopy machine company produces three types of laser printers— the Print Jet, the Print Desk, and the Print Pro— the sale of which earn profits of $60, $90, and $73, respectively. The Print Jet requires 2.9 hours of assembly time and 1.4 hours of testing time. The Print Desk requires 3.7 hours of assembly and 2.1 hours of testing. The Print Pro requires 3 hours of assembly and 1.7 hours of testing. The company wants to ensure that Print Desk constitutes at least 15% of the total production and Print Jet and Print Desk together constitute at least 40% of the total production. There are 3,600 hours of assembly time and 2,000 hours of testing time available for the month. What combination of printers should be produced to maximize profits?

Answer 1

Punt deste Perl plo 73 २.१ 3.7 3 Moderably Testing 2.1 34 20-15 Coe,+ +819) sito ²0.4 (Nitry + 3) Mon 6on, taon, +7323 Assembly 2.qx, +3.7 +385 3600 6 +2.10 +1.703 < 2000 Testing 0.15r, - -0.85 na to. 15 ng so 0.6 n, - 0.60 +0.48 z 20 > It is solved by simplen method 12=287 13 = 683 de 168

Max Z = 60X1 + 90 X2 + 73 X3 subject to 2.9 X1 + 3.7 X2 + 3 X3 < 3600 1.4 Xı + 2.1 X2 + 1.7 X3 S 2000 0.15X1 - 0.85X2 + 0.15x3 50 - 0.6 Xı - 0.6 X2 + 0.4 X3 50 and x1,x2,X3 = 0; The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate 1. As the constraint-1 is of type' S'we should add slack variable Si 2. As the constraint-2 is of type 's' we should add slack variable S2 3. As the constraint-3 is of type' S'we should add slack variable S3 4. As the constraint-4 is of type's' we should add slack variable S4 After introducing slack variables Max Z = 60X1 + 90X2 + 73X3 + 0S1 + 052 +0S3 + 054 subject to 2.9 *1 + 3.7 x2 + 3 X3 + S = 3600 1.4 X1 + 2.1 X2 + 1.7 X3 S2 = 2000 0.15X1 - 0.85X2 + 0.15 X3 + S3 = 0 - 0.6 Xi - 0.6 X2 + 0.4 X3 S4 = 0 and X1, X2, X3, S1, S2, S3, S420 Iteration-1 C 60 90 73 0 0 0 0 MinRatio Xg B Св Хв 21 X2 .83 Si S2 S3 S4 X2 3600 Si 0 3600 2.9 3.7 3 1 0 0 0 3.7 = 972.973 S2 0 2000 1.4 (2.1) 1.7 0 1 0 0 2000 2.1 = 952.381 — S3 0 0 0.15 -0.85 0.15 0 0 1 0 S4 0 0 -0.6 -0.6 0.4 0 0 0 1 Z = 0 Z; 0 0 0 0 0 0 0 Z; - C; -60 -90 1 -73 0 0 0 0

Negative minimum 2; -C;is -90 and its column index is 2. So, the entering variable is xz. Minimum ratio is 952.381 and its row index is 2. So, the leaving basis variable is Sz- .: The pivot element is 2.1. Entering = X2, Departing = S2, Key Element = 2.1 + Rz(new) - Rz(old) = 2.1 + R (new) = R. (old) - 3.7Rz(new) + Rz(new) = Rz(old) + 0.85R2(new) + R.(new) = R4(old) + 0.6R (new) Iteration 2 C 60 90 73 0 0 0 0 B XB xi x2 X3 Si S, S; S4 MinRatio Хв X3 S 0 76.1905 0.4333 0 0.0048 1 -1.7619 0 0 76.1905 0.0048 = 16000 X2 90 952.381 0.6667 1 0.8095 0 0.4762 0 0 952.381 = 1176.4706 0.8095 Sz 0 809.5238 0.7167 0 0.8381 0 0.4048 1 0 809.5238 = 965.9091 0.8381 S4 0 571.4286 -0.2 0 (0.8857) 0 0.2857 0 1 571.4286 0.8857 = 645.1613 - Z = 85714.2857 60 90 72.8571 0 42.8571 0 0 2 Z;-C; 0 0 -0.1429 1 0 42.8571 0 0

Minimum ratio is 645.1613 and its row index is 4. So, the leaving basis variable is S4- .: The pivot element is 0.8857. Entering = x3, Departing = 54, Key Element = 0.8857 + R.(new) = R. (old) + 0.8857 + R, (new) = R(old) - 0.0048R_(new) + R, (new) = RZ (old) - 0.8095R_(new) + Rz(new) = R3(old) - 0.8381R (new) Iteration-3 C 60 90 73 0 0 0 0 MinRatio Хв B Св XB 21 X2 X 3 Si S, S; S4 X1 Si 0 73.1183 (0.4344) 0 0 1 - 1.7634 0 -0.0054 73.1183 = 168.3168 → 0.4344 90 430.1075 0.8495 1 0 0 0.2151 0 -0.914 0.914 430.1075 0.8495 = 506.3291 Sz 0 268.8172 0.9059 0 0 0 0.1344 1 -0.9462 268.8172 0.9059 = 296.7359 X3 73 645.1613 -0.2258 0 1 0 0.3226 0 1.129 Z = 85806.4516 z; 59.9677 90 73 0 42.9032 0 0.1613 Z; - C; -0.03231 0 0 0 42.9032 0 0.1613

Negative minimum 2;-C; is -0.0323 and its column index is 1. So, the entering variable is x1- Minimum ratio is 168.3168 and its row index is 1. So, the leaving basis variable is S. :: The pivot element is 0.4344. Entering = x1, Departing = Si, Key Element = 0.4344 + Ri(new) = R1 (old) - 0.4344 + R2(new) = Rz(old) - 0.8495R(new) + R3(new) = R3(old) - 0.9059R; (new) + R_(new) = R (old) + 0.2258R1 (new) Iteration-4 60 90 73 0 0 0 0 C Xg B Св xi X2 X3 Si S; Sz S4 MinRatio X1 60 168.3168 1 0 0 2.302 -4.0594 0 -0.0124 X2 90 287.1287 0 1 0 -1.9554 3.6634 0 -0.9035 Sz 0 116.3366 0 0 0 -2.0854 3.8119 1 -0.935 73 683.1683 0 0 1 0.5198 -0.5941 0 1.1262 Z = 85811.8812 2 60 90 73 0.0743 42.7723 0 0.1609 Z; -C; 0 0 0 0.0743 42.7723 0 0.1609 Since all Z -0,20