1 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. CHAPTER 15 15.1 (a) Define xa = amount of product A produced, and xb = amount of product B produced. The objective function is to maximize profit, 45 20abPx x Subject to the following constraints 20 5 9500 {raw materials}abxx 0.04 0.12 40 {production time}abxx 550 {storage}abxx , 0 {positivity}abxx (b) To solve graphically, the constraints can be reformulated as the following straight lines 1900 4 {raw materials}baxx 333.3333 0.333333 {production time}baxx 550 {storage}baxx The objective function can be reformulated as (1/ 20) 2.25baxPx The constraint lines can be plotted on the xa-xb plane to define the feasible space. Then the objective function line can be superimposed for various values of P until it reaches the boundary. The result is P 22,250 with xa 450 and xb 100. Notice also that material and storage are the binding constraints and that there is some slack in the time constraint. 01002003000 200 400 600xboptimumxatimestoragematerialP = 22,500P = 15,000P = 7,500 (c) The simplex tableau for the problem can be set up and solved as Basis P xa xb S1 S2S3Solution Intercept P 1 -45 -20 0 0 0 0 S1 0 20 5 1 0 0 9500 4752 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. S2 0 0.04 0.12 0 1 0 40 1000 S3 0 1 1 0 0 1 550 550 Basis P xa xb S1 S2S3Solution Intercept P 1 0 -8.75 2.25 0 0 21375 xa 0 1 0.25 0.05 0 0 475 1900 S2 0 0 0.11 -0.002 1 0 21 190.9091 S3 0 0 0.75 -0.05 0 1 75 100 Basis P xa xb S1 S2S3Solution Intercept P 1 0 0 1.666667 0 11.66667 22250 xa 0 1 0 0.066667 0 -0.33333 450 S2 0 0 0 0.005333 1 -0.14667 10 xb 0 0 1 -0.06667 0 1.333333 100 (d) An Excel spreadsheet can be set up to solve the problem as The formulas in column D are The Solver can be called and set up as Before depressing the Solve button, depress the Options button and check the boxes to “Assume Linear Model” and “Assume Non-Negative.”3 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. The resulting solution is In addition, a sensitivity report can be generated as (e) The high shadow price for storage from the sensitivity analysis from (d) suggests that increasing storage will result in the best increase in profit. 15.2 (a) The LP formulation is given by 12 3Maximize 150 175 250Zxx x {Maximize profit}4 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. subject to 12371115 154xx x {Material constraint} 12 310 8 12 80xx x {Time constraint} 19x {“Regular” storage constraint} 26x {“Premium” storage constraint} 35x {“Supreme” storage constraint} 123,, 0xxx {Positivity constraints} (b) The simplex tableau for the problem can be set up and solved as Basis Z x1 x2 x3S1S2S3S4S5Solution Intercept Z 1 -150 -175 -250 0 0 0 0 0 0 S1 0 7 11 15 1 0 0 0 0 154 10.2667 S2 0 10 8 12 0 1 0 0 0 80 6.66667 S3 0 1 0 0 0 0 1 0 0 9 S4 0 0 1 0 0 0 0 1 0 6 S5 0 0 0 1 0 0 0 0 1 5 5 Basis Z x1 x2 x3S1S2S3S4S5Solution Intercept Z 1 -150 -175 0 0 0 0 0 250 1250 S1 0 7 11 0 1 0 0 0 -15 79 7.18182 S2 0 10 8 0 0 1 0 0 -12 20 2.5 S3 0 1 0 0 0 0 1 0 0 9 S4 0 0 1 0 0 0 0 1 0 6 6 x3 0 0 0 1 0 0 0 0 1 5 Basis Z x1 x2 x3S1S2S3S4S5Solution Intercept Z 1 68.75 0 0 0 21.88 0 0 -12.5 1687.5 S1 0 -6.75 0 0 1 -1.375 0 0 1.5 51.5 34.3333 x2 0 1.25 1 0 0 0.125 0 0 -1.5 2.5 -1.66667 S3 0 1 0 0 0 0 1 0 0 9 S4 0 -1.25 0 0 0 -0.125 0 1 1.5 3.5 2.33333 x3 0 0 0 1 0 0 0 0 1 5 5 Basis Z x1 x2 x3S1S2S3S4S5Solution Z 1 58.3333 0 0 0 20.83 0 8.33 0 1716.7 S1 0 -5.5 0 0 1 -1.25 0 -1 0 48 x2 0 0 1 0 0 0 0 1 0 6 S3 0 1 0 0 0 0 1 0 0 9 S5 0 -0.8333 0 0 0 -0.083 0 0.67 1 2.3333 x3 0 0.83333 0 1 0 0.083 0 -0.67 0 2.6667 (c) An Excel spreadsheet can be set up to solve the problem as5 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. The formulas in column E are The Solver can be called and set up as The resulting solution is In addition, a sensitivity report can be generated as6 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior
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