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RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 400.000000 400.000000 100.000000 3 400.000000 133.333328 200.000000 THE TABLEAU ROW (BASIS) X1 X2 X3 X4 SLK 2 1 ART 6.000 0.000 24.000 0.000 18.000 2 X2 -0.600 1.000 1.700 0.000 0.400 3 Х4 0.900 0.000 0.200 1.000 -0.100 ROW SLK 3 0.000 7200.000 2 -0.300 40.000 3 0.200 40.000 1.1 What is the optimal objective function value? 1.2 How many units of X1, X2, X3 and X4 should be produced in order to maximise profits? Or give the optimal product mix from the printout. 1.3 What is the status of the variables. Are they basic or non-basic? 1.4 Give the reduced costs of the non-basic variables. 1.5 Assume that 20 units of X3 have been produced by mistake. What is the resulting decrease in profit? 1.6 Which constraints are binding? 1.7 What will happen to the objective function if the right-hand side of constraint one is increased by one unit? 1.8 In what range can the profit margin per unit of X1 vary without changing the optimal basis? 1.9 What is the marginal value of increasing the production capacity of Workshop 1 and Workshop 2? 1.10 In what range can the capacity of Workshop 1 vary without changing the optimal basis?

RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 400.000000 400.000000 100.000000 3 400.000000 133.333328 200.000000 THE TABLEAU ROW (BASIS) X1 X2 X3 X4 SLK 2 1 ART 6.000 0.000 24.000 0.000 18.000 2 X2 -0.600 1.000 1.700 0.000 0.400 3 Х4 0.900 0.000 0.200 1.000 -0.100 ROW SLK 3 0.000 7200.000 2 -0.300 40.000 3 0.200 40.000 1.1 What is the optimal objective function value? 1.2 How many units of X1, X2, X3 and X4 should be produced in order to maximise profits? Or give the optimal product mix from the printout. 1.3 What is the status of the variables. Are they basic or non-basic? 1.4 Give the reduced costs of the non-basic variables. 1.5 Assume that 20 units of X3 have been produced by mistake. What is the resulting decrease in profit? 1.6 Which constraints are binding? 1.7 What will happen to the objective function if the right-hand side of constraint one is increased by one unit? 1.8 In what range can the profit margin per unit of X1 vary without changing the optimal basis? 1.9 What is the marginal value of increasing the production capacity of Workshop 1 and Workshop 2? 1.10 In what range can the capacity of Workshop 1 vary without changing the optimal basis?

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter5: Network Models
Section5.5: Shortest Path Models
Problem 31P
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RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 400.000000 400.000000 100.000000 3 400.000000 133.333328 200.000000 THE TABLEAU ROW (BASIS) X1 X2 X3 X4 SLK 2 1 ART 6.000 0.000 24.000 0.000 18.000 X2 -0.600 1.000 1.700 0.000 0.400 3 X4 0.900 0.000 0.200 1.000 -0.100 ROW SLK 3 0.000 7200.000 2 -0.300 40.000 3 0.200 40.000 1.1 What is the optimal objective function value? 1.2 How many units of X1, X2, X3 and X4 should be produced in order to maximise profits? Or give the optimal product mix from the printout. 1.3 What is the status of the variables. Are they basic or non-basic? 1.4 Give the reduced costs of the non-basic variables. 1.5 Assume that 20 units of X3 have been produced by mistake. What is the resulting decrease in profit? 1.6 Which constraints are binding? 1.7 What will happen to the objective function if the right-hand side of constraint one is increased by one unit? 1.8 In what range can the profit margin per unit of X1 vary without changing the optimal basis? 1.9 What is the marginal value of increasing the production capacity of Workshop 1 and Workshop 2? 1.10 In what range can the capacity of Workshop 1 vary without changing the optimal basis?
Transcribed Image Text:RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 400.000000 400.000000 100.000000 3 400.000000 133.333328 200.000000 THE TABLEAU ROW (BASIS) X1 X2 X3 X4 SLK 2 1 ART 6.000 0.000 24.000 0.000 18.000 X2 -0.600 1.000 1.700 0.000 0.400 3 X4 0.900 0.000 0.200 1.000 -0.100 ROW SLK 3 0.000 7200.000 2 -0.300 40.000 3 0.200 40.000 1.1 What is the optimal objective function value? 1.2 How many units of X1, X2, X3 and X4 should be produced in order to maximise profits? Or give the optimal product mix from the printout. 1.3 What is the status of the variables. Are they basic or non-basic? 1.4 Give the reduced costs of the non-basic variables. 1.5 Assume that 20 units of X3 have been produced by mistake. What is the resulting decrease in profit? 1.6 Which constraints are binding? 1.7 What will happen to the objective function if the right-hand side of constraint one is increased by one unit? 1.8 In what range can the profit margin per unit of X1 vary without changing the optimal basis? 1.9 What is the marginal value of increasing the production capacity of Workshop 1 and Workshop 2? 1.10 In what range can the capacity of Workshop 1 vary without changing the optimal basis?
Question 1 A factory can produce four products denoted by X1, X2, X3 and X4. Each product must be processed in each of two workshops. The processing times (in hours per unit produced) are given in the following table: X1 X2 X3 X4 |Workshop 1 Workshop 2 3 4 8 2 5 8 There are 400 hours of labour available in each workshop. The profit margins are R48, R72, R120 and R108 per unit of X1, X2, X3 and X4 produced respec- tively. Everything that is produced can be sold. Thus, maximising profits, the following linear program can be used: Маx Z = 48X1 + 72X2 120X3 108X4, subject to 3X1 4X2 8X3 6X4 400 6X1 2X2 5X3 8X4 400 X1,X2,X3 2 0. Using LINDO, we get the final tableau: LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1) 7200.000 VARIABLE VALUE REDUCED COST X1 0.000000 6.000000 X2 40.000000 0.000000 X3 0.000000 24.000000 X4 40.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 18.000000 3) 0.000000 0.000000 NO. ITERATIONS= 1 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE URRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X1 48.000000 6.000000 INFINITY X2 72.000000 0.000000 14.117646 X3 120.000000 24.000000 INFINITY X4 108.000000 180.000000 0.000000 VI VI
Transcribed Image Text:Question 1 A factory can produce four products denoted by X1, X2, X3 and X4. Each product must be processed in each of two workshops. The processing times (in hours per unit produced) are given in the following table: X1 X2 X3 X4 |Workshop 1 Workshop 2 3 4 8 2 5 8 There are 400 hours of labour available in each workshop. The profit margins are R48, R72, R120 and R108 per unit of X1, X2, X3 and X4 produced respec- tively. Everything that is produced can be sold. Thus, maximising profits, the following linear program can be used: Маx Z = 48X1 + 72X2 120X3 108X4, subject to 3X1 4X2 8X3 6X4 400 6X1 2X2 5X3 8X4 400 X1,X2,X3 2 0. Using LINDO, we get the final tableau: LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1) 7200.000 VARIABLE VALUE REDUCED COST X1 0.000000 6.000000 X2 40.000000 0.000000 X3 0.000000 24.000000 X4 40.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 18.000000 3) 0.000000 0.000000 NO. ITERATIONS= 1 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE URRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X1 48.000000 6.000000 INFINITY X2 72.000000 0.000000 14.117646 X3 120.000000 24.000000 INFINITY X4 108.000000 180.000000 0.000000 VI VI
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