A company produces two products (product 1 and product 2) on two machines (machine 1 and...

   

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A company produces two products (product 1 and product 2) on two machines (machine 1 and machine 2). Amount of machine time and labor time which depends on the machine and the product, and the cost of producing a unit of each product are shown in the following table. 480 hours of labor time is available this month and machine times available are 200 hours for machine 1 and 200 hours for machine 2 this month. At least 200 units of product 1 and at least 250 units of product 2 must be produced this month. Also, at least half of product 1 must be made on machine 1, and at least half of product 2 must be made en machine 2. Formulate an LP to minimize the cost of meeting the monthly demands. + Machine time (hours) Labor time (hours) Production cost per unit s.t. Product 1 Product 2 Machine 1 Machine 2 Machine 1 Machine 2 0.7 0.8 0.75 0.85 1.2 0.7 1.5 $2.50 $0.50 $4.00 Decision variables, Let x= units of product i made on machine j. The mathematical model Min Z=1.5x1+ 2.5x12 +0.5x21 +4x22 0.7x11 +0.75x21 200 0.8x12 +0.85x22 200 0.75 $1.50 (constraint 1) (constraint 2) 0.75*11+1.2x12 +0.7x21+1.5x22 480 (constraint 3) *11+x122200 (constraint 4) *21 +*22 250 (constraint 5) 0.5x11-0.5x12 20 (constraint 6) 0.5*22-0.5x21 20 (constraint 7) x/20 (i = 1,2; j = 1,2) (constraint 8) Write the GAMS code of the given mathematical model. Answer the following questions. 1. What are the optimum Z value and decision variable values? (For Z, *11*12*21*22 specify the values separately) 2. Which one(s) of the constraint(s) is(are) binding? Which is (are) non-binding? Why? 3. What are the shadow prices for all constraints? What do they mean for this model? Specify one by one. 4. What are the reduced costs for all variables? What do they mean for the model? Specify one by one. 5. If machine time available were 150 hours for machine 2, would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values? 6. If labor time available were 380 hours, would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values? 7. If the production cost for machine 1 product 2 were 2.5, would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values? 8. If demand for product 2 were 300, would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values? Answers A company produces two products (product 1 and product 2) on two machines (machine 1 and machine 2). Amount of machine time and labor time which depends on the machine and the product, and the cost of producing a unit of each product are shown in the following table. 480 hours of labor time is available this month and machine times available are 200 hours for machine 1 and 200 hours for machine 2 this month. At least 200 units of product 1 and at least 250 units of product 2 must be produced this month. Also, at least half of product 1 must be made on machine 1, and at least half of product 2 must be made en machine 2. Formulate an LP to minimize the cost of meeting the monthly demands. + Machine time (hours) Labor time (hours) Production cost per unit s.t. Product 1 Product 2 Machine 1 Machine 2 Machine 1 Machine 2 0.7 0.8 0.75 0.85 1.2 0.7 1.5 $2.50 $0.50 $4.00 Decision variables, Let x= units of product i made on machine j. The mathematical model Min Z=1.5x1+ 2.5x12 +0.5x21 +4x22 0.7x11 +0.75x21 200 0.8x12 +0.85x22 200 0.75 $1.50 (constraint 1) (constraint 2) 0.75*11+1.2x12 +0.7x21+1.5x22 480 (constraint 3) *11+x122200 (constraint 4) *21 +*22 250 (constraint 5) 0.5x11-0.5x12 20 (constraint 6) 0.5*22-0.5x21 20 (constraint 7) x/20 (i = 1,2; j = 1,2) (constraint 8) Write the GAMS code of the given mathematical model. Answer the following questions. 1. What are the optimum Z value and decision variable values? (For Z, *11*12*21*22 specify the values separately) 2. Which one(s) of the constraint(s) is(are) binding? Which is (are) non-binding? Why? 3. What are the shadow prices for all constraints? What do they mean for this model? Specify one by one. 4. What are the reduced costs for all variables? What do they mean for the model? Specify one by one. 5. If machine time available were 150 hours for machine 2, would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values? 6. If labor time available were 380 hours, would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values? 7. If the production cost for machine 1 product 2 were 2.5, would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values? 8. If demand for product 2 were 300, would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values? Answers

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The given LP problem can be formulated as follows Minimize Z 15x1 25x2 05x1 4x2 Minimize the cost of meeting the monthly demands Subject to the following constraints 07x1 075x2 200 Constraint 1 Machin... View the full answer