Why were the ratios disregarded when forming the model.












1












$begingroup$


I came across this question in a textbook and I dont fully understand why they omitted one of the conditons when they were forming the model for the LPP.



A manufacturer produces three models I, II and III of a certain product. He uses two
types of raw materials (A and B) of which 5000 and 8000 units respectively are available. Raw material of type A requires 3, 4 and 6 units of each model. Whereas type B requires 6, 4 and 8 of model I, II and III respectively.



The labour time of each unit of model I is twice that of model
II and three times of model III. The entire labour force of the factory can produce equivalent of 3000 units of model I. A market survey indicates that the minimum demand of three models is 600, 400 and 350 units respectively.



However, the ratios of number of units produced must be
equal to 3 : 2 : 5. Assume that the profit per unit of models I, II and III are Rs 80, 50, and 120 respectively. Formulate this problem as linear programming model to determine the number of units of each product which will maximize the profit.



Solution given as 



x1 <= 600

x2 <= 400

x3 <= 350

3x1 + 4x2 + 6x3 <= 5000

6x1 + 4x2 + 8x3 <= 8000

x1 + 0.5x2 +.333x3 <= 3000

 x1,x2,x3 >= 0


from the question , i thought we would need another constraint such as 



3/10x1 + 2/10x2 +5/10x3 = 1


to cater for the ratio of units that must be produced.



Why was there no mention of the ratio in the solution ?prog






linear-programming operations-research







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    I came across this question in a textbook and I dont fully understand why they omitted one of the conditons when they were forming the model for the LPP.



    A manufacturer produces three models I, II and III of a certain product. He uses two
    types of raw materials (A and B) of which 5000 and 8000 units respectively are available. Raw material of type A requires 3, 4 and 6 units of each model. Whereas type B requires 6, 4 and 8 of model I, II and III respectively.



    The labour time of each unit of model I is twice that of model
    II and three times of model III. The entire labour force of the factory can produce equivalent of 3000 units of model I. A market survey indicates that the minimum demand of three models is 600, 400 and 350 units respectively.



    However, the ratios of number of units produced must be
    equal to 3 : 2 : 5. Assume that the profit per unit of models I, II and III are Rs 80, 50, and 120 respectively. Formulate this problem as linear programming model to determine the number of units of each product which will maximize the profit.



    Solution given as 



    x1 <= 600
    
x2 <= 400
    
x3 <= 350
    
3x1 + 4x2 + 6x3 <= 5000
    
6x1 + 4x2 + 8x3 <= 8000
    
x1 + 0.5x2 +.333x3 <= 3000
    
 x1,x2,x3 >= 0


    from the question , i thought we would need another constraint such as 



    3/10x1 + 2/10x2 +5/10x3 = 1


    to cater for the ratio of units that must be produced.



    Why was there no mention of the ratio in the solution ?prog






    linear-programming operations-research







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      I came across this question in a textbook and I dont fully understand why they omitted one of the conditons when they were forming the model for the LPP.



      A manufacturer produces three models I, II and III of a certain product. He uses two
      types of raw materials (A and B) of which 5000 and 8000 units respectively are available. Raw material of type A requires 3, 4 and 6 units of each model. Whereas type B requires 6, 4 and 8 of model I, II and III respectively.



      The labour time of each unit of model I is twice that of model
      II and three times of model III. The entire labour force of the factory can produce equivalent of 3000 units of model I. A market survey indicates that the minimum demand of three models is 600, 400 and 350 units respectively.



      However, the ratios of number of units produced must be
      equal to 3 : 2 : 5. Assume that the profit per unit of models I, II and III are Rs 80, 50, and 120 respectively. Formulate this problem as linear programming model to determine the number of units of each product which will maximize the profit.



      Solution given as 



      x1 <= 600
      
x2 <= 400
      
x3 <= 350
      
3x1 + 4x2 + 6x3 <= 5000
      
6x1 + 4x2 + 8x3 <= 8000
      
x1 + 0.5x2 +.333x3 <= 3000
      
 x1,x2,x3 >= 0


      from the question , i thought we would need another constraint such as 



      3/10x1 + 2/10x2 +5/10x3 = 1


      to cater for the ratio of units that must be produced.



      Why was there no mention of the ratio in the solution ?prog






      linear-programming operations-research







      share|cite|improve this question











      $endgroup$




      I came across this question in a textbook and I dont fully understand why they omitted one of the conditons when they were forming the model for the LPP.



      A manufacturer produces three models I, II and III of a certain product. He uses two
      types of raw materials (A and B) of which 5000 and 8000 units respectively are available. Raw material of type A requires 3, 4 and 6 units of each model. Whereas type B requires 6, 4 and 8 of model I, II and III respectively.



      The labour time of each unit of model I is twice that of model
      II and three times of model III. The entire labour force of the factory can produce equivalent of 3000 units of model I. A market survey indicates that the minimum demand of three models is 600, 400 and 350 units respectively.



      However, the ratios of number of units produced must be
      equal to 3 : 2 : 5. Assume that the profit per unit of models I, II and III are Rs 80, 50, and 120 respectively. Formulate this problem as linear programming model to determine the number of units of each product which will maximize the profit.



      Solution given as 



      x1 <= 600
      
x2 <= 400
      
x3 <= 350
      
3x1 + 4x2 + 6x3 <= 5000
      
6x1 + 4x2 + 8x3 <= 8000
      
x1 + 0.5x2 +.333x3 <= 3000
      
 x1,x2,x3 >= 0


      from the question , i thought we would need another constraint such as 



      3/10x1 + 2/10x2 +5/10x3 = 1


      to cater for the ratio of units that must be produced.



      Why was there no mention of the ratio in the solution ?prog






      linear-programming operations-research




      linear-programming operations-research






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 15 at 21:41









      callculus

      18.2k31427




      18.2k31427










      asked Jan 15 at 20:17









      uhexosuhexos

      83




      83






















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          $begingroup$

          I agree that the ratio constraints are missing. Additionally the relation signs of the minimum demand constraints are wrong:




          ... that the minimum demand of three models is $600, 400$ and $350$
          units respectively.




          $x_1 geq 600$



          $x_2 geq 400$



          $x_3 geq 350$




          However, the ratios of number of units produced must be equal to $3 : 2 : 5$




          Here you need two separate constraints such as



          $2x_1=3x_2$



          $5x_2=2x_3$



          A third constraint is not necessary since from this two constraints it can be deduced that $frac{10}{3}x_1=2x_3$.



          You can check it by plugging in some values, for instance $x_1=600$, $x_2=400$, $x_3=1000$






          share|cite|improve this answer









          $endgroup$













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            $begingroup$

            I agree that the ratio constraints are missing. Additionally the relation signs of the minimum demand constraints are wrong:




            ... that the minimum demand of three models is $600, 400$ and $350$
            units respectively.




            $x_1 geq 600$



            $x_2 geq 400$



            $x_3 geq 350$




            However, the ratios of number of units produced must be equal to $3 : 2 : 5$




            Here you need two separate constraints such as



            $2x_1=3x_2$



            $5x_2=2x_3$



            A third constraint is not necessary since from this two constraints it can be deduced that $frac{10}{3}x_1=2x_3$.



            You can check it by plugging in some values, for instance $x_1=600$, $x_2=400$, $x_3=1000$






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              I agree that the ratio constraints are missing. Additionally the relation signs of the minimum demand constraints are wrong:




              ... that the minimum demand of three models is $600, 400$ and $350$
              units respectively.




              $x_1 geq 600$



              $x_2 geq 400$



              $x_3 geq 350$




              However, the ratios of number of units produced must be equal to $3 : 2 : 5$




              Here you need two separate constraints such as



              $2x_1=3x_2$



              $5x_2=2x_3$



              A third constraint is not necessary since from this two constraints it can be deduced that $frac{10}{3}x_1=2x_3$.



              You can check it by plugging in some values, for instance $x_1=600$, $x_2=400$, $x_3=1000$






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                I agree that the ratio constraints are missing. Additionally the relation signs of the minimum demand constraints are wrong:




                ... that the minimum demand of three models is $600, 400$ and $350$
                units respectively.




                $x_1 geq 600$



                $x_2 geq 400$



                $x_3 geq 350$




                However, the ratios of number of units produced must be equal to $3 : 2 : 5$




                Here you need two separate constraints such as



                $2x_1=3x_2$



                $5x_2=2x_3$



                A third constraint is not necessary since from this two constraints it can be deduced that $frac{10}{3}x_1=2x_3$.



                You can check it by plugging in some values, for instance $x_1=600$, $x_2=400$, $x_3=1000$






                share|cite|improve this answer









                $endgroup$



                I agree that the ratio constraints are missing. Additionally the relation signs of the minimum demand constraints are wrong:




                ... that the minimum demand of three models is $600, 400$ and $350$
                units respectively.




                $x_1 geq 600$



                $x_2 geq 400$



                $x_3 geq 350$




                However, the ratios of number of units produced must be equal to $3 : 2 : 5$




                Here you need two separate constraints such as



                $2x_1=3x_2$



                $5x_2=2x_3$



                A third constraint is not necessary since from this two constraints it can be deduced that $frac{10}{3}x_1=2x_3$.



                You can check it by plugging in some values, for instance $x_1=600$, $x_2=400$, $x_3=1000$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 15 at 21:44









                callculuscallculus

                18.2k31427




                18.2k31427






























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