3
Question
A chemical company produces two products,X and Y.Each unit of product X requires 3hrs on operation I and 4hrs on operation II,while each unit of product Y requires 4hrs on operation I and 5hrs on operationII.
Total available time for operations I and II is 20hrs and 26 hrs respectively.The production of each unit of product Y also results in two units of a by-product Z at no extra cost.
Product X sells at profit of Rs.10/unit, while Y sells at profit of Rs.20/unit.By-product Z brings a unit profit of Rs.6 if sold;in case it cannot be sold, the destruction cost is Rs.4 per unit.Forecasts indicate that not more that 5 units of Z can be sold.Formulate the L.P model to determine the quantities of X and Y to be produced,keeping Z in mind, so that the profit earned is maximum.
Total available time for operations I and II is 20hrs and 26 hrs respectively.The production of each unit of product Y also results in two units of a by-product Z at no extra cost.
Product X sells at profit of Rs.10/unit, while Y sells at profit of Rs.20/unit.By-product Z brings a unit profit of Rs.6 if sold;in case it cannot be sold, the destruction cost is Rs.4 per unit.Forecasts indicate that not more that 5 units of Z can be sold.Formulate the L.P model to determine the quantities of X and Y to be produced,keeping Z in mind, so that the profit earned is maximum.
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Solution
Formulation of the L.P model:
Let the number of units of products X,Y and Z produced be x1,x2,xz where
xz=number of units of Z produced.
=number of units of Z sold+number of units of Z destroyed.
=x3+x4(say)
Objective is to maximize the profit.Objective function(profit function) for productsX and Y is linear because their profits (Rs.10 per unit and Rs.20 per unit) are constants irrespective of the number of units produced.A graph between the total profit and quantity produced will be a straight line.However, a similar graph for product Z is non-linear since it has slope +6 for first part,while a slope of −4 for the second.However,it is piecewise linear,since it is linear in the regions0 to 5 and 5 to 2Y.Thus,splitting xz in to two parts,viz.the number of units of Z sold (x3) and number of units of Z destroyed (x4) makes the objective function for product Z also linear.
Thus, the objective function is
maximize Z=10x1+20x2+6x3−4x4.
Constraints are
on the time available on operationI:3x1+4x2≤20
on the time available on operationII:4x1+5x2≤26
on the number of units of product Z sold:x3≤5
on the number of units of product Z produced:2Y=Z
or 2x2=x3+x4 or −2x2+x3+x4=0,
where x1,x2,x3,x4, each≥0
Let the number of units of products X,Y and Z produced be x1,x2,xz where
xz=number of units of Z produced.
=number of units of Z sold+number of units of Z destroyed.
=x3+x4(say)
Objective is to maximize the profit.Objective function(profit function) for productsX and Y is linear because their profits (Rs.10 per unit and Rs.20 per unit) are constants irrespective of the number of units produced.A graph between the total profit and quantity produced will be a straight line.However, a similar graph for product Z is non-linear since it has slope +6 for first part,while a slope of −4 for the second.However,it is piecewise linear,since it is linear in the regions0 to 5 and 5 to 2Y.Thus,splitting xz in to two parts,viz.the number of units of Z sold (x3) and number of units of Z destroyed (x4) makes the objective function for product Z also linear.
Thus, the objective function is
maximize Z=10x1+20x2+6x3−4x4.
Constraints are
on the time available on operationI:3x1+4x2≤20
on the time available on operationII:4x1+5x2≤26
on the number of units of product Z sold:x3≤5
on the number of units of product Z produced:2Y=Z
or 2x2=x3+x4 or −2x2+x3+x4=0,
where x1,x2,x3,x4, each≥0
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