20. Photo Chemicals produces two types of photographdeveloping uids at a cost of $1.00 per gallon. Let x1 = gallons of product 1 xz : gallons of product 2 Photo Chemicals management requires that at least 30 gallons of product 1 and at least 20 gallons of product 2 be produced. They also require that at least 80 pounds of a perish able raw material be used in production. A linear programming formulation of the problem is as follows: Min 111 'i' 11'2 S.[. 1x1 2 30 Minimum product 1 1x2 2 20 Minimum product 2 1x1 + 2x2 2 80 Minimum raw material xhxz 2 0 a. Write the dual problem. 1). Solve the dual problem. Use the dual solution to show that the optimal production plan is X] = 30 andxz = 25. c. The third constraint involves a management request that the current 80 pounds of a perishable raw material be used. However, after learning that the optimal solution calls for an excess production of ve units of product 2, management is reconsidering the raw material requirement. Specically, you have been asked to identify the cost effect if this constraint is relaxed. Use the dual variable to indicate the change in the cost if only 79 pounds of raw material have to be used. The dual problem is as follows Jqu'hrrr- 303,11 -+- 20;\" -+ 809;; sf 1511+ 1.91:; S l lyg -+- 2!}; S I 91-92- yrs 2 0 The solution of the dual problem is as follows:- We use solver to solve this dual problem. Howver, it can be solved easily by simultaneous equation or simplex method. The solution is y1=O.5 y2=O, y3=0.5 The sensitivity report is as below Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefcient Increase Decrease $D$4 0,5 0 30 10 30 EM 0 -5 20 5 1E+30 $F$4 0.5 0 80 1E+30 10 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $G$5 1 30 1 1E+30 0.5 $G$7 1 25 1 1 1 As we can see that the allowable increase and decrease for coefficient of y3 is infinity and 10 respectively. So the optimal soultion for dual variables will not change even if coefficient of y3 is decreased by 10 or increased indefinitely. So if raw material of 79 is now being used, the y3 value still will be 0.5. However the objective function will change which is nothing but the cost of production. The change is objective function is (8079) *0.5 =O.5 Dollars