Question

Orders 2,800 2,600 3,000 Car frames 2,200 1,600 2,000 Truck frames 5. Production scheduling. Labor and material costs for manu- facturing two guitar models are given in the table: Material Cost Labor Cost Guitar Model $20 $30 $30 $40 (A) If a total of $3,000 a week is allowed for labor and material, how many of each model should be produced each week to use exactly each of the allocations of the $3,000 indicated in the following table? Weekly Allocation Labor $1.800 $1,750 $1,720 Material $1,200 $1.250 $1,280 (B) Is it possible to use an allocation of $1,600 for labor and $1,400 for material? Of $2,000 for labor and $1,000 for material? Explain.

Answer 1

A) Total cost per Model Material cost Unit Labor cost 30 $ 20 $ 50 40 | $ 30 | $ 70 Weekly allocation 1,750 $ 1,250 $ 3,000 $ Labor cost 1,800 $ 1,200 $ 3,000 $ 1,720 Material cost 1,280 Total cost 3,000 Assume: Number of Model A Number of Model B For week 1 the allocation can be written as following linear equations: 30x+40y = 1,800 20x+30y = 1,200 Equation 1 Equation 2 (1800 -40y)/30 (1200 -20x)/30 = 40- 0.6667x

Now put value of y from Equation 2 in Equation 1 (1800 - 40(40 - 0.6667x))/30 60 - 1.3333(40 - 0.6667x) 60 - 53.3320 + 0.8889x 6.6688 + 0.8889x 6.6688/0.1111 60 round upto 4 decimal places Now put value of x=60 in equation 2 to find y (1200 - 20*60)/30 For week 2 the allocation can be written as following linear equations: 30x+40y = 1,750 20x+30y = 1,250 (1750 -40y)/30 (1250 -20x)/30 Equation 1 = 41,6667- 0.6667x Equation 2

Now put value of y from Equation 2 in Equation 1 (1720 - 40(42.6667 - 0.6667x))/30 57.3333 - 1.3333(42.6667 - 0.6667x) 57.3333 - 56.8875 + 0.8889x 0.4458 + 0.8889x 0.4458/0.1111 4.0126 round upto 4 decimal places Now put value of x-4.0126 in equation 2 to find y y= (1280 - 20*4.0126)/30 round upto 4 decimal places 39.9916

B) 30x+40y = 1,600 20x+30y = 1,400 (1600 -40y)/30 (1400 -20x)/30 (1600 - 40(46.6667 - 0.6667x))/30 53.3333 - 1.3333(46.6667 - 0.6667x) =46.6667 - 0.6667x 53.3333 - 62.2207 +0.8889x -8.8874 + 0.8889x =-8.8874/0.1111 (79.99) round upto 4 decimal places Now put value of x=- 79.99 in equation 2 to find y (1400 - (20*-79.99))/30 round upto 4 decimal places 99.99333333 Conclusion - allocation not possible as it results in negative units for Model A Similarly it can be checked for $2000 & $1,000 labour and material cost