Question

Problem 4-21 (Algorithmic) Greenville Cabinets received a contract to produce speaker cabinets for a major speaker manufacturer. The contract calls for the production of 3,900 bookshelf speakers and 4,700 floor speakers over the next two months, with the following delivery schedule: Model Month 1 Month 2 Bookshelf 2,400 1,500 Floor 1,800 2,900 Greenville estimates that the production time for each bookshelf model is 0.8 hour and the production time for each floor model is hour. The raw material costs are $13 for each bookshelf model and $15 for each floor model. Labor costs are $25 per hour using regular production time and $36 using overtime. Greenville has up to 2,700 hours of regular production time available each month and up to 1,300 additional hours of overtime available each month. If production for either cabinet exceeds demand in month 1, the cabinets can be stored at a cost of $8 per cabinet. For each product, determine the number of units that should be manufactured each month on regular time and on overtime to minimize total production and storage costs. If required, round your answers to the nearest whole number Regular Time Manufacture Month 1 Month 2 Model Bookshelf Floor Overtime Manufacture Model Month 1 Month 2 Bookshelf Floor

Answer 1

B1R = Quantity of bookshelf speakers produced using regular time labor in month 1.

B2R = Quantity of bookshelf speakers produced using regular time labor in month2.

F1R = Quantity of floor speakers produced using regular time labor in month 1.

F2R = Quantity of floor speakers produced using regular time labor in month 2.

B1O = Quantity of bookshelf speakers produced using overtime labor in month 1.

B2O = Quantity of bookshelf speakers produced using overtime labor in month 2.

F1O = Quantity of floor speakers produced using overtime labor in month 1.

F2O = Quantity of floor speakers produced using overtime labor in month 2.

IB = Month 1 ending inventory for bookshelf speakers.

IF = Month 1 ending inventory for floor speakers.    

The Decision Variables are B1R, B2R_, F1R, F2R, ………….. IF, whose production and storage costs have to be minimized.

Table for Labor Costs per Unit:

Model	Regular (in $)	Overtime (in $)
Bookshelf speakers	0.8 x 25 = 20	0.8 x 36 = 28.8
Floor speakers	1 x 25 = 25	1 x 36 = 36

The Objective Function is:

Minimize Z = 20 B1R + 20 B2R + 25 F1R + 25 F2R + 28.8 B1O + 28.8 B20 + 36 F1O + 36 F2O + 13 B1R + 13 B2R + 15 F1R + 15 F2R + 13 B1O + 13 B20 + 15 F1O + 15 F2O + 8 IB + 8 IF.

Or,

Minimize Z = 33 B1R + 33 B2R + 40 F1R + 40 F2R + 41.8 B1O + 41.8 B20 + 51 F1O + 51 F2O + 8 IB + 8 IF.

Constraints:

Regular Time Production - Month 1:-   0.8 B1R + 1 F1R  $\leq$ 2,700 hours.

Regular Time Production - Month 2:-   0.8 B2R + 1 F2R  $\leq$ 2,700 hours.

Overtime Production - Month 1:-   0.8 B1O + 1 F1O  $\leq$ 1,300 hours.

Overtime Production - Month 2:-   0.8 B2O + 1 F2O  $\leq$ 1,300 hours.

Bookshelf Speaker Production - Month 1:-   1 B1R + 1 B1O – 1 IB = 2,400 units.

Bookshelf Speaker Production - Month 2:-   1 IB + 1 B2R + 1 B2O = 1,500 units.

Floor Speaker Production - Month 1:-   1 F1R + 1 F1O – 1 IF = 1,800 units.

Floor Speaker Production - Month 2:-   1 IF+ 1 F2R + 1 F2O = 2,900 units.

B1R, B2R_, F1R, F2R, ………….. IF $\geq$ 0 (Non-negativity constraints)

Optimal Solution:

Model	Regular Time Manufacture
	Month 1	Month 2
Bookshelf speakers	2,400	1,500
Floor speakers	780	1,500

Model	Overtime Manufacture
	Month 1	Month 2
Bookshelf speakers	0	0
Floor speakers	1,120	1,300

C D. G L M N H Greenville Cabinets Problem Variables IB 3. B1R F1R F2R B10 F10 Total Cost B2R B20 F20 IF 780 0 2400 1500 1500 0 1120 1300 100 344120 4. C 5. 6 Constraints RHS Regular Time Production Month 1 Regular Time Production Month 2 Overtime Production Month 1 Overtime Production Month 2 Bookshelf Speaker Production - Month 1 Bookshelf Speaker Production Month 2 Floor Speaker Production Month 1 2700 2700 2700 2700 9 1120 1300 10 1300 1300 2400 2400 11 12 1500 1500 13 1800 1800 14 Floor Speaker Production - Month 2 2900 2900