Question

Greenville Cabinets received a contract to produce speaker cabinets for a major speaker manufacturer. The contract calls for the production of 4,100 bookshelf speakers and 4,900 floor speakers over the next two months, with the following delivery schedule:

Model	Month 1	Month 2
Bookshelf	2,500	1,600
Floor	1,900	3,000

Greenville estimates that the production time for each bookshelf model is 0.8 hour and the production time for each floor model is 1 hour. The raw material costs are $14 for each bookshelf model and $16 for each floor model. Labor costs are $26 per hour using regular production time and $37 using overtime. Greenville has up to 2,800 hours of regular production time available each month and up to 1,400 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 $9 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. If an amount is zero, enter "0".

Regular Time Manufacture
Model	Month 1	Month 2
Bookshelf	_______	_______
Floor	_______	_______

Overtime Manufacture
Model	Month 1	Month 2
Bookshelf	_______	_______
Floor	_______	_______

Please fill out all the blanks! Thank you!!!

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 month 2.

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 26 = 20.8	0.8 x 37 = 29.6
Floor speakers	1 x 26 = 26	1 x 37 = 37

The Objective Function is:

Minimize Z = 20.8 B1R + 20.8 B2R + 26 F1R + 26 F2R + 29.6 B1O + 29.6 B20 + 37 F1O + 37 F2O + 14 B1R + 14 B2R + 16 F1R + 16 F2R + 14 B1O + 14 B20 + 16 F1O + 16 F2O + 9 IB + 9 IF.

Or,

Minimize Z = 34.8 B1R + 34.8 B2R + 42 F1R + 42 F2R + 43.6 B1O + 43.6 B20 + 53 F1O + 53 F2O + 9 IB + 9 IF.

Constraints:

Regular Time Production - Month 1:-   0.8 B1R + 1 F1R   2,800 hours.

Regular Time Production - Month 2:-   0.8 B2R + 1 F2R   2,800 hours.

Overtime Production - Month 1:-   0.8 B1O + 1 F1O   1,400 hours.

Overtime Production - Month 2:-   0.8 B2O + 1 F2O   1,400 hours.

Bookshelf Speaker Production - Month 1:-   1 B1R + 1 B1O – 1 IB = 2,500 units.

Bookshelf Speaker Production - Month 2:-   1 IB + 1 B2R + 1 B2O = 1,600 units.

Floor Speaker Production - Month 1:-   1 F1R + 1 F1O – 1 IF = 1,900 units.

Floor Speaker Production - Month 2:-   1 IF+ 1 F2R + 1 F2O = 3,000 units.

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

Optimal Solution:

Model	Regular Time Manufacture
	Month 1	Month 2
Bookshelf speakers	2,500	1,600
Floor speakers	825	1,600

Model	Overtime Manufacture
	Month 1	Month 2
Bookshelf speakers	0	0
Floor speakers	1,182	1,400

Total Cost = 34.8(2500) + 34.8(1600) + 42(825) + 42(1600) + 0 + 0 + 53(1182) + 53(1400)+ 0 +9(100) = 382276

Greenville Cabinets Problem
Variables		B1R	B2R	F1R	F2R	B10	B20	F10	F20	IB	IF	Total Cost
		2500	1600	825	1600	0	0	1182	1400	0	100	382276
Constraints		RHS
Regular time production - Month 1	2800	2800
Regular time production - Month 2	2800	2800
Over time production - Month 1	1182	1400
Over time production - Month 2	1400	1400
Bookshelf speaker production - Month 1	2500	2500
Bookshelf speaker production - Month 2	1600	1600
Floor speaker production - Month 1	1900	1900
Floor speaker production - Month 2	3000	3000