Question

6 Sample Final-exam.pdf S No Shde Title anie Portfolio E-Leaning Moodle | Ge X x x x ng.gju.edu.jo/pluginfile.php/142960/mod resource/content/1/Sample % 20Final-exam.pdf #Dropbox- Sketchy.. - OneDrive 1/18/20: 2Page Q3) 10points A manufacturer produces three models 1, II, and I, of a certain product using raw materials A, and B. the following table gives the data for the problem: Requirements per unit Availability Raw material 4000 A 2 3 5 4 6000 B 2 7 Minimum demand 200 150 200 Profit per unit (S) 30 50 20 The labor time per unit of model () is twice that of (II) and three times that of (II). The entire labor force of the factory can produce the equivalent of 1500 units of model (I). Market requirements specify the ratios: 2:3:5 for the production of the three respective models. Formulate the problem as an LP model.

Answer 1

Let,

x1 = number of Model I to be produced
x2 = number of Model II to be produced
x3 = number of Model III to be produced

Objective is ton maximize profit so objective function = Max 30x1+20x2+50x3

Subject to,

2x1+3x2+5x3 <= 4000 (raw material - A)
4x1+2x2+7x3 <= 6000 (raw material - B)

Minimum demand
x1>= 200
x2>= 200
x3>= 150

Ratio requirement

x1 = 2k,x2 = 3k and x3 = 5k

so, x1/2 = x2/3

or, 3x1-2x2 = 0

x2/3 = x3/5

or, 5x2 - 3x3 = 0

Labor force constraint

x1+1/2x2+1/3x3 <= 1500 [as labor force is equivalent with 1500 units of model I]

or, 6x1+3x2+2x3 <= 9000

x1,x2,x3 >= 0 (Non-negativity constraint)

Solving in solver we get,

x1 = number of Model I to be produced = 210.5263 = 211 (Rounded to nearest whole number as number of model I is an integer)
x2 = number of Model II to be produced = 315.7895 = 316 (Rounded to nearest whole number as number of model II is an integer)
x3 = number of Model III to be produced = 526.3158 = 526  (Rounded to nearest whole number as number of model III is an integer)

and maximized profit = 38947.37

Solver screenshot

	x1	x2	x3
	210.5263	315.7895	526.3158				Objective function	38947.37
raw material - A	2	3	5	4000	<=	4000
raw material - B	4	2	7	5157.895	<=	6000
(Minimum demand - Model I)	1			210.5263	>=	200
(Minimum demand - Model II)		1		315.7895	>=	200
(Minimum demand - Model III)			1	526.3158	>=	150
Ratio requirement	3	-2		-1.1E-13	=	0
Ratio requirement		5	-3	4.55E-13	=	0
labor force constraint	6	3	2	3263.158	<=	9000