Question

You ara in charga of purchasas at tha sudent-run used-hock supply program at yaur callaga, and you must dedda how many introductary calculus, history, and marketing texts shuld ba purchasad from studants for rasala. Dun to budgat limitations, yau cannat purchase more than 1100 of these textbooks each semester. There are also shelf-space lim tations: Calculus texts occupy 2 units of shelf space each, history books 1 unit each, and marketing texts 2 units each, and you can spare at most 1,900 units of shelf space for tha taxts. tf the usad book program makes a profit of $10 on each calculus text, S4 on each history text, and S8 an aach marketing taxt, how many of each typa of text should you purchasa to maximiza profit? HINT [See Exampla 3.1 900 unts calculus texts) history textis) marketing text(s) What is the maximum profit the program can maka in a semester? Need Help?Read It

Answer 1

let x1 denote number of calculus textbooks

x2 denote number of history textbooks

x3 denote number of marketing textbooks

Max Z = 10 x1 + 4 x2 + 8 x3

subject to

x1 + x2 + x3 ≤ 1100 2 x1 + x2 + 2 x3 ≤ 1900

and x1,x2,x3≥0;

After introducing slack variables

Max Z = 10 x1 + 4 x2 + 8 x3 + 0 S1 + 0 S2

subject to

x1 + x2 + x3 + S1 = 1100 2 x1 + x2 + 2 x3 + S2 = 1900

and x1,x2,x3,S1,S2≥0

Iteration-1		Cj	10	4	8	0	0
B	CB	XB	x1	x2	x3	S1	S2	MinRatio XBx1
S1	0	1100	1	1	1	1	0	1100/1=1100
S2	0	1900	(2)	1	2	0	1	1900/2=950→
Z=0		Zj	0	0	0	0	0
		Zj-Cj	-10↑	-4	-8	0	0

Entering =x1, Departing =S2, Key Element =2

Iteration-2		Cj	10	4	8	0	0
B	CB	XB	x1	x2	x3	S1	S2	MinRatio
S1	0	150	0	12	0	1	-1/2
x1	10	950	1	12	1	0	1/2
Z=9500		Zj	10	5	10	0	5
		Zj-Cj	0	1	2	0	5

Hence, optimal solution is arrived with value of variables as :
x1 = 950, x2 = 0, x3 = 0

Max Z = 9500

so maximum profit is $9500 and number of calculus texts is 950 , history and marketing text is 0