Kingdom of Saudi Arabia
Ministry of Higher Education
Majmaah University Faculty of Science-Zulfi
Department of Computer Science &
Information
Operational Research
A. Problems on Conditional Extrema (Lagrange Method)
1. Extremize ‫( أوجد القيم المثلى‬find the max. and min.) of F= 400 x y z2
given that x2 + y2 + z2 = 1 .
2. Find the maximum product ‫ حاصل ضرب‬of three numbers, such that the
sum ‫ مجموع‬of the 1st number, twice the 2nd and 3-times the 3rd is 108.
3. Find the height ‫ ارتفاع‬and base radius ‫ نصف قطر القاعدة‬of a right ‫قائم‬
circular ‫ دائري‬cone ‫ مخروط‬that may be inscribed ‫ يمكن وصفه‬inside ‫داخل‬
a sphere of radius A.
4. Show ‫ بين‬how ‫ كيف‬to divide ‫ تقسم‬a robe ‫ حبل‬of length L, to two
separated ‫ منفصلين‬shapes ‫ شكلين‬a square ‫ مربع‬and circle ‫دائرة‬. The sum
of the areas ‫ مساحات‬of the square and the circle is maximum.
B. Linear Systems of Equations (Simple Row Operations)
1. A factory has three machines A, B, and C that contribute in manufacturing
three products I, II, and III. Product I requires 2 hrs. on A, and one hr. on B,
and C. Product II requires 2 hrs. on B and one hr. on A and C. Product III
requires one hr. on B and 2 hrs. on A and C. If the machines A, B, and C are
operated for 100 hrs., 150 hrs., and 125 hrs. respectively per week. How many
items of the products I, II, and III are produced per week.
Dr. Eng. Moustafa Reda AbdALLAH Eltantawi Good 1/2 Luck
Operational Research – Model Problems for 1st Mid-Term Exam
2. A company of car decoration uses four materials A, B, C, and D to decorate four
models of cars I, II, III, and IV. Model I consumes 2 units, 2 units, 4 units, and
3 units of the materials A, B, C, and D respectively. Model II consumes, only,
one unit from the materials A, and C. Model III consumes 2 units of the
materials A, and D and one unit of the materials B, and C. Model IV consumes 3
units of the materials A and B, 2 units of material C and one unit of material D.
If 105 units, 75 units, 100 units and 65 units are consumed of the materials A,
B, C, and D respectively per week. How many cars of each model (of the four
ones) are decorated per week.
3. Some factory produces three items of home furniture. The factory uses three
substances: Wood, Glass, and Rubber. The 1st item requires 5 units of Wood, 2
units Glass and 3 units of Rubber. The 2nd item requires one unit of Wood, 4
units Glass and 6 units of Rubber. The 3rd item requires 7 units of Wood, 3
units Glass and 2 units of Rubber. If 900 units of Wood, 800 units of Glass, and
1100 units of Rubber are consumed per week, determine the number produced
of each item per week.
4. A stationery factory produces three kinds of pencils: Mars, Venus, and Jupiter.
The company uses Graphite and Wood as manufacturing materials, besides the
machine working, in the production of these pencils.
Each dozen of Mars needs: 6 units of Graphite, 2 units of Wood, and 5 units of
machine working, and each dozen of Venus needs: 4 units of Graphite, 7 units of
Wood, and 2 units of machine working. Also, each dozen of Mars needs: 8 units
of Graphite, 11 units of Wood, and 9 units of machine working.
If 8900 units of Graphite, 8480 units of Wood, and 7150 units of machine
working are consumed per day. Find the number of dozens of each kind
produced per day
Good Luck and Best Wishes
Dr. Eng. Moustafa Reda AbdALLAH Eltantawi Good 2/2 Luck
Operational Research – Model Problems for 1st Mid-Term Exam