Instructor: Assoc. Prof. Oya KARAŞAN
Teaching Assisstant : Seçil Sözüer
IE202- Recitation 1
Fall 2013-2014
October 10, 2013
Question 1.) Willy Wonka’s Candy Company Produces three types of Candy :
1.Wonka Bars
2.Bottle Caps
3.Giant Sweet Tarts
In order to produce the different type of candies, Willy can run three different production
processes as described below. Each process involves blending different types of sugars in
the Magical Factories Mixer.
Process 1:
Running Process 1 for one hour:
Costs: $5
Requires: Two barrels of sugar type A and three barrels of sugar type B
Output: Two Wonka Bars and one packet of Bottle Caps
Process 2:
Running Process 2 for one hour:
Costs: $4
Requires: One barrel of sugar type A and three barrels of sugar type B
Output: Three packets of Bottle Caps
Process 3:
Running Process 1 for one hour:
Costs: $1
Requires: Two barrels of sugar type B and three packets of Bottle Caps
Output: Two Wonka Bars and one packet of Giant Sweet Tarts
Incidentally, the barrels are really tiny, and the candy is very sweet.
Each week we can purchase:
 200 barrels of sugar type A at $2 Per Barrel
 300 barrels of sugar type B at $3 Per Barrel
Assume that they can sell everything that they can produce.
 Wonka Bars are sold at $9 per bar.
 Bottle Caps are sold at $10 per packet.
 Giant Sweet Tarts are sold at $24 per packet.
Assume that 100 hours of mixing time are available.
a. Formulate an LP whose solution will maximize Willy Wonka’s Profits.
b. Assume that instead of having 200 barrels of sugar type A and 300 barrels of sugar
type B available that you can order a total of 500 Barrels. Show how to modify your
LP formulation in Part A to account for this revised problem.
c. Suppose that instead of selling the three candies separately, they can only be sold as
part of a box consisting of one Wonka Bar, two packets of Bottle Caps, and one pack of
Giant Sweet Tarts. Each Wonka Box sells for $54. Modify your LP formulation in part A
to model this new scenario.
Question 2) In the planning of the monthly production for the next six months a
company must, in each month, operate either a normal shift or an extended shift (if it
produces at all). A normal shift costs £100,000 per month and can produce up to 5,000
units per month. An extended shift costs £180,000 per month and can produce up to
7,500 units per month. Note here that, for either type of shift, the cost incurred is fixed by
a union guarantee agreement and so is independent of the amount produced.
It is estimated that changing from a normal shift in one month to an extended shift in the
next month costs an extra £15,000. No extra cost is incurred in changing from an
extended shift in one month to a normal shift in the next month.
The cost of holding stock is estimated to be £2 per unit per month (based on the stock
held at the end of each month) and the initial stock is 3,000 units (produced by a normal
shift). The amount in stock at the end of month 6 should be at least 2,000 units. The
demand for the company's product in each of the next six months is estimated to be as
shown below:
Month
Demand
1
6,000
2
6,500
3
7,500
4
7,000
5
6,000
6
6,000
Production constraints are such that if the company produces anything in a particular
month it must produce at least 2,000 units. If the company wants a production plan for
the next six months that avoids stockouts, formulate their problem as an integer program.
Question 3) A large department store has decided to stay open for business on a
continuous 24 hour basis. The store manager has divided the 24-hour day into six equal
4-hour periods and determined the following minimum personnel requirement for each
period:
Store personnel can report for work at the beginning of any of the above time periods and
will work for 8 consecutive hours. The store manager wants to know the minimum
number of employees to begin work at each 4-hour segment to minimize the total number
of employees. Formulate an integer linear program modeling this problem instance.
Question 4) Esra Cosmetics (EC) produces perfume, which requires chemical and labor.
Two production processes are available: Process 1 transforms 1 unit of labor and 2 units
of chemicals into 3 cl of high quality perfume. Process 2 transforms 2 units of labor and 3
units of chemicals into 7 cl of low quality perfume. It costs EC 3 TL to purchase one unit
of labor and 2 TL to purchase one unit of chemicals. Each month, up to 2000 units of
labor and 3500 units of chemicals can be purchased. One cl of high quality perfume sells
for 5 TL and one cl of low quality perfume sells for 3 TL. The monthly demand for high
quality perfume is 1000cl and that for low quality perfume is 1800cl.
a) If the quantity of high quality perfume is more than its corresponding demand, EC can
use this excess production to meet the demand for low quality perfume. In such a case,
EC can only charge the sales price of low quality perfume. On the other hand, the
demand of high quality perfume can only be met by producing high quality perfume.
Formulate a linear programming problem that maximizes the total profit of EC while
meeting the demand and production requirements.
b) Assume that there is a 3rd process that can transform 1 cl of low quality perfume to 1
cl of high quality perfume using 0.4 units of labor and 0.3 units of chemicals. Formulate a
linear programming problem that maximizes the total profit of EC while meeting the
demand and production requirements. (For this part, please ignore the demand
substitution specified in part a).
Question 5. Consider the problem of locating a new machine to an existing layout
consisting of four machines.
These machines are located at the following x1 and x2 coordinates: (3, 1), (0, -3), (-2, 2),
and (1, 4).
Let the coordinates of the new machine be (x1, x2). Formulate the problem of finding an
optimal
location as a linear program for each of the following cases:
a. The sum of the distances from the new machine to the four machines is minimized.
Use the street distance: for example, the distance from (x1, x2) to the first machine
located at (3, 1) is
|x1 - 3|+|x2 - 1|.
b. Because of various amounts of flow between the new machine and the existing
machines, reformulate the problem where the sum of the weighted distances is
minimized, where the weights corresponding to the four machines are 6, 4, 7, and 2,
respectively.
c. In order to avoid congestion, suppose that the new machine must be located in the
square {(x1, x2) : -1 ≤ x1 ≤ 2, 0 ≤ x2 ≤ 1}. Formulate parts (a) and (b) with this added
restriction.
d. Suppose that the new machine must be located so that its distance from the first
machine does not exceed 2. Formulate the problem with this added restriction.
Question 6. For a telephone survey, a marketing research group needs to contact at least
150 wives, 120 husbands, 100 single adult males, and 110 single adult females. It costs
$2 to make a daytime call and (because of higher labor costs) $5 to make an evening call.
The table below lists the results that can be expected. For example, 30% of all daytime
calls are answered by a wife, and 15% of all evening calls are answered by a single male.
Because of limited staff, at most half of all phone calls can be evening calls.
Person Responding
Wife
Husband
Single Male
Single Female
None
Percentage of Daytime
Calls
30
10
10
10
40
Percentage of Evening Calls
30
30
15
20
5
Ignore the integer constraints and assume that everything is deterministic. Formulate a
linear program to minimize the cost of completing the survey.
Question 7. A production manager is planning the scheduling of three products on 4
machines. Each product can be manufactured on each of the machines. The unit
production cost (in $) are summarized.
PRODUCT
1
2
3
MACHINE
2
4
7
10
1
4
6
12
3
5
5
8
4
7
6
11
The time (in hours) required to produce each unit product on each of the machines is
summarized.
MACHINE
PRODUCT
1
2
3
1
0.3
0.2
0.8
2
0.25
0.3
0.6
3
0.2
0.2
0.6
4
0.2
0.25
0.5
Suppose that 4000, 5000, and 3000 units of the products are required, and that the
available machine hours are 1500, 1200, 1500, and 2000 respectively. Formulate the
scheduling problem as a linear program.