END395 – OPERATIONS RESEARCH II
SELF STUDY 1 QUESTIONS
1. Comment whether the following statements are TRUE or FALSE. Explain your answers in
detail.
a) The feasible region of an LP is a convex set.
b) For a maximization problem, the optimal objective function value of the LP relaxation
of an IP is smaller than the optimal objective function value of the corresponding IP.
c) For a minimization problem, any feasible solution to an IP has a smaller objective
function value than the optimal objective function value of the corresponding IP.
d) The feasible region of the LP relaxation contains the feasible region of the
corresponding IP.
e) If the upper and lower bound to an IP model are equal, it can be said that optimal
solution to that model is found.
f) The feasible region of the LP relaxation covers the feasible region of the corresponding
IP.
2. According to the graph given below:
a)
b)
c)
d)
Provide an objective function whose optimal solution is (0,1).
Provide an objective function whose optimal solution is (0,0).
Provide an objective function whose optimal solution is (1,2).
Is there any objective function that yields infeasible solution? If your answer is yes,
provide the objective function.
3. A firm sells item 1 at a profit of $2/unit and item 2 at a profit of $5/unit. Three units of raw
material are needed to produce 1 unit of item 1, and 6 units of raw material are needed to
produce 1 unit of item 2. A total of 120 units of raw material are available. If any of item 1 is
produced, a setup cost of $10 is incurred, and if any of item 2 is produced, a setup cost of $20
is incurred. Formulate an IP to maximize profits.
4. Cutting Stock Problem: Assume there are two standard wooden rods with lengths 1.2 meters
and 1 meter. Currently, there are twenty 1.2-meter (1.2m) rods and twenty-five 1-meter (1m)
rod in the storage. There is a demand for sixty 0.8-meter (0.8m), forty 0.5-meter (0.5m) and
seventy five 0.3-meter (0.3m) rods. To obtain the desired lengths we can either cut existing rods
at a cost of 0.5 TL per cut, or purchase new rods at a cost of 2 TL, 1.5 TL and 1.1 TL for the
0.8m, 0.5m and 0.3m rods, respectively. Formulate the problem to minimize the total cost for
obtaining the required rods.
5. A product can be produced on five different machines. Each machine has a fixed setup cost,
variable production costs per-unit-processed, and a production capacity given in table. A total
of 2000 units of the product must be produced. Formulate an IP to minimize the total cost.
Machine
1
2
3
4
5
Fixed Cost
6000
2500
2300
2100
7000
Variable Cost per Unit
30
20
16
60
28
Capacity
900
1000
1200
1600
1800
6. A firm produces three types of clothing: shirts, shorts, and pants. Manufacturing of each
type of clothing requires the appropriate type of machinery available. The machinery needed to
manufacture each type of clothing must be rented at the following rate: Shirt machinery, $200
per week; Shorts machinery, $150 per week; Pants machinery, $100 per week. Manufacturing
of each type of clothing also requires the amounts of cloth and labor shown in the Table 1. Each
week 150 hours of labor and 160 meters of cloth are available. In addition, the variable unit
cost and selling price for each type of clothing are shown in the Table 2. Formulate an IP whose
solution will maximize firm’s weekly profit.
Table 1. Resources to produce cloths by clothing type
Clothing Type
Shirt
Shots
Pants
Labor (Hours)
3
2
6
Cloth (Meter)
4
3
4
Table 2. Sales prices and variable costs by clothing type
Clothing Type
Shirt
Shots
Pants
Sales Price
3
2
6
Variable Cost
4
3
4
soruyu anladim x neden if olan bir decision variable oluyor onu anlamadm bir de big m kisiti neden öyle oluyor anlamadim
7. TS receives credit card payments from four regions of Turkey (Karadeniz, Akdeniz, İç
Anadolu and Doğu Anadolu). The average daily value of payments mailed by customers from
each province is as follows: Karadeniz, $70000; Akdeniz, $50000; İç Anadolu, $60000; Doğu
Anadolu, $40000. TS must decide where customers should mail their payments. Because TS
can earn 20% annual interest by investing these revenues, it would like to receive payments as
quickly as possible. TS is considering setting up operations to process payments (often referred
to as lockboxes) in four different cities: Trabzon, Antalya, Nevşehir, and Ankara. The average
number of days (from time payment is sent) until a check clears and TS can deposit the money
depends on the city to which the payment is mailed, as shown in Table below. For example, if
a check is mailed from the Karadeniz to Ankara, it would take an average of 8 days before TS
could earn interest on the check. The annual cost of running a lockbox in any city is $50,000.
Formulate an IP that TS can use to minimize the sum of costs due to lost interest and lockbox
operations. Assume that each region must send all its money to a single city and that there is no
limit on the amount of money that each lockbox can handle.
From/To
Karadeniz
Akdeniz
İç Anadolu
Doğu Anadolu
City 1 (Trabzon) City 2 (Antalya)
2
6
6
2
8
5
8
5
City 3 (Nevşehir) City 4 (Ankara)
8
8
5
5
2
5
5
2