ITY512SE OPERATIONS RESEARCH
ASSIGNMENT 2
Özgür Kabak, Ph.D. October 23, 2013
November 7, 2013
 You can constitute a group of max. three students to submit HW. Names and numbers of the group members should be indicated on the cover. No cheating will be tolerated!
 You can ask your questions to Özgür Kabak ( ozgurkabak@gmail.com
).
 Any homework brought after the class hour on November 7, 2013 will not be accepted .
 Please send the QM for Windows file (1 st question) and EXCEL file (2 nd question) to Özgür Kabak (
 Submit your solutions as printouts to the lecturer. E-mail attachments are not accepted. ozgurkabak@gmail.com
).
QUESTIONS
1.
(50 Points) The ATK Oil Company controls two oil fields. Field 1 can produce up to 65 million barrels of oil per day, and field 2 can produce up to 75 million barrels of oil per day. At field 1, it costs $4 to extract and refine a barrel of oil; at field 2, the cost is $3. ATK sells oil to three countries: US, Canada and China. The shipping cost per barrel is shown in the following table. Each day, US is willing to buy up to 45 million barrels (at $6.6 per barrel), Canada is willing to buy 35 million barrels (at $7.4 per barrel) and China is willing to buy up to 40 million barrels (at $7.1 per barrel). a.
Formulate a balanced transportation problem to maximize ATK’s profits. b.
Find bfs for the problem using NWC method, minimum cost method, and Vogel’s method. (Notice that the minimum cost and Vogel’s methods should be revised according to a maximization problem)
From ($)
Field 1
Field 2
US
0.9
1.5
To ($)
Canada
1.5
2.5
China
2.1
1.6 c.
Write down the LP for the balanced transportation problem formulated in part (a), d.
Solve LP formulated in part (c) using QM for Windows software. Interpret the solution (Please send the QM file to ozgurkabak@gmail.com
)
2.
(70 points) AEK-White case and related LP formulation is given in the following. a) Solve the given LP using MS Excel Solver (Please send the Excel file to ozgurkabak@gmail.com
). b) Summarize and interpret the results.
AEK-White
AEK-White is a supplier of an automobile factory that makes seven products (PR1 to PR7) on the following machines: four grinders, two vertical drills, three horizontal drills, one borer, and one planer. Each product yields a certain contribution to profit (defined as TL/unit selling price minus cost of raw materials). These quantities (in
TL/unit) together with the unit production times (hours) required on each process are given below. A dash indicates that a product does not require a process.
PR1 PR2 PR3 PR4 PR5 PR6 PR7
Contribution to profit (TL /unit) 8 4 7 4 2 6 9
Grinding (hours)
Vertical drilling (hours)
0.9
0.15
0
0.25
0
0
0.6
0.35
1.5
0
0.8
0.7
1.4
0.1
Horizontal drilling (hours)
Boring (hours)
0.4
0.25
0.2
0.1
1.2
0.07
0
0.4
0
0.5
0.2
0
0.9
0.4
Planning (hours) 0 0 0.1 0 0.5 0 0.5
In the present month (January) and the five subsequent months certain machines will be down for maintenance.
These machines will be:
January: 1 grinder
February: 2 horizontal drills

March: 1 borer
April: 1 vertical drill
May:
June:
1 grinder and 1 vertical drill
1 planer and 1 horizontal drill
There are marketing limitations on each product in each month. These are:
PR1 PR2 PR3 PR4 PR5 PR6 PR7
January
300 200 200 0 300 100 400
February 500 300 500 600 100 300 600
March 400 200 300 400 300 0 300
April
100 300 0 600 300 300 200
May 500 200 600 100 200 100 0
June
200 400 400 100 100 600 100
It is possible to store up to 100 of each product at a time at a cost of TL0.5 per unit per month. There are no stocks at present but it is desired to have a stock of 50 of each type of product at the end of June.
The factory works a 6 day week with two shifts of 8 hours each day. It may be assumed that each month consists of only 24 working days.
The linear programming model to maximize the total profit of AEK-White is formulated as follows:
LP MODEL
Decision Variables:
P ij
: production amount of product i at month j , i = 1..7, j = 1,…,6
S ij
: sales amount of product i at month j , i = 1..7, j = 1,…,6
V ij
: inventory of product i at the end of month j , i = 1..7, j = 1,…,6
Objective function:
Max
∑ 7 𝑖=1
∑ 6 𝑗=1
𝐶𝑜𝑛𝑡 𝑖
𝑆 𝑖𝑗
− 0.5 ∑ 7 𝑖=1
∑ 6 𝑗=1
𝑉 𝑖𝑗
Where Cont i
is unit contribution of product i to profit.
Constraints:
Inventory, production, sales balance:
V i ( j -1)
+ P ij
= S ij
+ V ij
 i , j
Marketing limitations:
S ij

M ij
 i , j, where M ij
is the marketing limitation on product i in month j .
Machine capacities:
∑ 7 𝑖=1
𝑃 𝑖𝑗
𝑅 𝑖𝑘
≤ 𝐴𝑉 𝑗𝑘
∗ 384  j,k where R ik
is unit production time (hours) of product i required on machine k .
AV jk
is the number of available machine k in month j.
Maximum storage:
V ij

100
 i , j
Desired stock levels:
V i 6
= 50
 i .
Sign restrictions: All decision variables

0.