ITY512EOPERATIONSRESEARCH
ASSIGNMENT1-ANSWERS
1) a)
Decision variables:
x1 = number of Milas carpet to produce
x2 = number of Uşak carpet to produce
Objective function:
Max Z = 1200x1 + 600 x2
Constrains:
x1 £ 6
x2 £ 4
6 x1 + 8 x2 £ 48
x1,x2 ³ 0
b) See the graphic below. There are two candidate points for solution: point A, B
x1
x2
Z
A
6
3/2
8100*
*
B
8/3
4
5600
Point A gives the maximum Z. ATK-White should produce 6 Milas caprpets and 3/2 of
Uşak carpets to make maximum profit of 8100TL per month.
x2
x1 £ 6
B
Feasibleregion A
x2£4
x1
6 x1 + 8 x2 £ 48
c) The feasible region does not change. For the profit per Milas carpet decreases from 1200TL
to 800TL the Z values are calculated as follows:
x1
x2
Z
A
6
3/2
5700**
B
8/3
4
4534
The optimal x1, x2 values do not change, profit will decrease to 5700TL.
For the profit per Milas carpet decreases from 1200TL to 400TL the Z values are calculated
as follows:
x1
x2
Z
A
6
3/2
3300
B
8/3
4
3467**
The optimal x1, x2 values will change to (4, 8/3). Profit will decrease to 3467TL.
2) a)
Decision variables:
𝑥𝑖𝑗 : amount of raw oil i purchased in month j ( i= 1 (Çanakkale), 2 (İzmir), 3 (Antalya), 4
(Hatay); j = 1 (October), …., 7 (April))
𝑦𝑖𝑗 : amount of raw oil i blended in month j
𝑡𝑖𝑗 : amount of raw oil i stocked at the end of month j
Objective function:
.
1
𝑀𝑎𝑥145
.
1
𝑦,- −
,/0 -/0
1
.
𝑐,- 𝑥,- − 7
,/0 -/0
𝑡,,/0 -/0
where cij is the raw oil prices for raw oil i in mouth j.
Subject to the following constraints:
Hardness constraint:
1
3
1
𝑦,- ≤
1
ℎ, 𝑦,- ≤ 6
,/0
,/0
𝑦,- 𝑓𝑜𝑟𝑗 = 1, … ,7.
,/0
where hi is the harness of raw oil i.
Production line capacity:
𝑦1𝑗 + 𝑦2𝑗 ≤ 220𝑓𝑜𝑟𝑗 = 1, … ,7.
𝑦3𝑗 + 𝑦4𝑗 ≤ 300𝑓𝑜𝑟𝑗 = 1, … ,7
Balancing purchased – stocked – blended:
𝑥𝑖𝑗 + 𝑡𝑖 𝑗−1 = 𝑦𝑖𝑗 + 𝑡𝑖𝑗 𝑓𝑜𝑟𝑖 = 1, … 4.
where 𝑡𝑖0 = 250, 𝑖 = 1 … 4 (initial stock)
At the end of April, required raw oil stock:
𝑡𝑖7 = 250𝑓𝑜𝑟𝑖 = 1, … 4.
Storage capacity:
4
𝑡𝑖𝑗 ≤ 1000𝑓𝑜𝑟𝑗 = 1, … ,7.
𝑖=1
Sign restrictions:
𝑥𝑖𝑗 , 𝑦𝑖𝑗 , 𝑡𝑖𝑗 ≥ 0𝑓𝑜𝑟𝑖 = 1, … 4; 𝑗 = 1, … 7.
b) When the model is solved the maximum profit will be found as 160.640 TL. See the excel
file for the formulation of the model in Excel Solver. Values of the decision variables will be
as follows:
𝑥𝑖𝑗 October
November
December
January
February
March
April
𝑦𝑖𝑗 October
November
December
January
February
March
April
𝑡𝑖𝑗 October
November
December
January
February
March
April
Çanakkale İzmir
Antalya Hatay
0
0
0
0
640
0
0
400
0
0
0
0
0
0
850
0
0
0
0
0
650
250
0
300
0
0
0
550
Çanakkale İzmir
Antalya
Hatay
220
0
50
250
200
20
0
300
220
0
200
100
30
190
300
0
220
0
300
0
200
20
0
300
200
Çanakkale
30
470
250
220
0
450
250
20
İzmir
250
230
230
40
40
270
250
0
Antalya
200
200
0
550
250
250
250
300
Hatay
0
100
0
0
0
0
250
According to the results, for instance, ATK-Brown will not purchase any raw oil in October
while the production of October will be made from the raw oil at the stocks. The purchasing is
mostly made on the moths when the sales price is low. For instance, Çanakkale oil will be
purchased on November and when its price is at lowest levels, 80 TL, 105 TL, respectively.
Production line capacities seems like the most restrictive limitaiton on the production, that is
all capacities are fully utilized.