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Introduction to Optimization

Week 8: Problem Sheet 7

Problem 1

In three warehouses $A_1$, $A_2$, and $A_3$ there are units of one type of product. The quantities available are:

  • $A_1$: 140 units,
  • $A_2$: 180 units,
  • $A_3$: 160 units.

These products must be shipped by truck to four distributing warehouses $B_1$, $B_2$, $B_3$, $B_4$, and $B_5$ with fixed demands:

  • $B_1$: 60 units,
  • $B_2$: 70 units,
  • $B_3$: 120 units,
  • $B_4$: 130 units,
  • $B_5$: 100 units.

The total supply equals the total demand (480 units). The shipping costs from $A_i$ to $B_j$ are given in the table below. Use the Northwest Corner Rule and the minimal element method to determine any feasible shipment plan.

Shipping Cost Table

Source \ Destination B1 B2 B3 B4 B5
$A_1$ 2 8 9
3
$A_2$ 4 7 1 4
$A_3$ 3 7 2

Solution

Northwest Corner Rule
Source \ Destination B1 (60) B2 (70) B3 (110) B4 (70) B5 (100)
$A_1$ 2
3
$A_2$ 7
4
$A_3$ 2

Total cost: $S = 2 \cdot 60 + 3 \cdot 70 + 4 \cdot 10 + 1 \cdot 110 + 4 \cdot 70 + 7 \cdot 60 + 2 \cdot 100 = 1380$

Minimal Element Method
Source \ Destination B1 (60) B2 (70) B3 (120) B4 (80) B5 (40)
$A_1$ 2
7
$A_2$ 3
4
$A_3$ 1

Total cost: $S = 2 \cdot 60 + 7 \cdot 70 + 1 \cdot 120 + 2 \cdot 80 + 7 \cdot 50 + 1 \cdot 40 = 1380$

Problem 2

In three warehouses $A_1$, $A_2$, and $A_3$ there are units of one type of product. The quantities available are:

  • $A_1$: 160 units,
  • $A_2$: 140 units,
  • $A_3$: 170 units.

These products must be shipped to four distributing warehouses $B_1$, $B_2$, $B_3$, and $B_4$ with fixed demands:

  • $B_1$: 120 units,
  • $B_2$: 50 units,
  • $B_3$: 190 units,
  • $B_4$: 110 units.

The shipping costs from $A_i$ to $B_j$ are given in the table below. Use the Northwest Corner Rule and the minimal element method to determine any feasible shipment plan.

Shipping Cost Table

Source \ Destination B1 (120) B2 (50) B3 (190) B4 (110)
$A_1$ 7
8
$A_2$ 5
9
$A_3$ 1

Solution

Northwest Corner Rule
Source \ Destination B1 (120) B2 (40) B3 (130) B4 (110)
$A_1$ 7
8
$A_2$ 5
9
$A_3$

Total cost: $S = 7 \cdot 120 + 8 \cdot 40 + 5 \cdot 10 + 9 \cdot 130 + 3 \cdot 60 + 6 \cdot 110 = 3230$

Minimal Element Method
Source \ Destination B1 (120) B2 (160) B3 (50) B4 (90)
$A_1$
7
$A_2$
4
$A_3$

Total cost: $S = 1 \cdot 160 + 4 \cdot 120 + 8 \cdot 20 + 2 \cdot 50 + 3 \cdot 30 + 6 \cdot 90 = 1530$

Problem 3

Use Vogel’s Approximation Method to determine any feasible shipment plan.

Shipping Cost Table (same as Problem 2)

Source \ Destination B1 (120) B2 (50) B3 (190) B4 (110)
$A_1$ 7
8
$A_2$ 5
9
$A_3$ 1

Solution

By Columns

Source \ Destination B1 B2 B3 (140) B4 (110)
$A_1$ 4
7 (120)
$A_2$
5 (30)
$A_3$
1 (60)

By Rows

Source \ Destination B1 B2 B3 (140) B4 (50)
$A_1$
7
5 (20)
$A_3$

Total cost: $S = 1 \cdot 50 + 2 \