ASSIGNMENT 1
SPRING SEMESTER 2024
ES 344 - OPTIMIZATION MODELING
1. Consider the following minimum cost transportation problem:
From To
1
2
4
3
10
Supply
6
A
100
8
16
6
B
300
14
18
10
C
Demand
300
200
300
200
700
(a) Use north-west corner method to find an initial feasible solution.
(b) Use minimum cost method to find an initial feasible solution.
(c) Use the transportation Simplex method to find an optimal solution.
2. Refer to previous problem 1. Suppose we must ship 100 units on the A-1 route. What
would be the optimal solution now?
3. Again refer to previous problem 1. Because of road construction, the route B-3 is
now unacceptable. Resolve the initiable problem.
4. Digital Solutions supplies goods to three customers, who each require 30 units. The
company has two warehouses. Warehouse 1 has 40 units available, and warehouse
2 has 30 units available. The costs of shipping 1 unit from warehouse to customer
are shown in the following table:
To
From
Customer 1 Customer 2
Customer 3
Warehouse 1
$15
$35
$25
Warehouse 2
$10
$50
$40
There is a penalty for each unmet customer unit of demand: With customer 1, a
penalty cost of $90 is incurred; with customer 2, $80; and with customer 3, $110.
Formulate a balanced transportation problem to minimize the sum of shortage
and shipping costs.
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5. Refer to previous problem 4. Use the transportation simplex algorithm to solve the
problem. Interpret your answer.
6. Again refer to previous problem 4. Now suppose that extra units could be purchased
and shipped to either warehouse for a total cost of $100 per unit and that all
customer demand must be met. Formulate a balanced transportation problem to
minimize the sum of purchasing and shipping costs. Use the transportation simplex
algorithm to solve the problem. Interpret your answer.
7. Based on your solution of problems 4, 5 and 6, formulate the best policy for Digital
Solutions to meet its customer demands.
8. The distribution system for the Herman Company consists of three plants, two
warehouses, and four customers. Plant capacities and shipping costs per unit (in
dollars) from each plant to each warehouse are as follows:
Plant
Warehouse 1 Warehouse 2 Capacity
1
4
7
450
2
8
5
600
3
5
6
380
Customer demand and shipping costs per unit (in dollars) from each warehouse to
each customer are
From
Customer 1
Customer 2 Customer 3
Customer 4
Warehouse 1
6
4
8
4
Warehouse 2
3
6
7
7
Demand
300
300
300
400
The objective is to have an optimal shipping plan that will meet the demands at
minimum cost.
(a) Develop a network representation of this problem.
(b) Formulate a linear programming model of the problem.
(c) Develop a balanced transportation model (in tableau form) of this problem.
(d) Solve the problem using Transportation Algorithm and interpret your solution.
9. Refer to previous problem 8. Now suppose that shipments between the two warehouses are permitted at $2 per unit and that direct shipments can be made from
plant 3 to customer 4 at a cost of $7 per unit. How will you incorporate thesechanges
in the above three models?
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