EGR 549
Spring 2003
Problem 1 / Page 371
Sensitivity Analysis Minimization Transportation Problem
8/1/2017
By: Kenneth Tang
Page 371, problem 1
A company 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 Table. 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.
Warehouse 1
Warehouse 2
Demand
Penalty
Customer 1
$15
$10
30
$90
Customer 2
$35
$50
30
$80
Customer 3
$25
$40
30
$110
Supply
40
30
20
a). NWC Method to find bfs
Initial Transportation Tableau:
The total cost of shipment for initial bfs is $4400.
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EGR 549
Spring 2003
Problem 1 / Page 371
Sensitivity Analysis Minimization Transportation Problem
(a)
Use WinQSB software to solve the problem. Input data.
Notice that we had to enter dummy supply, just because of including penalty costs.
(b)
Output from WinQSB.
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8/1/2017
By: Kenneth Tang
EGR 549
Spring 2003
(c)
Problem 1 / Page 371
Sensitivity Analysis Minimization Transportation Problem
Range of Optimality
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8/1/2017
By: Kenneth Tang
EGR 549
Spring 2003
(d)
Range of Feasibility
(e)
Sensitivity Analysis
Problem 1 / Page 371
Sensitivity Analysis Minimization Transportation Problem
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8/1/2017
By: Kenneth Tang
EGR 549
Spring 2003
Problem 1 / Page 371
Sensitivity Analysis Minimization Transportation Problem
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8/1/2017
By: Kenneth Tang
EGR 549
Spring 2003
Problem 1 / Page 371
Sensitivity Analysis Minimization Transportation Problem
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8/1/2017
By: Kenneth Tang
EGR 549
Spring 2003
Problem 1 / Page 371
Sensitivity Analysis Minimization Transportation Problem
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8/1/2017
By: Kenneth Tang
EGR 549
Spring 2003
Problem 1 / Page 371
Sensitivity Analysis Minimization Transportation Problem
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8/1/2017
By: Kenneth Tang
EGR 549
Problem 1 / Page 371
Spring 2003
Sensitivity Analysis Minimization Transportation Problem
(f)
Write a report for a manager and explain important findings.
8/1/2017
By: Kenneth Tang
Based on the data, the following shipment distribution should be used to minimize the total shortage and shipment costs. The Warehouse 1 should
be shipping 10 units to Customer 2 and 30 units to Customer 3; Warehouse 2 should be shipping 30 units to Customer 1. By doing so, the total
shortage and shipment costs will be minimized to $3,000.
Although the current solution does not include the shipment from Warehouse 1 to Customer 1, these can be done if the shipment cost is increased
from $15 to $20. It should be noticed that the company should not be shipping any quantities from Warehouse 2 to Customer 2 and 3. This can be
done only if the shipment cost is free of charge, which is impossible in this situation in the real world. Moreover, the capacity of the Warehouse 1
and 2 should have at least 40 and 30, respectively. On the other hand, the Customer 1, 2, and 3 should have storages with the capacity of at least
30, 10, and 30, respectively.
Since the shadow price of the 40 units supply for the Warehouse 1 is -$45, which mean an extra unit of supply will decrease the total shortage and
shipment cost to $45. Thus, the company should be willing to pay up to $45 for an extra unit of supply for Warehouse 1. On the other hand, for the
Customer 1, because of the shadow price is $40; Customer 1 should not be paying for any extra unit of products, because this will increase the
total shipment cost. Likewise, it is same explanation for other productions. In short, Customer 2 and 3 should not be paying extra unit of product.
However, reducing the demand for those Customers is recommended in order to minimize the total shipment cost.
Let discuss one of the shipment by changing the unit costs for the shipment from Warehouse 1 to Customer 3. The sensitivity analysis table and
graph indicate that increasing the unit cost beyond $45 is not recommended. This is due to the fact that it will not improve the shortage and
shipment cost as well as the extra supplies will not be demanded. If the shipment cost is raised to $1, i.e. $26, 40 units of supply are still optimal to
be shipped. In this situation, the optimal range of increasing is within $25 to $45. Other shipping production can be analyzed with the same
fashion. On the other hand, 40 units supply from Warehouse 1 are allowing increasing within 40 units to 60 units. Beyond the range can be
jeopardized the other supplies and the total shipment cost will not be optimized. More analysis is available upon request.
In addition, it was determined that 20 units were not met from customer 2. Therefore, they would suffer a penalty of $1600.
From
Ware House 1
Ware House 1
Ware House 2
To
Customer 2
Customer 3
Customer 1
Quantity
10
30
30
No product is being shipped from Warehouse 1 to customer 1. If the unit cost of shipment of transporting products is reduced by $20 then
products can be shipped from Warehouse 1 to customer 1. But this is not possible since at present the cost of shipment is $15. Reducing it by $20
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EGR 549
Problem 1 / Page 371
8/1/2017
Spring 2003
Sensitivity Analysis Minimization Transportation Problem
By: Kenneth Tang
would mean a negative shipment cost which is absurd (it means profit!). Hence under no circumstance could the product be shipped from
Warehouse 1 to customer 1.
(The above paragraph is important to pay attention. The software performs some mathematical calculation. It is not intelligent to know
what each figure means. It is up to us, intelligent beings, to interpret the output properly. Such as this case a cost of 15-20=-5 will only
means that some shipment company will pay us $5 per unit as well as ship our units!!! Which will not happen. So Ronita’s conclusion is
what I expect and is correct.)
Since the total demand is greater than the total supply, the company must increase their capacity to produce more to meet the extra demand
otherwise they will have to pay a penalty of $1600. Customer 2 must look for a new source to meet its unfulfilled demand.
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