Bank Problem with Application of WinQSB for Transportationand Report to Manager

Source of the problem: page 86, W. L. Winston, "Operations Research, Application and Algorithms", 4th Edition, Thomson Learning, 2004, ISBN: 0-534-38058-1

Page 371 Problem 6

By: Parisay

Spring 2006

Problem statement:

A bank has two sites at which checks are processed. Site 1 can process 10,000 checks per day, and site 2 can process 6,000 checks per day. The bank processes three types of checks: vendor, salary, and personal. The processing cost per check depends on the site (see Table 11). Each day, 5,000 checks of each type must be processed. Formulate a balanced transportation problem to minimize the daily cost of processing checks.

From/To / Site1 / Site 2 / Supply
Vendor / 5 / 3 / 5000
Salary / 4 / 4 / 5000
Personal / 2 / 5 / 5000
Demand / 10000 / 6000

Input to WinQSB:

Output from WinQSB:

Summary solution:

Notice this summary solution (none Zero solution) does not have some detail information. Therefore, it is better to use the complete solution

Complete solution:

Notice: Unfilled_Demand is a node that WinQSB adds to indicate dummy supply point. Dummy supply point will be added when total capacity of supplies is less than total capacity of demands. Here X(salary-site 2) and X(Unfilled_Demand – site 1) are both BV!! Although they both have a solution equal to zero (Reduced cost is zero).

Report to manager, Part 1 (based on the solution, table above):

The minimum total daily cost will be $45,000 if the following plan is implemented. All 5000 vendor checks should be processed at site 2.All 5000 salary checks should be processed at site1.All 5000 personal checks should be processed at site1. There will be an extra capacity (unused capacity) at site 2 as much as 1000 checks. All the capacity of site 1 is used.

Alternate solution:

Notice: Click on Result option of top menu and you will see there is an option called “alternate solution”. Click on it and you will get the table above. That is this problem has two solutions. It helps with flexibility in decision making by a manager. Here X(Unfilled_Demand – site 2) is a BV!!

Report to manager, Part 2 (based on the alternate solution, table above):

There is also another optimal solution that provides some flexibility in terms of decision making. Both solutions (original and alternate) are summarized below. Both will provide the minimum total cost of $45000.

Solution 1 / Solution 2
Site 1 / Process 5000 personal and 5000 salary / Process 5000 personal and 4000 salary
Site 2 / Process 5000 vendor / Process 5000 vendor and 1000 salary
Unused capacity / 1000 at site 2 / 1000 at site 1

Sensitivity analysis on Cij (range of optimality based on the original solution):

Analysis: Here vendor-site1 is a NBV. If we want to process some vendor checks at site1 we need to reduce cost of processing from $5 to any value less than $3. Similarly, we can reduce cost of processing personal checks at site2 from $5 to less than $2. Here salary-site2 is a BV. The cost of processing salary at site 2 can be increased from $4 to $7 and still have the same processing plans, however, total cost will be more. Similarly, salary-site1 is BV, but cannot increase processing cost (range is $2 to $4). Also, vendor-site2 and personal-site1 are BV. They both can be increased up to $4and still have the same processing plans, however, total cost will be more.

Report to manager, Part 3 (based on range of optimality, table above):

If we want to process some vendor checks at site1 we need to reduce cost of processing from $5 to any value less than $3. Similarly, we can reduce cost of processing personal checks at site2 from $5 to less than $2. If we increase the processing cost of salary-site1 the total processing plan will change. The cost of processing salary at site 2 can be increased from $4 to $7 and still have the same processing plans, however, total cost will be more. Also, vendor-site2 and personal-site1 can be increased up to $4and still have the same processing plans, however, total cost will be more.

Sensitivity analysis on Si and Dj (range of feasibility based on the original solution):

Analysis: Capacity of vendor, personal, site 1, and site 2 are similar to a binding constraint for an LP (SP not zero). Capacity of personal checks has a negative shadow price which means if we increase personal check capacity by 1 the total cost will be reduced by 2, which is desirable. The allowable maximum value indicates that this capacity can increase up to 10000 checks. Similar situation holds for vendor checks, however the allowable minimum and maximum are equal and are 5000 which means we cannot change this capacity unless we want the total solution changes (Please refer to sensitivity analysis for LP. This is similar to the out of range for a binding constraint). We need to be careful how to interpret shadow price here. At this time I do not know how WinQSB deals with penalty cost for an unfulfilled demand. For example, the software indicates that, in last line, there is a shadow price of 4 for site 2. However,

site 2 has extra capacity and should be a nonbinding constraint with zero shadow price.

Report to manager, Part 4 (based on range of feasibility, table above):

Study of sensitivity analysis indicates that the only useful change is to increase personal checks up to 10000. For each addition personal check there will be $2 reduction on total cost.

You can use WinQSB to obtain a sensitivity graph/table. From “result” in top menu select perform parametric analysis and enter the range for graphing as below

Selection of parameters for sensitivity analysis on Cij

Sensitivity graph for cost of shipment from vendor to site 1:

Sensitivity graph for cost of shipment from salary to site 1:

Sensitivity table for cost of shipment from salary to site 1:

Analysis: Comparing the above table and table for range of optimality (Cij) we can notice that the above table is the result of solving this problem for four different ranges of processing cost. Ranges are: below $2, $2-4 (as in original problem), $4-6, and above $6. The processing plan for each range will be different from any other range (you can try it yourself). Notice that for values more than $6 the graph is flat, indicating that this cost will not have effect on solution. That is salary-site1 will become NBV with no effect on optimal solution.

Report to manager, Part 5 (based on sensitivity graph/table, table above):

This graph/table indicates the effect of changes in cost of salary-site1 on total cost. However, the detail of processing plan will be different at some levels of cost and we need to solve problem again if we are interested in such information. The graph indicates that if the processing cost is more than $6 then salry-site1 will not be in processing plan.

Final report to manager: Just put together all five parts of report.

Last updated: 4-29-06

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