IE 416
Operations Research I
Extra Credit-Transshipment Problem
Presented to :
Dr. Sima Parisay
Revised 12/3/2011
Submitted by:
Hashmat Amin
Valerie Bastian
Gonzalo Keymolent
Po Chuen (Boris) Law
Problem Statement:
Optimal Ovens, Incorporated makes home toaster ovens at their production plants in Wisconsin and Alabama.
They ship completed ovens to warehouses in either Memphis or Pittsburgh. Next, they are distributed to
customer facilities in Fresno, Peoria, and Newark. The two warehouses also use company trucks to ship ovens
between warehouses at a cost of $2 per unit shipment.
The task is to plan OOI’s distribution of ovens over the next month. Up to 1000 units can be shipped from each
plant during this period. At this time, no units are stored in the warehouses. Fresno, Peoria, Newark require 450,
500, and 610 ovens respectively.
Unit Costs Tables:
From/To
Wisconsin
Alabama
From/To
Memphis
Pittsburgh
Memphis
7
4
Fresno
25
29
Pittsburgh
8
7
Peoria
5
8
Newark
17
5
Our task is to find a shipment plan that will minimize the total cost.
We will also perform two sensitivity analyses:
-unit cost of shipment sensitivity
-capacity sensitivity
Next, we will solve the problem again, now considering that Alabama has a 500 unit capacity.
Finally, we will include a Report to Manager.
Problem Formulation:
The company has 2 production plants, 2 warehouses, and 3 customer facilities. This means there are 7 nodes to
consider.
Graphical Representation of Problem:
Input to WinQSB:
This is the input table for WinQSB to find an optimal shipment plan with a minimum total cost.
Wisconsin and Alabama have supply capacities of 1000 ovens each because they can each ship 1000 ovens.
The demand capacities are zero for these nodes because they are the production plants (that do not demand any
ovens).
Memphis and Pittsburgh are limited to send and receive at most 2000 ovens (simply by the fact that the
production plants can only supply up to 1000 ovens each). However, we cannot enter a supply capacity of 2000
for each of these nodes, because it will affect the total supply capacity (The computer will “think” that there are
6000 ovens of supply available, which is not accurate). It’s best to use “M” (a very large number). The total
demand capacity would be affected in the same way, and that is why we use “M” for the demand capacities at
these nodes. (Zero capacity should work the same.)
Fresno, Peoria, and Newark have demand capacities of 450, 500, and 610 ovens, respectively. Their supply
capacities are zero because they are the customer facilities (that do not supply any ovens).
The unit costs are given; however, “M” is used to eliminate illogical shipments. “M” (a very large number) will
be ignored by the software, because there are cheaper routes to select. It will choose the cheaper route because it
is a minimization problem.
Solution:
WinQSB Output Table: (It is better to have the complete solution here.)
Solution Summary:
The total minimum cost for this shipment plan is $25,380.
From
Wisconsin Plant
Alabama Plant
Memphis Warehouse
Memphis Warehouse
Memphis Warehouse
Pittsburgh Warehouse
To
Pittsburgh Warehouse
Memphis Warehouse
Fresno Facility
Peoria Facility
Pittsburgh Warehouse
Newark Facility
Units Shipped
560
1000
450
500
50
610
The required demand at all three customer facilities are met. At the Wisconsin plant, there is overproduction by
440 ovens. Either need to reduce production in Wisconsin by 440 ovens (only produce 560 ovens) or create
another customer facility to accommodate 440 ovens.
Both our warehouses are utilized. The Memphis warehouse requires a capacity of 1000 units and the Pittsburgh
warehouse requires a capacity of 610 units for this shipment plan.
WinQSB has provided an optimal solution; however, as engineers we should look further into the information
available, and find ways to reduce the total cost even further. We will perform sensitivity analysis to achieve
this.
Final conclusions will be discussed in the Report to Manager.
Sensitivity Analysis: Changes in Unit Cost
How sensitive is the objective function value to changes in the unit cost? We can find out by using the WinQSB
solution to show the Range of Optimality Table.
Range of Optimality Table:
Now we can select one unit cost (of shipment) to analyze further and determine its effect on the objective value
(the total minimum cost).
If we choose to analyze a change in the shipping cost of a basic variable, it means we are considering a route
that we currently ship through. If we also choose to respect the allowable minimum and maximum values, this
means we will keep the same basis (same routes), and our solution values (the amount we ship in these same
routes) will be the same. The total cost will change based on the new unit cost for the route we consider. If a
change in the unit shipping cost is outside the range, it is expected that the shipping routes, the amount that ship
between the routes, and also the total cost will change.
Or we can choose to analyze the shipping cost of a non-basic variable, which means we are considering the cost
of shipping through a route we are not utilizing. If we choose to consider the reduced cost (which is applicable
to non-basic variables) it means we are selecting a cost outside the allowable range. Reduced cost is how much
the unit cost will need to decrease in order for the non-basic variable to become a basic variable (in other words,
reduced cost is how much the unit cost will need to decrease in order to utilize a shipment in this route). The
total cost will change based on changes in the new shipment plan and the new unit cost.
If a change in unit cost is made within the range (the reduced cost ignored) of a non-basic variable, we expect
no changes in the shipment plan or the total cost. This is because if you don’t ship through a route in the first
place, changing the unit cost within the range still means you don’t ship through the route, so you still cannot
expect a change in total cost. It is better to provide one example.
We chose to perform a sensitivity analysis on the unit cost of flows from the Memphis Warehouse to the Fresno
Facility. We choose this one because we feel the largest changes can be made here. Please refer to the Changes
in Unit Cost Table.
Changes in Unit Cost Table: Good Job 
From
To
Units
Shipped
Current
Unit Cost
Proposed
New Unit
Cost (a 2%
reduction)
Change in unit
cost (2%
reduction)
Wisconsin
Plant
Pittsburgh
Warehouse
560
$8
8* (0.98)=
$7.84
8-7.84=$0.16
560(.16)=
Or
$89.60
8*(0.02)=$0.16
Alabama Plant
Memphis
Warehouse
Fresno Facility
1000
$4
$3.92
$0.08
$80.00
450
$25
$24.50
$0.50
$225
Peoria Facility
500
$5
$4.90
$0.10
$50
Pittsburgh
50
Warehouse
Newark Facility 610
$2
$1.96
$0.04
$2.00
$5
$4.90
$0.10
$61.00
Memphis
Warehouse
Memphis
Warehouse
Memphis
Warehouse
Pittsburgh
Warehouse
How much
we can
potentially
save
We need to make a decision on what change in unit cost can provide the greatest savings. One may say choose
the most expensive route we have, and reduce the cost. One may say, choose the route that ships the most units,
and reduce the cost there. Although these are GOOD ways to reduce the total minimum cost, there is a
BETTER way.
Please notice in the Changes in Unit Cost Table, Alabama to Memphis ships the most units (1000). However, if
we reduce the cost by 2%, we only save $80. There are two better options. A 2% reduction from Wisconsin to
Pittsburgh saves almost $90, and a 2% reduction from Memphis to Fresno saves $225.
Since we want to minimize costs, we want to choose Memphis to Fresno route to reduce the cost.
Once we have made this decision, we need to check if we can make this 2% reduction in cost (from $25 to
$24.50). The Range of Optimality Table will tell us what is the minimum and maximum amount the unit cost
can be to keep the same routes and shipment amounts (same basis and solution). In this case, the table says the
allowable range is from -M to 26. So 25 is within this range, meaning this will be a good place to perform
sensitivity analysis on.
Mathematically, the software says we can reduce the cost to $0 or $1 and still keep this solution, however,
practically speaking such large reductions are unrealistic to achieve.
This is why we feel that reducing the cost by 2% (from $25 to $24.50) is a much more feasible and realistic way
to approach this sensitivity analysis. We can calculate this effect on the objective value manually or we can reenter the new unit cost into WinQSB to get a new z value (total minimum cost).
Z= original Z + (change in cost)*(units shipped)
=25,380 + (-$0.50)*(450)
=$25,380 - $225
= $25,155 total minimum cost if we make this 2% reduction
Sensitivity Analysis: Changes in capacity
Next, we want to investigate how sensitive the objective function value (the total cost, minimized) is to changes
in the capacity.
We can find out by using the WinQSB solution to show the Range of Feasibility Table.
Range of Feasibility Table:
Now we can select one capacity (of either the supply or the demand) to analyze further and determine its effect
on the objective value (the total minimum cost).
The nodes with a shadow price indicate binding constraints. This is because the shadow price indicates how
much the objective function value will change, when a change is made in the [binding] capacity. We need to
remember that the idea of a binding constraint is that even when small changes occur, we expect the objective
value to be affected. Of course that also means that when small changes occur in non-binding constraints we
don’t expect to see an immediate or drastic change in the objective function value (which is why we see shadow
price of zero, to represent this characteristic of a non-binding constraint).
Shadow price means that for every unit increase in capacity (with respect to the allowable range) the shadow
price is added to the objective function value. (Adding shadow price (of positive or negative value) is how
WinQSB handles this calculation). Equally, for every unit decrease in capacity, we expect shadow price to be
subtracted from the objective function value.
That being said, it is to our advantage (minimizing total cost) to perform sensitivity analysis on nodes in which
we have a negative shadow price (if we increase the capacity) or a positive shadow price (if we reduce the
capacity).
We are interested in performing a sensitivity analysis on the Fresno Facility node. We choose this node because
it has the highest shadow price, and we can make changes to the demand capacity that respects the allowable
range. This way, we can reduce the demand capacity here and benefit from reducing our objective function
value.
That being said, the allowable range is from 0 to 450 ovens. Mathematically, we can reduce the capacity to zero
(that is, close down this facility) because it will actually save us $31 per oven. Basically, the objective function
value will reduce to $11,430 (at a savings of $13,950). But we need to look at this practically.
The objective function value is to reduce total costs. We can close down all facilities and stop doing business all
together; this would reduce all our costs to zero! Of course this is not what we should suggest. Instead, we want
to perform a sensitivity analysis on what would happen if we reduce our demand capacity by 5%.
Currently the demand is 450 ovens in Fresno, we suggest a 5% reduction (23 ovens less), so that the demand
will be 427 ovens. What is the effect on total minimum cost?
Z= original Z + (change in capacity)*(shadow price)
=25,380 + (-23 ovens)*($31)
=$25,380 - $713
= $24,667 total minimum cost if we make this 5% reduction
**Although we can manually calculate the change in the total minimum cost, we need to re-input this
information into WinQSB because changes in capacity mean it’s possible the shipment plan can change (both
the routes and shipment amounts).
WinQSB output, considering the capacity change:
Note: If we had suggested making changes for both supply AND demand capacities at the same time, we cannot
use the current feasibility table to manually answer that effect and we had to change WinQSB input.
Sensitivity Analysis: Consider both changes (This was not expected from this class. Changes should be
implemented one by one if manual analysis is expected.)
We previously made conclusions about reducing unit cost by 2% from Memphis to Fresno shipments, and also
reducing Fresno capacity by 5%. We know from our calculations what these changes individually will reduce
our total costs by ($255 from unit cost reductions and $713 from capacity reductions). However, to see the
combined effect we need to input these changes into WinQSB to see more exactly the total minimum cost, as
well as the new shipment plan.
WinQSB input:
WinQSB output:
Solution Summary:
The total minimum cost for the shipment plan (considering both changes in the sensitivity analysis) is
$24,453.50
Here is the shipment plan:
From
Wisconsin Plant
Alabama Plant
Memphis Warehouse
Memphis Warehouse
Memphis Warehouse
Pittsburgh Warehouse
To
Pittsburgh Warehouse
Memphis Warehouse
Fresno Facility
Peoria Facility
Pittsburgh Warehouse
Newark Facility
Units Shipped
537
1000
427
500
73
610
The required demand is met at all three customer facilities. The Wisconsin plant overproduces by 463 ovens.
Therefore, it is recommended that production is reduced by 463 ovens or another customer facility is created to
accommodate 463 ovens.
Both warehouses are utilized. The Memphis warehouse requires a capacity of 1000 units and the Pittsburgh
warehouse requires a capacity of 646 units for this shipment plan.
Final conclusions will be discussed in the Report to Manager.
Problem Formulation:
Assume the same original problem statement, but now consider that the capacity of the Alabama plant is only
500 ovens (previously 1000).
Input to WinQSB:
WinQSB Output Table:
Solution Summary:
The total minimum cost for the shipment plan is $24,910.
Here is the shipment plan:
From
Wisconsin Plant
Wisconsin Plant
Alabama Plant
Memphis Warehouse
Memphis Warehouse
Pittsburgh Warehouse
To
Memphis Warehouse
Pittsburgh Warehouse
Memphis Warehouse
Fresno Facility
Peoria Facility
Newark Facility
Ovens Shipped
390
610
500
390
500
610
The required demand is not met at all three customer facilities. The Fresno Facility is not meeting demand by 60
ovens. Therefore, it is recommended that production be increased by 60 ovens at one of the plants, or another
supplier is found to accommodate 60 ovens.
Both warehouses are utilized. The Memphis warehouse requires a capacity of 890 units and the Pittsburgh
warehouse requires a capacity of 610 units for this shipment plan.
Final conclusions will be discussed in the Report to Manager.
Report to Manager:
Dear Manager,
The total minimum cost for our shipment plan is $24,453.50
Here is the shipment plan:
From
Wisconsin Plant
Alabama Plant
Memphis Warehouse
Memphis Warehouse
Memphis Warehouse
Pittsburgh Warehouse
To
Pittsburgh Warehouse
Memphis Warehouse
Fresno Facility
Peoria Facility
Pittsburgh Warehouse
Newark Facility
Units Shipped
537
1000
427
500
73
610
We have met required demand at all three customer facilities; however, we have reduced the demand from 450
to 427 ovens in Fresno to reduce our costs. Our Wisconsin plant overproduces by 463 ovens. Therefore, it is
recommended that production is reduced by 463 ovens or another customer facility is created to accommodate
463 ovens.
Both warehouses are utilized. The Memphis warehouse requires a capacity of 1000 units and the Pittsburgh
warehouse requires a capacity of 646 units for this shipment plan.
These are the suggestions we make based on two investigations. First, we moved to reduce to the unit cost of
shipments from Memphis to Fresno by 2% and secondly, we looked into reducing the capacity of Fresno by 5%.
Please see how these changes affected our final plan (final plan has lowest total cost)
From
To
Units Shipped (originally)
Wisconsin Plant
Alabama Plant
Memphis Warehouse
Memphis Warehouse
Memphis Warehouse
Pittsburgh Warehouse
Pittsburgh Warehouse
Memphis Warehouse
Fresno Facility
Peoria Facility
Pittsburgh Warehouse
Newark Facility
560
1000
450
500
50
610
Units Shipped (after
improvements)
537
1000
427
500
73
610
The original shipment plan cost $25,380 and our improvements reduced the total cost even further to
$24,453.50.
We want you to keep in mind we made these suggestions based on very realistic, conservative figures. There
may be more room to reduce total cost in the future if we are able to lower the unit cost of shipping between
Memphis and Fresno (down to any value) because our the cost of our plan is based off a 2% reduction.
We also considered the option of reducing the Alabama capacity to 500; however the total cost here is $24,910.
This plan comes at a higher cost than or other suggestion.
Our final solution, presented graphically:
Unit cost from Memphis to Fresno shown in red, the change is a result of SA, which is different from the
original problem statement.
Also
Capacity of Fresno shown in red, the change is a result of SA, which is different from the original problem
statement.