Linear Programming Formulation And Sensitivity Analysis
Operations Planning for the
Plainview Oil Company
The Plainview Oil Company is an international producer, refiner, transporter and
distributor of oil, gasoline and petrochemicals. Plainview Oil is a holding company with
subsidiary operating companies that are wholly or partially owned. A major problem for
Plainview Oil is to coordinate the actions of these various subsidiaries into an overall
corporate plan, while at the same time maintaining a reasonable amount of operating
autonomy for the subsidiary companies.
To deal with this dilemma, the logistics department at Plainview Oil Headquarters
develops an annual corporate-wide plan which details the pattern of shipments among
the various subsidiaries. The plan is not rigid but provides general guidelines and the
plan is revised periodically to reflect changing conditions. Within the framework of this
plan, the operating companies can make their own decisions and plans. This corporatewide plan is presently done on a trial and error basis. There are two problems with this
approach. First, the management of the subsidiaries often complains that the plan does
not reflect properly the operating conditions under which the subsidiary operates. The
plan sometimes calls for operations or distribution plans that are impossible to
accomplish. And secondly, corporate management is concerned that the plan does not
optimize for the total company.
The technique of linear programming seems a possible approach to aid in the annual
planning process, that will be able to answer at least in part, the two objections above. In
addition the building of such a model will make it possible to make changes in plans
quickly when the need arises. Before embarking on the development of a world-wide
model, Plainview Oil asks you to build a model of the Far Eastern operations for the
Far Eastern Operations
The details of the 2020 planning model for the Far Eastern Operations are now described.
There are two sources of crude oil, Saudi Arabia and Borneo. The Saudi crude is relatively
heavier (24 API), and the Far Eastern sector could obtain as much as 60,000 barrels per
day at a cost of $18.50 per barrel during 2020. A second source of crude is from the Brunei
fields in Borneo. This is a light crude oil (36 API). Under the terms of an agreement with
the Netherlands Petroleum Company in Borneo, a fixed quantity of 40,000 b/d of Brunei
crude, at a cost of $19.90 per barrel is to be supplied during 2020. There are two subsidiaries that have refining operations. The first is in Australia,
operating a refinery in Sydney with a capacity of 50,000 b/d throughput. The company
markets its products throughout Australia, as well as having a surplus of refined
products available for shipment to other subsidiaries.
The second subsidiary is in Japan, which operates a 30,000 b/d capacity refinery.
Marketing operations are conducted in Japan, and excess production is available for
shipment to other Far Eastern subsidiaries.
In addition, there are two marketing subsidiaries without refining capacity of their own.
One of these is in New Zealand and the other is in the Philippines. Their needs can be
supplied by shipments from Australia, Japan, or the Plainview Oil subsidiary in the
United States. The latter is not a regular part of the Far Eastern Operations, but may be
used as a source of refined products.
Finally, the company has a fleet of tankers that move the crude oil and refined products
among the subsidiaries.
The operation of a refinery is a complex process. The characteristics of the crudes
available, the desired output, the specific technology of the refinery, etc., make it difficult
to use a simple model to describe the process. In fact, management at both Australia and
Japan have complex linear programming models involving approximately 300 variables
and 100 constraints for making detailed decisions on a daily or weekly basis.
For annual planning purposes the refinery model is greatly simplified. The two crudes
(Saudi and Brunei) are input. Two general products are output – (a) gasoline products
and (b) other products such as distillate, fuel oil, etc. In addition, although the refinery
has processing flexibility that permits a wide range of yields, for planning purposes it
was decided to include only the values at highest and lowest conversion rates (process
intensity). Each refinery could use any combination of the two extreme intensities.
Capacity utilization varies between intensity levels. These yields are shown in Table 1.
The incremental costs of operating the refinery depend somewhat upon the type of crude
and process intensity. These costs are shown in Table 1. Also shown are the incremental
transportation costs from either Borneo or Saudi Arabia.TABLE 1: REFINERY OPERATIONS
Capacity (b/d of input) 50,000 30,000
Transportation Cost ($/b) 0.65 0.7
High Process Intensity ($/b) 1.19 1.26
Yield of Gasoline 0.31 0.3
Yield of Distillate 0.61 0.62
Low Process Intensity ($/b) 0.89 0.88
Yield of Gasoline 0.19 0.18
Yield of Distillate 0.73 0.74
Transportation Cost ($/b) 0.15 0.25
High Process Intensity ($/b) 0.93 0.91
Yield of Gasoline 0.36 0.35
Yield of Distillate 0.58 0.59
Low Process Intensity ($/b) 0.61 0.55
Yield of Gasoline 0.26 0.25
Yield of Distillate 0.69 0.7
Marketing is conducted in two home areas (Australia and Japan) as well as in the
Philippines and New Zealand. Demand for gasoline and distillate in all areas has been
estimated for 2020 and is shown in Table 2.TABLE 2: 2020 DEMAND (THOUSANDS OF B/D)
Area Gasoline Distillate
Australia 9 21
Japan 3 12
Philippines 5 8
New Zealand 5.4 8.7
Total 22.4 49.7
Variable costs of supplying gasoline or distillate to New Zealand and the
TABLE 3: VARIABLE COSTS OF SHIPMENT OF
GASOLINE/DISTILLATE IN $ /BBL
From: To: New
Australia 0.3 0.45
Japan 0.3 0.6
Tankers are used to bring crude from Saudi Arabia and Borneo to Australia and Japan
and to transport refined products from Australia and Japan to the Philippines and New
Zealand. The variable costs of these operations are included above.
However, there is a limited capacity of tankers available. The fleet has a capacity of 6.5
equivalent (standard sized) tankers.
The amount of capacity needed to deliver one barrel from one destination to another
depends upon the distance traveled, port time, and other factors. The table below lists the
fraction of one standard sized tanker needed to deliver 1,000 b/d over the indicated
routes. TABLE 4: TANKER USAGE FACTORS (FRACTION OF STANDARD
SIZED TANKER NEEDED TO DELIVER 1000 B/D
Between Australia Japan
Saudi Arabia 0.12 0.11
Borneo 0.05 0.045
Philippines 0.02 0.01
New Zealand 0.01 0.06
United States Supply
United States operations on the West Coast expect a surplus of 12,000 b/d of distillate
during 2020. The cost of distillate at the loading port of Los Angeles is $20.70 per barrel.
There is no excess gasoline capacity. The estimated variable shipping costs and tanker
requirements of distillate shipments from the United States are:
Table 5: US SUPPLIER SHIPPING DETAILS
Variable Cost of
shipments ($ per bbl.)
(Fraction of standard sized tanker
needed to deliver 1000 b/d)
New Zealand 2.1 0.18
Philippines 1.65 0.15Questions
1. Formulate a linear program which can be used to generate a comprehensive plan
for the whole Far Eastern operations. Clearly define every variable used in your
formulation. Solve your linear program using Excel solver.
How many barrels of crude should Plainview Oil purchase from Saudi Arabia for
its Far Eastern operations? How much crude should be refined in Australia? How
much in Japan? Provide tables showing the quantities of gasoline and distillate
shipped from each of the two refineries and from the U.S. to each of the four
2. Use sensitivity analysis to answer the following questions:
• What is the marginal value of increasing supply from Brunei fields? Can
this marginal value be used to estimate the total savings for Far Eastern
operations if 41,000 b/d are supplied from Brunei fields instead of 40,000?
• What is the marginal value of increasing the tanker fleet? Can this
marginal value be used if we want to increase the size to 7 tankers (from
the current size of 6.5)? Explain.
• What is the additional cost to Far Eastern operations if demand for
gasoline in the Philippines increases to 5,200 b/d? What is the minimum
price of gasoline in the Philippines that would make it profitable for
Plainview Oil to consider such an increase in distribution?
• Currently, it is not economical to ship US distillate to the Philippines.
What is the cost of US distillate at which Plainview Oil should consider
starting such shipments?
• Plainview Oil is planning a three day shutdown of its Australian refinery
for maintenance purposes in the coming year. It has storage facilities and
at least two weeks of inventories, so a refinery shutdown for a few days
will not disrupt distribution. What would be the cost of a shutdown of the
refinery in Australia for three days per year? What about a similar
shutdown in Japan? [Hint: Treat shutting down 3 days per year as shutting
down for a small fraction of the day, every day, over the course of the year.]
3. Several opportunities present themselves to the Plainview Oil company (see the
attached memos). Consider combinations of these options and prepare a
recommendation. Motivate your results.Memos
Memo to: Plainview Oil Headquarters
From: Australian Affiliate
Re: Supplements to Annual Plan
Since submitting data for annual planning purposes, two additional opportunities
have arisen. We would like to include these in the plans.
A. Bid on Gasoline Contract with Australian Government
The government of Australia will submit to bid a contract for 1.5 thousand b/d of
gasoline. We expect we could win this bid at a price of $26.40 per barrel. Estimated
costs per barrel as follows:
Variable Costs (crude, refining, transportation) $24.90
Allocated Overhead .80
At these costs, the contract would have a profit contribution of $0.70 per barrel. We
would like permission to bid on the contract. We hope that the linear programming
wizards who are working for your logistics department will not contradict us!
B. Expansion of Australian Refinery
For the past two years, the Australian refinery has been operating at full capacity. We
request authorization for capital expenditures to increase the refinery capacity to 55
There are several reasons for the need for this expansion:
1. Australia can supply the current requirements in New Zealand and the
Philippines more cheaply than can Japan.
2. The proposed bid on Australian government gasoline contract (above).
3. We understand the New Zealand affiliate is considering increasing its
requirements by 4.5 thousand b/d. (See below.)
The cost of this expansion is 4.0 million dollars. To recover this investment, we need
an annual savings of $702,000. This assumes a cost of capital rate of 20%. Depreciation
tax effects are included. With these considerations, the $702,000 savings per year is
equivalent to the $4 million investment.Memo to: Plainview Oil Headquarters
From: New Zealand Affiliate
Re: Supplement to Annual Plan
Negotiations have been begun with the NOZO Oil Company in New Zealand. This
company is a distributor, with sales of 1.6 thousand b/d of gasoline and 3.2 thousand
b/d of distillate. If negotiations are successfully completed, these requirements would
be added to current requirements for New Zealand, making total requirements of:
Gasoline: 7.0 thousand b/d
Distillate: 11.9 thousand b/d
The anticipated revenue (after subtracting variable marketing costs) for this
acquisition are $30.20 per barrel for gasoline and $24.60 per barrel for distillate. The
purchase cost of NOZO oil is expected to be about $21.0 million. On an annual basis,
this would require $3.5 million per year incremental profit to justify the purchase.
Memo to: Plainview Oil Headquarters
From: Tanker Affiliate
Re: Supplement to Annual Plan
We have been made aware that expansions in requirements are being considered in
New Zealand and Australia. We are currently operating the tanker fleet at capacity.
Additional requirements will increase the transport requirements both for crude and
refined products. This will necessitate spot chartering unless additional tanker
capacity is added.
We can lease on a long time basis additional tankers at a rate of $4.8 thousand dollars
per day per 1 unit tanker equivalent. We propose a lease of 0.5 equivalent tanker units
giving us a total capacity of 7.0 equivalent units. The cost of this would be $2.4
thousand per day or $876,000 per year. If you prefer other arrangements, we are
willing to discuss them.
Memo to: Plainview Oil Headquarters
From: Borneo Office
Re: Supplement to Annual Plan
We have just been offered the opportunity to increase our contract with our supplier
of Brunei crude. They are willing to supply us an additional 5,000 b/d at a cost of
$20.65 per barrel. Should we accept the offer? If not, is there a counter offer that we