Reconsider the case study presented in the supplement to Chap. $8($on the CD$-$ROM$)$ involving the Texago Corp. site selection problem.

Texago management has tentatively chosen St$.$ Louis as the site of the new refinery. However, management now is addressing the question of whether the capacity of the new refinery should be made somewhat larger than originally planned.

While analyzing the site selection problem, the task force had been told to assume that the new refinery would have the capacity to process $120$ million barrels of crude oil per year. As indicated in Table $3,$ this then would increase the total ca$-$ pacity of all the corporation$$s refineries from $240$ million bar$-$ rels to $360$ million barrels. According to marketing forecasts, Texago would be able to sell all its finished product once this new capacity becomes available, but no more. Therefore, the choice of $120$ million barrels as the capacity of the new re$-$ finery would enable all the corporation$$s refineries to operate at full capacity while also fully meeting the forecasted de$-$ mand for Texago$$s products.

However, to prepare for possible future increases in de$-$ mand beyond the current forecasts, management now wants to also consider the option of enlarging the plans for the new refinery so that it would have the capacity to process

$150$ million barrels of crude oil annually. Although this would force the corporation$$s refineries collectively to op$-$ erate below full capacity by $30$ million barrels for awhile, the extra capacity then would be available later if Texago continues to increase its market share. This might well be worthwhile since the capital and operating costs incurred by enlarging the plans for the new refinery would be far less $($perhaps $40$ percent less$)$ than constructing and oper$-$ ating another refinery later to process only $30$ million bar$-$ rels of crude oil per year. Furthermore, management feels that this extra capacity might be needed within a few years.

The extra capital needed to increase the capacity of the new refinery by $30$ million barrels is estimated to be $$1\mathrm{.}2$ bil$-$ lion. The cost of carrying this extra capital would be $$100$ million per year, although this figure could change depend$-$ ing on future interest rates. If some of this extra capacity is used at the new refinery, the total operating cost for the

refinery would be somewhat larger than the amount shown in Table $6,$ but decreasing the production rate by the same amount at another refinery would decrease its total operating cost by a comparable amount. Since the operating cost per million barrels of crude oil processed is roughly the same at all the refineries, including the new one, the total operating cost for processing $360$ million barrels should not be sub$-$ stantially affected by the allocation of this work to the refiner$-$ ies. However, management feels that having some flexibility for where to allocate this work might enable substantially re$-$ ducing the cost of shipping crude oil and finished product. Since Table $7$ indicates that the total annual shipping cost for crude oil and finished product would be $$2\mathrm{.}92$ billion with St$.$ Louis as the site for the refinery, management hopes that substantial reductions can be achieved in this way.

Figures $4$ and $8$ show the optimal shipping plans for crude oil and finished product, respectively, when the new refinery is in St$.$ Louis and has a capacity of processing $120$ million barrels of crude oil per year. Management now is asking the task force to analyze the situation under the option of in$-$ creasing this capacity to $150$ million barrels. In particular, management wants the following questions addressed. Under the new option, how should the shipping plan for crude oil in Fig. $4$ change, and how much reduction in the total shipping cost would be achieved? How should the shipping plan for finished product in Fig. $8$ change, and how much reduction in the total shipping cost would be achieved? Finally, assum$-$ ing that the differences in operating costs shown in Table $6$ would continue to apply under the new option, would the fi$-$ nancial comparison of the three sites given in Table $7$ be al$-$ tered substantially if this option were to be adopted?

As the head of the task force, you have decided to lead the way by executing the following steps with the new op$-$ tion. Thus, each of the following parts assumes that the ca$-$ pacity of the new refinery will be $150$ million barrels instead of the $120$ million barrels assumed in the original Texago case study.

$($a$)$ Formulate and solve a model to find an optimal plan for ship$-$ ping $360$ million barrels of crude oil per year from the oil fields to the refineries, including the new one in St$.$ Louis, where the amount of crude oil each refinery will receive $($up to its

$15$

$16$ ADDITIONAL CASES

capacity$)$ is based on minimizing the total annual cost for these shipments. $($Hint: If you are using a spreadsheet model, you can save some time in this and subsequent parts by usi