A gas company is planning a natural gas export project. This...

A gas company is planning a natural gas export project. This involves three component projects, namely to develop the gas field, construct a pipeline to a large coastal city, and develop a gas liquidification and export facility at the end of the pipeline. In the first case, gas would be sold on-site to another company. If the pipeline is constructed, the company could find domestic markets for all its production. A gas liquidification and export facility would allow substantially higher gas prices to be obtained through exporting. $$\$ 160 \mathrm{M}$$ is available for investment. Gas-field development would cost $$\$ 50 \mathrm{M}$$, pipeline construction $$\$ 60 \mathrm{M}$$, and development of a liquidification plant and an export facility $$\$45M$$. Respective NPV payoffs are $$\$20M$$ if gas is sold on-site, $$\$ 10 \mathrm{M}$$ for sales domestically and $$\$ 30 \mathrm{M}$$ if gas is exported. Set up a MILP model and solve this problem, taking account of the contingency relationship between the three component projects.

Key Concepts

MILP Formulation

A Mixed Integer Linear Programming (MILP) model is used to optimize decision making when some decisions must be made in discrete (often binary) terms. In an MILP, the objective function and constraints are linear, but some of the decision variables are required to take integer values. This formulation is particularly useful for project selection problems where complex, interdependent decisions need to be made subject to limited resources.

Binary Decision Variables

Binary decision variables are used to indicate whether a project is undertaken or not by taking values of either 1 (selected) or 0 (not selected). They allow the model to capture yes/no decisions, and they are key to modeling discrete choices in situations where projects or investments can only be implemented in an ‘all-or-nothing’ manner.

Objective Function

The objective function in an MILP model is a linear combination of decision variables that represents the performance measure to be maximized or minimized—in many cases, this is the net present value (NPV), profit, or cost. In project selection problems, the objective function typically sums the NPVs of the selected projects to determine the best overall combination under the given constraints.

Budget Constraint

A budget constraint represents the limited amount of resources or capital available for investment in projects. It is formulated as a linear inequality where the sum of the costs of the selected projects must not exceed the available budget. This constraint ensures that only feasible combinations of projects, that do not exceed the resource limitation, are considered in the solution.

Contingency (Logical) Relationships

Contingency or logical relationships are modeled in an MILP through additional constraints that define dependencies between projects. For instance, a project that can only be implemented if another project is undertaken is reflected by constraints that force a binary variable representing one project to be zero if its prerequisite is not selected. This ensures that the sequence or conditional nature of the projects is correctly captured in the model.

Transcript

00:01 For this problem, we are working with an oil refinery that manufactures two products, gasoline and fuel oil.

00:08 And if you read through the problem, the first thing we want to answer is what is the best way to maximize our revenue? so this is a maximization problem.

00:20 And this isn't too surprising.

00:22 Most companies want to maximize the amount of money that they're bringing in.

00:25 So what is my revenue? well, we're told that gasoline is $250 a gallon, fuel oil is $2 .00 per day.

00:33 Gallon.

00:34 So if i'm going to put together a revenue equation, i'm going to say that's $2 .50 times my gasoline, you know, a number of gallons of gasoline, and $2 per gallon of fuel oil.

00:47 And i'm going to mark these.

00:49 It's always a good idea to get in the habit of marking your variables on a word problem because they can be a little long.

00:56 And by the time you get to the end, you want to remember what you did when you're first setting things up.

01:01 Okay, so here is my revenue equation.

01:04 What are my constraints? what do i need to keep in mind? well, first, we're told that we have to have at least two gallons of gasoline manufactured for every gallon of fuel oil.

01:15 So my gasoline, the amount of gasoline we produce has to be at least twice as much as the amount of fuel oil.

01:24 So that's our first one.

01:25 Next, we're told we need to have at least three million gallons a day.

01:31 Well, that means fuel oil has to be at least three.

01:38 And i'm going to just put a mark here.

01:40 I'm going to put this in millions of gallons, my constraints.

01:45 So when i'm done, my constraints will give me numbers, and those numbers are going to be in the millions.

01:50 That doesn't affect this top one because that's just a proportion between them.

01:54 So whether it's single gallons or millions of gallons, the proportion of that first constraint doesn't change.

02:00 So that one is still good just the way it is.

02:02 So the amount of fuel oil has to be at least 3 million gallons.

02:08 What about my gasoline? well, gasoline goes the other way.

02:12 It is no more than 6 .4 million.

02:15 So there's that constraint.

02:17 No more than 6 .4 million gallons.

02:20 And i also need to take a look at how long it takes to ship these different products out because we're told that we only have four hours and 39 minutes available for shipping.

02:31 Now, if i put this in terms of minutes, that works out to 279 minutes.

02:39 And that's all i've got.

02:40 So whatever i do for shipping has to be less than or equal to 279 minutes.

02:45 Well, how long does it take to ship things? it takes 15 minutes to ship a million gallons of gasoline.

02:53 So 15 minutes per million gallons of gasoline.

02:56 And it's an hour to ship each gallon of fuel oil.

03:01 Again, putting that into minutes, 60 minutes.

03:04 Per million gallons of fuel oil.

03:07 Okay, so those are my constraints.

03:10 I am using the desmos graphing calculator application on my computer.

03:14 You can use any graphing thing you would like, calculator, whatever app on your computer you're comfortable with.

03:20 And as you can see, i've already entered the four constraints that we talked about.

03:24 And if i start to show these, you can see where they overlap.

03:30 In fact, our overlap, i'm going to mark the endpoints.

03:35 I've got four boundary points where all four of these constraints overlap.

03:40 Remember, these are in millions.

03:43 So we'll just need to remember that when we go to look at our answers.

03:47 So these are our four points.

03:50 Let's just put these in.

03:51 I'm doing these in no particular order.

03:54 First, we had six and three, 6 .2 and 3 .1, 6 .4 and 3 .05, and 6 .4 and 3 .3.

04:09 And again, those four points, came directly from those four boundary points there on our graph.

04:16 What is my revenue? i'm going to go back to that revenue equation, given these points.

04:22 Now remember, these are in millions.

04:25 So when i go to plug these in, i'm going to put these in as $6 .2 ,000, $3 .05 million into my calculator.

04:34 So my revenue that i get from my first point, 6 million gallons of gasoline, 3 million gallons of fuel oil gives me a revenue.

04:42 Of 21 million.

04:46 My second point gives me 21 ,700 ,000.

04:51 My third point, 22 ,000, 100 ,000.

04:57 And the fourth point is 22 million.

05:02 So this point gives me my maximum revenue...

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