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The Basic Math Programming Model and Graphical Approach
Lecture II
I. An Agricultural Example
A. Assume that the farmer can produce two outputs, cotton and corn, with 100 acres of land and 25 hours of labor. Further, assume that each crop has the following input use per acre:
Cotton Corn Labor .3. Land 1.0 1.
Finally, assume that the profit per acre is $50 for cotton and $10 for corn. B. Basic Construction of the Problem: The basic formulation of the problem involves answering three questions.
- What does the model seek to determine–What are the variables?
- What constrains the variables from taking on any value–Whar are the constraints?
- What is the objective? – What is the rule used to determine the best or optimum? C. In the current model determine each compone nt. D. Basic Mathematical Formulation 1 2 1 2 1 2 1 2
max50 10
.. .3 .2 25 100 , 0
x x s t x x x x x x
where x 1 is cotton and x 2 is corn. E. Graphical solution
- Basic Formulation of constraints 1 2 1 2
1 2
1 2 1 2
x x x x
x x
x x x x
Do these lines intersect?
2 2
2
x x
x
x
Professor Charles B. Moss
x 1
100
125 x 2
x 1 (^) = 100 − x 2 : Land
1 2
(^250 2) : 3 3
x = − x Labor
The region satisfying all constraints is called the feasible region or region of feasibility.
- Choosing the optimal point: Construction of the Iso-profit line. Let z be the profit from some allocation of cotton and corn. Specifically, z = 50 x 1 (^) + 10 x 2 Question: How do we know that the objective value increases as x 1 and x 2 increase? Using basic calculus, we differentiate the profit function and set it equal to zero yielding 1 2 1 2
dz dx dx dx dx
Thus, we know that profit increases as either x 1 or x 2 increases by the first equation. Second, we know what the effect of increasing x 1 on profit is given that we have to decrease x 2. For example, on the land constraint (where we have to trade one unit of x 1 for each unit of x 2 ) the effect on profit is –1/5. Alternatively, profit is maximized where the slope of the feasible region is equal to –1/5.
Professor Charles B. Moss
x 1
100
125 x 2
s 1 =
s 1 =
c. Why s 1 =5? For a second, what if x 1 =0 and x 2 =100. Then
2 1
1
1
1
x s
s
s
s
− ^ − =
Or, more appropriately, suppose s 1 =5. Then
1 2 (^ )
1 2
x x
x x
d. Two things to note: (i) We can transform an inequality system into an equality system by adding (subtracting) the appropriate slack (surplus) variables. (ii) The nonnegativity condition on the slack (surplus) variable limits the feasibility condition to everything below (above) the line e. The land constraint 1 2 2 1 2 2
x x s x x s
Professor Charles B. Moss
- Combining the two equality representations graphically
x 1
100
83.33 125 x 2
s 2 =16.
s 1 =
a. Thus, one alternative is to produce x 1 =83.33 acres and x 2 = acres. This exhausts the labor constraint (s 1 =0), but leaves land in excess (s 2 =16.67). b. Another alternative is to use all the land and let labor go slack: x 2 =100, x 1 =0, s 1 =5, and s 2 =0. B. Using Matrix Algebra
- Note that an important part of the solution process was determining the intersection of the two constraints. This process can be simplified using matrix algebra. a. Note that the system of equations 1 2 1 2
x x x x
can be expressed in matrix form as .3 .2 25 1 1 100
using row operations, we can multiply the first row by 10/ yielding 1 2 250 3 3 1 1 100
Subtracting the first row from the second, 1 2 250 3 3 0 1 50 3 3
Professor Charles B. Moss
- Next, nonlinear constraints have much the same effect as a nonlinear objective function
- Integer Programming: Integer programming is the branch of mathematical programming which searches for whole number solutions. This type of programming became popular when it became difficult to raise 1.236 steer calves. The feasible region is developed in much the same way as the basic problem, but only integer solutions are considered.
x 1
100
125 x 2
Professor Charles B. Moss
IV. Sensitivity
A. In and of itself, the solution of an optimum may be of little value. More important questions may involve the tendency of the solution to change as certain parameters of the model change. Three types of these changes are
- Changes in resource availability over which the same constraints are binding.
- What is the value of an additional unit of resource?
- By how much can a given objective function coefficient change before the solution changes? B. Adjusted agricultural problem:
- Additional constraints: Assume that the current government program restricts the producer to growing no more than 75 acres of cotton, and no less than 25 acres of cotton. a. Does the policy change the solution? Yes, given that the current solution for the producer to grow 83.33 acres of cotton, we know that the producer will reduce cotton by 8.33 acres. b. Graphically, the new feasible region can be depicted as:
x 1
100
125 x 2
Policy Constraint
c. The government policy changes the solution in that in the original solution the cotton was constrained by the labor constraint. By constraining the cotton acreage, labor is freed. The question then becomes, can this freed labor be used to increase profit? The answer is yes. (i) What is the quantity of labor freed?
1 1
s s
