Q U A N T I T A T I V E
M O D U L E
1. Students may select from eight LP applications given in the introduction. These include school bus scheduling, police patrol allocation, scheduling bank tellers, selecting product mix, picking blends to minimize cost, minimizing shipping cost, developing production schedules, and allocating space.
2. LP theory states that the optimum lies on a corner. All three solution techniques make use of the “corner point” feature.
3. The feasible region is the area bounded by the set of problem constraints. A feasible solution is any combination of x, y coordinates (or x1, x2 coordinates) that is in or on the feasible region.
4. Each LP problem that has been formulated correctly does have an infinite number of possible solutions. Any point within the feasible region is a solution that satisfies all constraints (although it is not necessarily optimal). In addition, for any problem in which the optimal solution lies on a constraint that is parallel to the objective function, all points along that constraint are also both feasible and optimal.
point in the feasible region that the line touches is the optimal corner point.
11. The corner point method examines the profit at every corner point, whereas the iso-profit line method draws a series of parallel profit lines until one line finally touches the last tip (corner point) of the feasible region. That last point touched is the optimal solution, so other corner points need not be tested.
12. When two constraints do not cross at an axis, we use simultaneous equations—there is only one point where two linear equations (constraints) cross.
13. (a) Adding a new constraint will reduce the size of the feasible region unless it is a redundant constraint. It can never make the feasible region any larger.
(b) A new constraint can only reduce the size of the feasible region; therefore the value of the objective function will either decrease or remain the same. If the original solution is still feasible, it will remain the optimal solution.
5. The objective function contains the profit or cost information that enables us to determine whether one solution is better than another solution. Our choice of best depends only on the objective.
ACTIVE MODEL EXERCISE
6. Before activity values can be placed into the objective, they must meet the constraints. Notice that the objective function has no minimum-required profit level unless it is included as a constraint.
1. By how much does the profit on Walkmans need to rise to make it the only product manufactured?
If the profit per walkman is more than $10 per unit then it is the only product that should be manufactured.
7. As long as the costs do not change, the diet problem always provides the same answer. In other words, the diet is the same every day. Unlike animals, people enjoy variety, and variety cannot be included as a linear constraint.
8. The number of feasible solutions is infinite. We only need to consider extreme points—corner points—to find the optimal solution. If we use isoprofit lines, we only need to examine one corner point to determine the optimal solution.
9. Shadow price or dual: the value of one additional unit of a resource, such as one more hour of a scarce labor resource or one more dollar to invest.
10. The iso-cost line is moved down in a minimization problem until it no longer intersects with any constraint equation. The last
ACTIVE MODEL B.1: LP Graph
2. By how much does the profit on Walkmans need to fall to stop manufacturing it?
At $6.66 and below we should not manufacture any
3. What happens to the profit as the number of assembly hours increases by 1 hour at a time? For how many hours does this hold true?
The profit rises by $.50 per hour until we reach 120 hours at which point the rise stops.
4. What happens if we can reduce the electronics time for
Watch-TVs to 2.5 hours?
The profit rises by $70.