NCC508-00#1

Prof. Byron L. NewberryOklahoma Christian UniversityManagement Science, The Art of Modeling with SpreadsheetsStephen G. Powell and Kenneth R. BakerWiley Publishing CompanyLinear Programming (Chapter 11)

No problem can stand the assault of sustained thinking.

Voltaire

Linear Programming Problems(Many Different Types, But All Similar)Allocation modelsMaximize objective (e.g., profit) subject to LT constraints on capacity

Covering modelsMinimize objective (e.g., cost) subject to GT constraints on required coverage

Blending modelsMix materials with different properties to find best blend

Network models (much more on this next chapter)Describe patterns of flow in a connected system

Model ClassificationLinear optimization or programmingObjective and all constraints are linear functions of the decision variables.

Nonlinear optimization or programming Either objective or a constraint (or both) are nonlinear functions of the decision variables.

Techniques for solving linear models are more powerful.

What do LP Problems Look Like?

Note the linear form in the end. Obviously nonlinear models would not share this feature.The first and most critical step is problem setup!Take the time to do it correctly!

A Graphical View of LP ModelsBeyond even two variables, graphical techniques become hard, but we can gain some insight.

A Two Variable Model Graphical Example

A Three Variable Model Graphical Example

Take the time to watch the video before proceeding (it is on Blackboard).

Solution to the Planting ProblemIn order to actually solve the problem, we will need numbers for our variables. The numbers below are simply values to allow solution.

$800x1 + $500x2 (Maximize this objective)

Subject to the following constraints

x1 + x2 200029x1 + 17x2 4280017x1 + 12x2 26000x1 0x2 0

We now solve by plotting the lines associated with each constraint (color coded in the plot). The solution area is shaded. The optimal solution will be at one of the constraint intersections (marked with circles). The optimal solution is marked YELLOW ($11,939).

So the process is to draw the lines and check the intersections. The easiest way to draw the lines is for each constraint equation, first set x1 to zero and solve for x2. Then set x2 to zero and solve for x1. With two points you can draw the line.

$10,000$11,200$11,939$11,808Patterns in Linear Programming SolutionsThe optimal solution tells a story about a pattern of the problem priorities.Leads to more convincing explanations for solutionsCan anticipate answers to what-if questionsProvides a level of understanding that enhances decision making

After optimization, you should always try to discern the qualitative pattern in the solution!

Tornado plots can be very useful forthis task.

Example #1We want to buy two different brands of feed and blend it to produce a quality, low-cost diet for turkeys. The details of the two feeds are given below:

Feed #1: 5 oz of A, 4 oz of B, and 0.5 oz of C for each pound of feed$0.02 / pound

Feed #2: 10 oz of A, 3 oz of B, and no C for each pound of feed$0.03 / pound

The final blend should provide a required MONTHY minimum of at least 90 oz of A, 48 oz of B, and 1.5 oz of C (per turkey) at minimal cost.

Setup the problem by hand (i.e. write the problem equations as on the earlier slide) and attempt to solve the LB problem GRAPHICALLY.

Example #1 Mathematical EquationsLet X1 = Feed #1 & X2 = Feed #2, we now need to write the mathematical relationships implied by the problem statement

Minimize: 2 X1 + 3 X2 Cumulative Cost

S.T.5 X1 + 10 X2 90 Constraint on Ingredient A4 X1 + 3 X2 48 Constraint on Ingredient B0.5 X1 1.5 Constraint on Ingredient CX1 0 Non-negative on X1X2 0 Non-negative on X2

Note that we have only 2 variables. We are limited to two variables for graphical solution by hand. Excel, however, is NOT limited to two variables.

Example #1 Solution GraphThe solution will be at one of the corners (or in the RARE event along an entire line)

Checking all corners, we see that the lowest cost (per bird) is 31.2 cents/month.

X1X2Some Special LP Cases(Graphics from Quantitative Analysis for Management by Render, Stair, and Hanna, 11th Ed, 2012)

If the objective of this problem is maximization we have issues the solution domain is unbounded to the rightIn this case NO SOLUTION is possible as the set of constraints are mutually exclusive

Even More Special Cases(Graphics from Quantitative Analysis for Management by Render, Stair, and Hanna, 11th Ed, 2012)

The constraint line to the far right is REDUNDANT and provides no additional useful informationIn this case MANY SOLUTIONS (mathematically infinite set) exist asthe isoline (Isoprofit in this case) is parallel to a constraint line

An Additional Practice ProblemIn this example, the problem is already formulated as mathematical equations. You are simply to GRAPHICALLY solve the problem.

Minimize: 24 X1 + 28 X2

S.T. 5 X1 + 4 X2 2000 X1 80 X1 + X2 300 X2 100 X1, X2 0Notice anything about this formulation? Something is REDUNDANT!

Practice Problem Solution(Graphics from Quantitative Analysis for Management by Render, Stair, and Hanna, 11th Ed, 2012)

Graphical Depiction of Constraint Equations and Solution SpaceDetermination of the best Solution using an IsoclineApproach