Unformatted text preview: Chapter 6
Chapter
Toward Proper Modeling Mathematical Modeling
Mathematical Part art, part science
Considerable room for judgment
Assumptions concerning variables, constraints, coefficients
Assumptions affect results
How do you know if the model is “right”? Structural Component Identification
Structural Variable identification Objective function specification Constraint setup Model structure Variable Identification
Variable Variables may be included for technical, accounting or convenience reasons
The variables are the “unknowns.” LP solution determines the level of the variables. Technical variables are the true unknowns. Accounting variables normally summarize some other activities in the model.
Convenience variables may help simplify the model structure or serve some similar purpose. One Variable or Several?
One There are cases when multiple variables must be defined for what may, at first, seem to be a single activity. (a) When more than one process can be used to produce the same output using different resource mixes; e.g., the production of an item using either of two different machines. (b) When different processes produce different outputs using common resources; i.e., one can use essentially the same resources to produce either 2 x 4 or 4 x 4 sawn lumber. (c) When products can be used in several ways; e.g., selling chickens that can be quartered or halved. We have seen that properly defining variables is essential in many types of common LP problems. One Variable instead of Two
Criteria may also be developed where two variables may be treated as one. The simplest case occurs when the coefficients of one variable are simple multiples of another (aij = Kaim and cj = Kcm). The second case occurs when one variable uniquely determines another; i.e., when n units of the first variable always implies exactly m units of the second. Objective Function
Objective Solution is only useful if the objective function is correctly specified. Identifies the optimal point.
Multiple objective models are discussed in the multiobjective and risk chapters.
Range analysis can be used to see for what values of the objective function the currently solution holds. Model Constraints
Model Technical constraints depict limited resources, intermediate products, or contractual requirements. Technical constraints may also express complementary, supplementary, and competitive relationships among variables.
Institutional constraints reflect external regulations imposed on the problem. Examples include credit limits or farm program participation requirements.
Subjective constraints are imposed by the decision maker or
modeler. These might include a hired labor limitation based on the decision maker's willingness to
supervise labor.
Convenience constraints facilitate model interpretation and may be included to sum items of interest. Model formulation constraints aid in problem depiction. These include constraints used in conjunction with approximations . Number of Constraints
Number Heady and Candler argue that multiple constraints are needed to depict availability of a resource whenever the marginal rate of factor substitution between resource usage in different time periods does not equal one. (Labor is an example – can’t substitute July labor for June labor.)
Constraints must be developed so that the resources available within a particular constraint are freely substitutable. Cases of imperfect substitution will require multiple constraints. Constraints and Solutions
Constraints An LP solution will include no more variables at a nonzero level than the number of constraints (including the number of upper and lower bounds). Thus, the number of constraints directly influences the number of nonzero variables in the optimal solution.
Subjective constraints should not be imposed before determining their necessity. Often, subjective constraints "correct" model deficiencies. But the cause of these deficiencies is frequently missing either technical constraints or omitted variables. Avoiding Improper Constraint Specifications LP model constraints have higher precedence than the objective function. The first major effort by any LP solver is the discovery of a feasible solution.
The modeler must question whether a constraint should be established so it always restricts the values of the decision variables. Often, it may be desirable to relax a constraint allowing resource purchases if the value of a resource becomes excessively high.
Modelers should be careful in the usage of minimum requirement constraints. Minimum requirements must be met before profit seeking production can proceed.
Third, judicious use should be made of equality constraints. Modelers should use the weakest form of a constraint possible. Consider
Consider
Consider the following example:
Max 3X + 2Y
s.t. X Y ? 0 X LE 10 Y LE 15
Where ? is the constraint type, X depicts sales and Y production. Suppose we have made a mistake and have specified the cost of production as a revenue item (i.e., the +2Y should be 2Y in the objective function). If the relation is an equality, then the optimal solution is X = Y = 10 and we do not discover the error. On the other hand, if the relation is LE then we would produce Y = 15 units while selling only X = 10 units and we would see the error. Model Structure
Model Model building is an iterative process.
People work in different ways.
I break a large model into smaller pieces, develop those, and then put them together.
Another alternative is to follow the steps laid out in McCarl and Spreen. Formulating an applied LP problem
Formulating
Development of Model Structure
Example: A profit maximizing firm produces 4 crops s.t. land and labor constraints. Crops are grown at different times of the year.
Crop1 is planted in the spring and harvested in the summer. Crop23 are planted in the spring and harvested in the fall. Crop4 is planted following crop1 and harvested in the fall. Setting Up this Model
Setting
1.
Make a table laying out potential variables across the top and constraints /objective function down the side. 2.
Enter profits for crops and resource coefficient and endowment. Schematic
Schematic
land
crop14
splabor
sumlab
falllab Crop1
c1
1
1
a11
a21
a31 Crop2
c2
1 Crop3
c3
1 a12
a22
a32 a13
a23
a33 Crop4
c4
1
1
a14
a24
a34 le
le
le
le
le LAND
0
SPLAB
SULAB
FALAB Proper Use of Variables and Constraints
Proper Reading down the column for a variable – these coefficients represent the simultaneous requirements for this variable.
Choice is modeled across variables, never within a variable. If there are two production alternatives, there must be two variables. If there are two possible uses for a decision item, there will need to be two variables (example selling tables or using tables to make sets).
Resources within a constraint are homogeneous. If there are different classes of the resource (e.g. good land and notsogood land), you will need separate constraints. Rules for Rows and Column
Rules
Rules 1. All coefficients in a row have common numerators – e.g. operating on the same units (consistently in lbs or consistently in kilos, not some of each). All coefficients in a column have common denominators (e.g. per unit of X produced). The Joint Product Problem
The A chicken is cut into four parts: breasts, legs, necks, and giblets.
Each chicken weighs 3 pounds.
There are 1,500 chickens available.
Breast
leg
neck
giblet % of chicken
0.50
0.35
0.10
0.50 %meat in part
0.75
0.60
0.20
0.00 Price of Part/lb
1.00
0.80
0.20
0.70 You Must Disassemble Down the Column
You
≤ Formulation 6.5(b) Chickens
(lbs.) Objective Function ($) Breast
Quarte
r (lbs.)
+ 1.00X1 Breast Quarter 0.50Y Leg Quarter 0.35Y Neck 0.10Y Giblets 0.05Y Chickens 1/3Y Leg
Quarte
r (lbs.)
+ 0.80X2 Neck
(lbs.)
+ 0.20X3 Giblet Maximize
s
(lbs.)
+ 0.70X4 + X1 le
+ X2 0 Le
Le + X3
+ X4 0
0 Le 0 Le 1500 Last constraints is 1/3 Y because chickens weigh 3 pounds. Equal to Y Le 4500 (total pounds), which would probably be the way I would write it. Look at a Common Incorrect Specification
Look
≤ Formulation 6.5(a) Chickens
(lbs.) Leg
Quarter
(lbs.) + 1.00X1 Objective function ($) Breast
Quarter
(lbs.) + 0.80X2 Neck
(lbs.)
+ 0.20X Giblets
(lbs.) Maximize + 0.70X4 3 Balance (lbs.) Y Chickens Available (birds) 1/3Y + 0.50X1 + 0.35X2 + 0.1X3 + 0.05X4 Le
le Here, we are trying to disassemble across a row. Look at what happens
if we set Y=1. If we have Y equal to 1, then X1 (breast quarters in lbs) can
equal up to 2 pounds, with all else zero. Alternatively, we could get about 3 pounds of leg quarters from this one pound of chicken, or10 pounds of necks. All would be feasible in this formulation. 0
1500 More than One Use
More
Now let us be able to sell mixed quarter packs, which
is an arbitrary combination of legs and breasts. And let us be able to debone the chicken and sell
the meat for $1.20 a pound.
Here, instead of having one use for some items,
there are various uses – selling them as is, deboning, or
putting them in packs. These alternative uses
require new variables. Each Use Requires a Variable
Each
Chicken
Objective
BQ
LQ BQ
1.0X1 0.5Y +D
0.35Y 8: Neck 0.1Y Giblets 0.5Y Chickens LQ Neck Giblet BQ
Meat LQ
Meat Neck
Meat + 0.8X1 + 0.2X3 + 0.7X4 X1 +
+ + X2 LQ in
Pack + 0.95Q3
+ Q1 M2 + Qtr. Pack + MQ
Sold Q2 1/3Y Meat X3 BQ in
Pack 1.2M4 M1
+ + Total
Meat + M3 X4
 0.75M1  0.6M2  0.2M3 + M4
 Q1  Q2 + ≤
≤
≤
≤
≤
≤
Q3 ≤ 0
0
0
0
1500
0
0 Common Incorrect Formulation of Packs
Alternative
Alternative
Chicken
Objective
BQ
LQ
Neck
Giblets
Chickens 0.5Y +
0.35Y
0.1Y
0.5Y
1/3Y BQ Neck LQ 1.0X1
X1 Giblet 0.8X1 + 0.2X3 + BQ
Meat + M2 X3 +
+ Total
Meat
+ 1.2M4 M1 X2
+ Neck
Meat 0.7X4
+ + LQ
Meat M3 X4
 0.75M1  0.6M2  0.2M3 + M4 MQ
Sold
+ 0.95Q3
+ .50Q3 ≤
+ .50Q3 ≤ ≤
≤
≤
≤ 0
0
0
0
1500
0 Meat This specification allows you to use BQ’s and LQ’s twice, once as direct sales or meat, the other as part of a quarter pack. Erroneous Imperfect Substitution
Erroneous Consider labor
It is paid one normal hourly rate and another “overtime” rate
Suppose a firm has 77 regular hours of labor at $10/hour and up to 27 more hours at $15 per hour. Assume the firm manufactures chairs, which take 10 hours of labor each, with an average of 7 hours regular labor and 3 hours overtime.
Chairs sell for $220 each. Incorrect Specification
Incorrect regular labor
overtime labor
rl limit
ot limit Sell
Chair
220
7
3 Regular
labor
10
1 Overtime
labor
15
1 1
1 le
le
le
le 0
0
77
27 It is easy to see why this is wrong. Even if only
one chair is produced, the company has to hire
overtime. Also, the overtime limit becomes binding
at 9 chairs, when in fact more chairs could be
produced using “regular” labor. Correct Model
Correct labor
reg labor
ot labor Sell
Chair
220
10 Regular
labor
10
1
1 Overtime
labor
15
1
1 le
le
le 0
77
27 In this specification, the cheaper labor would be used first. Numerical Model Analysis
Numerical
It can be useful to set the model (either in
your mind or with a special restriction) to
one unit of a decision variable and check that all the inputs and other
relationships make sense before running
the model “for real.” Data Development
Data Time frame: defines characteristics of data. All data must be consistent.
Uncertainty: Coefficients are almost never known with certainty
Data Sources: May come from statistical estimation or from deductive processes (economic engineering approach) Economic Engineering
Economic Deductive Approach
We compute profit per unit by taking yield*price and subtracting costs
An art – must always defend the assumptions in published work. Why that yield? Why that price? Sensitivity can be useful to test if results are robust Model Size
Model Used to be costly to add rows and columns to an LP. Parsimonious model building was thus necessary.
Computational time is now cheap.
Rows and columns can be added to save the modeler time in interpretation of results. For example, total balance of loans each period is an accounting identity which the modeler could calculate based on decision variables (starting balance + new loans – principal paid), but it is easier to put in a transfer row and let the computer do it for you. Writing Reports
Writing The raw computer output is almost never very useful to anyone other than the modeler.
Tables should be developed that make the model results clear to others.
Sensitivity analysis (changing key assumptions) is often needed for journal papers and must be reported as well. ...
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 Fall '08
 Duffy,P
 Constraints, objective function, Howard Staunton

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