Modeling LP

8/2/2019 Modeling LP

http://slidepdf.com/reader/full/modeling-lp 1/10

© Hun Myoung Park Tuesday, February 01, 2011

1

Linear Programming and Policy Modeling1

Linear Programming (LP), a type of mathematical programming, is pioneered by George B.Dantzig in the U.S. Air Force, who developed the simplex method in 1947 to solve military

logistics problems.2

LP has been widely used in the private sector (e.g., manufacture andaviation business) but less employed in the public sector largely due to its somewhat strongassumptions. But its model formulation and core concepts (e.g., reduced cost and shadow

price) are valuables to policy analysts and public managers.

1. Underlying Assumptions

LP has core underlying assumptions of certainty, divisibility, and linearity (proportionality,additivity, and homogeneity), which are seldom satisfied precisely in the real world. They arestrong enough in this sense. See pp. 78-81 of Albright & Winston (2005) and pp. 36-42 of Hillier & Lieberman (2010).

1.1 Certainty (deterministic or nonstochastic) Every piece of information (i.e., coefficients inthe objective function and constraints) is known. LP is deterministic without any risk anduncertainty; therefore, LP often uses “average” to summarize the reality assuming certainty.The environmental variables (the right-hand side values of constraints) are not probabilistic.The outcomes do not have any probability distributions. If outcomes are affected by weather (e.g., rain, temperature, humidity, and wind) and social events (e.g., war, strike, recession,and disease), the assumption is likely to be violated.

1.2 Divisibility The legitimate values of the decision variables are non-negative real numbersincluding integers. Integer programming, a variant of LP, can handle indivisiblility.

1.3 Linearity includes proportionality, additivity, and homogeneity that are closely relatedeach other.

• Proportionality means that the outcome variables are proportional to the level of thedecision variables. That is, the coefficients in constraints or objective function areconstants. If different workers had different productivities, for example, thisassumption (homogeneity) is violated.

• Additivity means that the production possibility frontier is a straight line. Thecontribution of a decision variable to the objective and any constraint is independentof other decision variables. The objective function and constraints are expressed by

linear combination of decision variables. This assumption rules out synergies or complementarities between decision variables (no interaction term is allowed in theobjective function and constraints).

• Homogeneity requires that all units of decision, outcome, and environmental variablesare assumed to be identical. For example, individual employees have the same

productivity; materials (e.g., gas, water, and timber) have the same quality; and eachtype of machines (e.g., snow blower and excavator) has the same function andefficiency.

1

This class note is based on David Good’s lecture and textbooks. But, this note SHOULD NOT be used as asubstitute of textbooks.2 The original name is Programming of Interdependent Activities in a Linear Structure.




2

2. Components of Mathematical Programming

2.1 Decision variables are the alternatives that are settable (controllable) by decision makers.These variables should appear in the “By Changing Cells” box of Excel Solver.

2.2 Objective function is a mathematical expression for a single goal (e.g., maximizing profitor minimizing cost) that you want to optimize: Max(Σ β i xi) or Min(Σ β i xi). This objectivefunction should appear in the “Set Target Cell” input box in Excel Solver.

2.3 Constraints describe environmental, technological, legal, and other factors which limityour choices. A constraint compares an outcome variable, which is determined by acombination of decision variables, on the left-hand side (LHS) and an associatedenvironmental variable on the right-hand side (RHS): Σδi xi ≤ ci, Σδi xi ≥ ci , or Σδi xi = ci.For example, the total amount of snow removed by sprinkler systems and snow blowersshould be larger than 9,000 tons per day (average daily snowfall on the major streets).

Constraints should appear in the “Subject to the Constraints” box in Excel Solver.

2.4 Nonnegativity restricts decision variables to having nonnegative values: xi ≥ 0. In general,it is not possible that a decision variable has a negative value (e.g., the number of snow

blowers mobilized is -10. Does this sentence make sense?). Excel Solver has an optionwhere you can choose “Assume Non-Negative.”

3. Types of Linear Programming

Product mix and blending problems are most common but there are variants of LP problems.

3.1 Product mix problem maximizes the value (profit) of the outputs produced given a set of resources available. For instance, “How do I have to use sprinkler systems and snow

blowers available to maximize the amount of snow removed?”

3.2 Blending problem minimizes the cost of production (value of resources) given a set of outputs determined by decision makers. For example, “How do I have to use sprinkler systems and snow blowers to minimize the cost of removing 9,000 tons?”

3.3 Transportation and assignment problems. In a transportation problem, decision variablesinvolve matching things from origins and destinations and the objective function is oftento minimize travel time or cost subject to supply constraints (resource limitations or

capacities) and demand constraints (minimum requirement of to be transported).Assignment problems always have exactly one unit for each demand (task) and exactly




3

one unit for each source supply (person), and their solutions are integer (integer programming)

3.4 Variants of LP

• Integer programming

• Binary programming• Goal programming

4. Steps in Formulating LP Problems

4.1 Evaluate the relevance of applying LP to the problem on hand. Does this problem fit astandard LP pattern (product mix or blending)? Example 3.2 in Albright & Winston(2005) has all pieces of necessary information (e.g., frame prices, availability of labor,metal, and glass, production function, and profit function) under certainty. This is aclassical LP product mix problem.

4.2 Determine decision variables and assign relevant symbols (e.g., x1, x2, x3… rather than x,yy, xyz). Ask yourself if you can control the values of policy variables or if they aredetermined by others (environments). In Example 3.2, the decision variables are thenumbers of frames 1 through 4 to be produced and are labeled as x1 through x4.

4.3 Define the criterion variable using a mathematical (linear) expression of decisionvariables (policy variables). This is an objective function. You may maximize the profit(product mix) or minimize the cost of production (blending). In Example 3.2, theobjective function is “maximizing the profit from producing frame 1 through 4” and itscorresponding expression is “Max(6x1 + 2x2 + 4x3 + 3x4).” Note that 6, 2, 4, and 3 are

unit profit for frame 1 through 4. For example, unit profit for frame 2 is 2 =$12.5-1(8.00)-2(.50)-2(.75) since one unit of frame 2 needs one unit of labor, two units of metal, andtwo units of glass. See page 85 for their computation.

4.4 Incorporate constraints that compare outcome variables, which are expressed usingdecision variables, on the left-hand side with corresponding environmental variables onthe right-hand side. For example, the constraint for labor says, “Do not use more labor than 4,000 hours in any circumstance for producing frames” and can be expressed as “2x1 + x2 + 3x3 + 2x4 ≤ 4,000.” The constraint for frame 1 says, “Do not produce more frame 1than 1,000 because of its limited market demand of 1,000)” and can be expressed as “x1 ≤ 1,000.” DO NOT forget to include nonnegativity constraints. As a result, the product mix

problem of producing frames is summarized as:

Max(6x1 + 2x2 + 4x3 + 3x4)

Subject to:2x1 + x2 + 3x3 + 2x4 ≤ 4,000 (labor constraint)4x1 + 2x2 + x3 + 2x4 ≤ 6,000 (metal constraint)6x1 + 2x2 + x3 + 2x4 ≤ 10,000 (glass constraint)x1 ≤ 1,000 (sales limit of frame 1)x2 ≤ 2,000 (sales limit of frame 2)x3 ≤ 500 (sales limit of frame 3)

x4 ≤ 1,000 (sales limit of frame 4)x1, x2, x3, x4 ≥ 0 (nonnegativity)




4

4.5 Check units of measurement so that coefficients and decision variables in the objective

function and constraints are commensurable. For example, the unit of 6 (coefficient of frame 1 in the objective function) is $ per labor hour (=$/hour) and unit of x is labor hour.Therefore, 6 × x1 results in $ (=$/hour × hour); Okay, that makes sense! Similarly, 3 × x 3

in the labor constraint returns labor hours (=labor hours/frame 3 × frame 3). If thecoefficient of a decision variable x1 is measured in million USD per frame 1 and the unitof x1 is 1,000 frame 1, you will be in trouble. What if a variable is measured in gallon andits coefficients in the objective function and constraints has some units per gram or pound(e.g., Killowatt/gram)? Also you have to adjust scales to avoid too large numbers or toosmaller numbers (e.g., 1billion or 1,000 million rather than 1,000,000,000); the former may result in the overflow problem, whereas the latter ends up getting misreadingnumbers due to “underflow.” In Example 3.2, the scales of coefficients (e.g., 6, 2, 4, 3 for the objective function) and resources (e.g., labor hour and ounce) are reasonable. See pp.80-81 of Albright & Winston (2005).

4.6 Translate the objective function (including decision variables) and constraints into theexpressions that computer software packages (e.g., SAS/OR, LINDO/LINGO, and ExcelSolver) can understand. Of course, you may solve LP problems using the graphicalmethod (identify the feasible set defined by constraints; find the corners of the feasibleset; evaluate the value of the objective function—isoprofit line— at each corner; anddetermine the corner which has the best value) or the simplex method by hand. For datainput in Excel, see pp. 64-66 of Albright & Winston (2005). In addition, see a SAS script

below to solve Example 3.2. SAS/OR by default assumes nonnegativity.

DATA product_mix;

INPUT _ID_ $ x1 x2 x3 x4 _TYPE_ $ _RHS_;

DATALINES;

object 6 2 4 3 MAX .

labor 2 1 3 2 LE 4000

Metal 4 2 1 2 LE 6000

glass 6 2 1 2 LE 10000

frame1 1 0 0 0 LE 1000

frame2 0 1 0 0 LE 2000

frame3 0 0 1 0 LE 500

frame4 0 0 0 1 LE 1000

RUN;

PROC LP; RUN;

4.7 Albright & Winston (2005) suggest an Excel input style for LP that is logical andsystematic. But my preferred style is one that arranges elements from 1) raw data (e.g.,

price and coefficients), 2) decision variables, 3) the objective functions, 4) constraints,and, if any, 5) intermediate computations as shown in the SAS data input above; this styleis more logical and systematic.3 See ProductMix2.xls available on the course Web page.

5. General Principles of LP Solutions

There are general rules regarding corner solutions. A corner is an intersection of constraintslines and appears on the edge of the feasible set. See pp. 62-63 of Albright & Winston (2005).

3

I don’t like the first part of their convention, “to enter enough range names but not to go overboard”. Mysuggestion is to use modeling concepts (e.g., “objective” instead of “Profit,” “decision” instead of “Frames_

produced,” “constraint1_3” instead of “Resource_used,” and “RHS1_3” instead of “Resources_available”).




5

5.1 The solution cannot be in the interior of the feasible set.

5.2 The solution must involve a corner of the feasible set (as opposed to other alternative points on the edge). The only exception is where the isoprofit line is parallel to one of theconstraints (multiple solutions).

5.3 If two corners have the same best value, then all combinations between them are alsosolutions. This case occurs when the isoprofit line has the same slope as one of theconstraints; the entire edge of the feasible set is optimal.

5.4 If we are searching at a corner, and the value of objective function goes down at alladjacent corners of the feasible set, you must be at the solution (this is the stopping rulethat the simplex method use).

Table 1. Excel Solver’s Answer Report (Example 3.2 of Albright & Winston)Target Cell (Max)

Cell Name Original Value Final Value

$F$32 Profit $9,200 $9,200

Adjustable Cells

Cell Name Original Value Final Value

$B$16 Frames produced 1000 1000

$C$16 Frames produced 800 800

$D$16 Frames produced 400 400

$E$16 Frames produced 0 0

Constraints

Cell Name Cell Value Formula Status Slack

$B$21 Labor hours Used 4000 $B$21<=$D$21 Binding 0

$B$22 Metal (oz.) Used 6000 $B$22<=$D$22 Binding 0

$B$23 Glass (oz.) Used 8000 $B$23<=$D$23 Not Binding 2000

$B$16 Frames produced 1000 $B$16<=$B$18 Binding 0

$C$16 Frames produced 800 $C$16<=$C$18 Not Binding 1200

$D$16 Frames produced 400 $D$16<=$D$18 Not Binding 100

$E$16 Frames produced 0 $E$16<=$E$18 Not Binding 1000

6. Interpretation of the Objective Function (Optimal Solution)

Under normal conditions, the final values of decision variables in the final tableau areoptimal solution that maximizes or minimizes the objective function.

6.1 Unique optimal solution. When implementing the optimal solution (a set of values of decision variables), you will get the optimal value (maximum profit or minimum cost) of the objective function. Table 1 suggests the optimal solution of producing 1,000 frame800 frame 2,400 frame 3, and zero frame 4 that will bring a profit of 9,200 to you. This

production consumes 4,000 labor hours, 6,000 ounces of metal, and 8,000 ounces of glass.

6.2 Reduced cost is relevant only for decision variables that have a zero value at the optimal

solution. If a decision variable has a positive value at the solution, its reduced cost willalways be zero. Hillier and Lieberman (2010) say, “[I]t is the minimum amount by which




6

the unit cost of activity j would have to be reduced to make it worthwhile to undertakeactivity j (increase x j from zero)” (p.234). From a theoretical view, Winston (2004) states,“The reduced cost for a nonbasic variable … is the maximum amount by which thevariable’s objective function coefficient can be increased before the current basis

becomes suboptimal, and it become optimal for the nonbasic variable to enter the basis”

(pp. 277-278). Reduced cost is about how much a coefficient of the objective functionwould have to decrease for you to start producing a particular output (more than or equalto 1 unit). In Table 2, the optimal value of frame 4 is zero and its reduced cost is -.2. 4 Thisreduced cost can be interpreted as 1) if you want to (forcefully) produce at least one frame4, its coefficient in the objective function should be reduced at least by -.2 (or increased

by .2) or 2) when you increase the objective function coefficient of frame 4 at least by .2,the new optimal solution includes more than or equal to 1 unit of frame 4. In this case, thesolution and objective value will change accordingly. See pp. 70-74 of Albright &Winston (2005) and pp.233-235 of Hillier & Lieberman (2010).

6.3 Ranging information of the coefficient of a decision variable in the objective function

provides an allowable increase and an allowable decrease (or lower and upper bounds) of a coefficient in the objective function. This information tells us how flexible objectivefunction coefficients are or how much a coefficient of the objective function can increaseor decrease while having the optimal solution remain unchanged. An allowable increaseand an allowable decrease are not necessarily symmetric. As long as coefficients of theobjective function stay within the range, the optimal solution to the problem remains thesame but the objective value might change due to the change in the coefficient . In Table 2,the current coefficient 6 of frame 1 can change from 4 (=6-2) to 6+1030 (6+1E+30,virtually positive infinity) without influencing the current optimal solution (a set of valuesof decision variables). Note that 2 is its allowable decrease and +1030 is its allowableincrease. The coefficient of frame 2 ranges from 1.75 (=2-.25) to 3 (=2+1) and frame 3coefficient from 3.5 (=4-.5) to 6 (4+2).

Table 2. Excel Solver’s Sensitivity Report (Example 3.2 of Albright & Winston)Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$16 Frames produced 1000 2.0 6 1E+30 2

$C$16 Frames produced 800 0.0 2 1 0.25

$D$16 Frames produced 400 0.0 4 2 0.5

$E$16 Frames produced 0 -0.2 3 0.2 1E+30

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$B$21 Labor hours Used 4000 1.2 4000 250 1000

$B$22 Metal (oz.) Used 6000 0.4 6000 2000 500

$B$23 Glass (oz.) Used 8000 0.0 10000 1E+30 2000

4 The reduced cost of 2.0 for frame 1 does not make any sense because the value of decision variable for frame 1is 1,000 (nonzero!). This Excel report seems misleading somewhat. In fact, the 2.0 should be the shadow price

of constraint of the market demand of frame 1. The interpretation is, “For 1 increase in the market demand of frame, the profit is expected to increase by $2, holding other variables constant.” Excel Solver also returnswrong allowable increase (infinity) and decrease (2); Quattro Pro Optimizer reports 400 and 600, respectively.




7

Corel Quattro Pro Optimizer produces similar result as Excel Solver does (SeeProductMix.qpw available on the course Web site). But the Optimizer does not assumenonnegativity and thus asks users to explicitly input the nonnegativety constraint. UnlikeExcel Solver, the Optimizer reports reduced prices for objective function coefficients under the Dual Value column. But the allowable increase (1,000) and decrease (250) of the

coefficient for frame 4 is questionable; Excel Solver’s allowable increase (.2) and decrease(infinity) seem more reasonable (correct).5

Table 3. Quattro Pro Optimizer’s Sensitivity Report (Example 3.2 of Albright & Winston)Solution Cell Starting Final

A:H17 7500 9200

Variable Cells Starting Final Gradient Increment DecrementA:B14..B14 200 1000 6 Infinite 2

A:C14..C14 2000 800 2 1 0.25

A:D14..D14 200 400 4 2 0.5

A:E14..E14 500 0 3 0.2 Infinite

Cell Value Constraint Binding? Slack Dual Value Right Value Increment DecrementA:B20..B20 4000 <=4000 Yes 0 1.2 4000 250 1000

A:B21..B21 6000 <=6000 Yes 0 0.4 6000 2000 500

A:B22..B22 8000 <=10000 No 2000 0 10000 Infinite 2000

A:B23..B23 1000 <=1000 Yes 0 2 1000 400 600

A:B24..B24 800 <=2000 No 1200 0 2000 Infinite 1200

A:B25..B25 400 <=500 No 100 0 500 Infinite 100

A:B26..B26 0 <=1000 No 1000 0 1000 Infinite 1000

A:B14..B14 1000 >=0 No 1000 0 0 1000 Infinite

A:C14..C14 800 >=0 No 800 0 0 800 Infinite

A:D14..D14 400 >=0 No 400 0 0 400 Infinite

A:E14..E14 0 >=0 Yes 0 -0.2 0 1000 250

7. Problematic Solutions

If a LP problem does not have a unique solution, it may have either no solutions (infeasiblesolution and unbounded solution) or unreliable solutions (multiple solutions and degeneracy).

5 In Excel, worksheets are referred to by sheet1!, sheet2!, …, whereas Quattro Pro by default use A:, B:, …They respectively use “:” and “..” to indicate a range of cells (e.g., B2:D5 in Excel and B2..D5 in Quattro Pro).




8

7.1 Infeasible solutions occur when no alternative satisfies all of the constraints. The problem

may be over-constrained. You are not given sufficient resources to carry that out when producing a certain amount of output. Or you might mistakenly enter constraints (e.g., ≤ instead of ≥) or incorrectly formulated the problem. If any nonzero artificial variable

appears in the set of basic feasible variables in the final tableau, it indicates infeasiblesolution. See pp. 81-82 of Albright & Winston (2005).

7.2 Unbounded solutions. The optimal values of at least one of the decision variables areinfinite. Unbounded solutions will occur when necessary constraints are excluded fromthe problem or constraints are incorrectly entered (e.g., ≤ instead of ≥). We can recognizethis solution when there is no positive coefficient in the pivot column in the tableau of any round except for the final tableau. In this case, the ranges of RHS and coefficients inthe objective function cannot be calculated.

7.3 Multiple solutions occur when the objective function slope is the same as the slopes for

one of the constraints. You will observe multiple solutions if one or more of the allowableincreases or decreases for the coefficients of the objective function are zero. In thiscircumstance, the slightest change in the coefficient would force us to a new corner andhave the solution change; that is, the solution is shaky.

7.4 Degeneracy (or degenerate solution) occurs if a corner which is defined by more than theminimum number of constraints also turns out to be the optimal corner. You can detectdegeneracy, 1) if a constraint which is binding (zero slack or surplus) also has a zeroshadow price, 2) if the ranging information for some constraints has a zero allowableincrease or decrease, and/or 3) if basic feasible variables have zero values in the finaltableau. Degeneracy is particularly a problem for shadow prices. If we have too many

binding constraints at the optimal corner, and we relax or tighten one of them, then thecombination of constraints which are binding will change, and shadow price informationis no longer reliable.

8. Interpretation of Constraints

Binding and non-binding constraints define the feasible set, whereas superfluous andredundant constraints do not.

8.1 Binding constraints limit the optimal solution by forming the boundary of the feasible set.

The solution lies right on the top of these binding constraint lines; resources of theseconstraints are fully utilized (zero slack or surplus). They form the bottle necks or primary limiting factors in the solution. Changing these constraints might produce adifferent solution to the problem. Therefore, the binding constraints are the mostimportant for public managers (decision makers). Excel Solver indicates if a constraint is“Binding” or “Not Binding” under the Status column. In Table 1, the labor and metalconstraints are binding and all available labor and metal were used for production (zeroslack). Similarly, the constraint for frame 1 is binding with zero slack.

8.2 Non-binding constraints are not satisfied as equality at the optimal solution. Not all theresources are utilized (slack or surplus is greater than zero). In Table 1, the glass

constraint is not binding and 2,000 ounces of glass were not used (nonzero slack).




9

Similarly, the constraints of frame 2 through 3 are not binding since their slacks are notzeros.

8.3 Superfluous constraints never touch the feasible set, whereas redundant constraints contain exactly the same limitation information as another constraint. Superfluous and

redundant constraints are not important, if not useless, because ignoring them would leadto exactly the same choice.

8.4 Slack and surplus. Slack is calculated for a “≤ type” constraint usually for resourcelimitations, while surplus is for a “≥ type” constraint usually for decision variables.6 Slack describes the amount of slack or unutilized resources (RHS-LHS) and surplus describesthe amount by which a minimum requirement has been exceeded (LHS-RHS). In Table 1,the optimal solution suggests that 4,000 hours of labor, 6,000 ounces of metal, and 8,000ounces of glass be utilized to maximize the profit of $9,200. Labor and metal uses reachtheir maximums but glass use does not. Therefore, the slack (remember, this is a ≤ typeconstraint) of labor and metal is zero; 0=4,000 (maximum value of RHS)-4,000 (LHS)

and 0=6,000-6,000. Similarly, the slack of glass use is 2,000=10,000-8,000. The slacks of constraints for frame 2 through 3 are 1,200 (=2,000-800), 100 (=500-100), and 1,000(=1,000-0), respectively. If the slack or surplus of a constraint is zero, the constraint is

binding (constraints for labor, metal, and frame 1). Otherwise, the constraint is not binding (glass and frame 2 through 4). Slack and surplus can never be negative at theoptimal solution.

9. Interpretation of Shadow Price

Shadow price (or dual value) asks how much better the objective function would be if theconstraint could be changed a little bit. This shadow price contains very importantimplications for decision making. Shadow price tells us something about the maximum weshould be willing to pay to relax a constraint by one unit. Relaxing a constraint means that thefeasible set becomes expanded and more alternatives will be considered. See pp.70-74 of Albright & Winston (2005) and pp.131-136 of Hillier & Lieberman (2010).

9.1 The shadow price can also be regarded as a rate at which the objective function changesas you add or delete resources to the constraint. This price is not a price of a product, of course, but the amount of improvement in the objective function that occurs when werelax a constraint by one unit. In other word, for each additional unit of a resource (RHSvalue of the constraint) the amount of profit of the objective function is expected to

increase by that shadow price. In Table 2, find that the shadow prices of labor and metalare 1.2 and .4, respectively; for instance, if you increase the maximum labor hours by 1hour, 4,001=4,000+1, the profit will increase by $1.2. Policy analyst and public managersshould consider labor and metal as key factors that influence the profit significantly. Theshadow price of glass is zero because of the presence of nonzero slack (unused glassavailable to you); increasing the maximum of glass does not influence the objectivefunction.

9.2 Complementary slackness. At the optimal solution to the LP problem, the shadow price of a nonbinding constraint with positive slack/surplus is zero. Conversely, if the shadow

6

Albright & Winston (2005) do not distinguish surplus from slack. Excel Solver indicates slack or surplus under Slack column. If a constraint has ≤ in the Formula column and a positive value in the Slack column, it is slack;if a constraint has ≥ and a positive value, it is surplus.




10

price of a constraint is not zero, then the constraint must be a binding constraint with zeroslack or surplus. In Table 1 and 2, labor and metal have nonzero shadow price; their slacks are zero; and they thus are binding constraints. By contrast, the glass has zeroshadow price and nonzero slack, and accordingly it is not a binding constraint.

9.3 Shadow price ranging information tells you the range of values for the RHS value of aconstraint which does not alter the combination of binding constraints, and which leavesthe shadow price information as valid. SAS/OR reports the range of RHS (RHS ± change)within which the combination of binding constraints remain unchanged, whereas ExcelSolver produces an allowable increase and an allowable decrease of RHS (± change only).In Table 2, find an allowable increase of 250 and an allowable decrease of 1,000 for thelabor constraint; that is, the maximum value (RHS) of labor may change from 3,000(=4,000-1,000) to 4,250 (=4,000-250) within which the current shadow price 1.2 remainsvalid. Similarly, the RHS value of the metal constraint may change from 5,500 (=6,000-500) to 8,000 (=6,000+2,000). If you increase the RHS beyond the range, the combinationof constraints which is binding at the solution would change and you do not know what

the new shadow price would be. DO NOT be confused with an allowable increase and anallowable decrease of a coefficient of the objective function and the RHS value of aconstraint (see the table below).

Table 4. Sensitivity Analysis for Objective Function Coefficients and RHS ValuesA Coefficient of a Decision Variable A Right-hand Side Value

Sensitivity Reduced cost Shadow priceCondition for

being nonzeroThe value of a decision variable is zeroat the optimal solution

The slack/surplus of a constraint is zero(binding constraint) at the optimal solution

Where? Objective function ConstraintsInterpretation(meaning)

Minimum amount of reduction of aobjective function coefficient to forcethe decision variable to be produced

Marginal change in the objective functionvalue for unit increase in the RHS value

Caution Not reliable in case of degeneracy

Allowable range Tolerable upper and lower bounds of the coefficient of a decision variable.

Tolerable upper and lower bounds of aRHS value

Symmetric? Allowable increase and decrease are not symmetric.Software issue SAS/OR reports the upper and lower bounds of the coefficient (or RHS): current

coefficient or RHS value±change (tolerable deviation or allowableincrease/decrease), while Excel returns allowable increase/decrease (±change) only

* All analyses are done under the ceteris paribus assumption (“holding all others things constant”).

9.4 All of the sensitivity analysis, such as shadow prices and shadow price ranginginformation assumes that only one of the RHS values is changing at a time, holding all

other variables being equal or constant (Ceteris Paribus assumption).

Modeling LP

Documents

Transcript of Modeling LP