Deck 4: Sensitivity Analysis and the Simplex Method

Question 1

A binding greater than or equal to ( $\ge$ ) constraint in a minimization problem means that

A) the variable is up against an upper limit.
B) the minimum requirement for the constraint has just been met.
C) another constraint is limiting the solution.
D) the shadow price for the constraint will be positive.

Answer 1

the minimum requirement for the constraint has just been met.

Question 2

Binding constraints have

A) zero slack.
B) negative slack.
C) positive slack.
D) surplus resources.

Answer 2

A

Question 3

When performing sensitivity analysis, which of the following assumptions must apply?

A) All other coefficients remain constant.
B) Only right hand side changes really mean anything.
C) The X₁ variable change is the most important.
D) The non-negativity assumption can be relaxed

Answer 3

A

Question 4

Risk Solver Platform (RSP) provides sensitivity analysis information on all of the following except the

A) range of values for objective function coefficients which do not change optimal solution.
B) impact on optimal objective function value of changes in constrained resources.
C) impact on optimal objective function value of changes in value of decision variables.
D) impact on right hand sides of changes in constraint coefficients.

Question 5

The allowable increase for a constraint is

A) how many more units of resource to purchase to maximize profits.
B) the amount by which the resource can increase given shadow price.
C) how much resource to use to get the optimal solution.
D) the amount by which the constraint coefficient can increase without changing the final optimal value.

Question 6

The allowable increase for a changing cell (decision variable) is

A) how many more units to produce to maximize profits.
B) the amount by which the objective function coefficient can increase without changing the optimal solution.
C) how much to charge to get the optimal solution.
D) the amount by which constraint coefficient can increase without changing the optimal solution.

Question 7

The allowable decrease for a constraint is

A) how many more units of resource to purchase to maximize profits.
B) the amount by which the resource can decrease given shadow price.
C) how much resource to use to get the optimal solution.
D) the amount by which constraint coefficient can increase without changing the final optimal value.

Question 8

The allowable decrease for a changing cell (decision variable) is

A) the amount by which the constraint coefficient can decrease without changing final optimal solution.
B) an indication of how many more units to produce to maximize profits.
C) the amount by which objective function coefficient can decrease without changing the final optimal solution.
D) an indication of how much to charge to get the optimal solution.

Question 9

Risk Solver Platform (RSP) provides all of the following reports except

A) Answer
B) Sensitivity
C) Cost performance
D) Limits

Question 10

The sensitivity analysis provides information about which of the following:

A) the impact of a change to an objective function coefficient.
B) the impact of a change in a resource level.
C) the impact of adding simple upper or lower bounds on a decision variable.
D) all of these.

Question 11

The slope of the level curve for the objective function value can be changed by

A) increasing the value of the decision variables.
B) doubling all the coefficients in the objective function.
C) increasing the right hand sides of constraints.
D) changing a coefficient in the objective function.

Question 12

The Cell Value column in the Solver Answer Report shows

A) which constraints are binding.
B) final (optimal) value assumed by each constraint cell.
C) objective function values.
D) Right hand sides of constraints.

Question 13

Benefits of sensitivity analysis include all the following except:

A) provides a better picture of how solutions change as model factors change.
B) fosters managerial acceptance of the optimal solution.
C) overcomes management skepticism of optimal solutions.
D) answers potential managerial questions regarding the solution to an LP problem.

Question 14

When a manager considers the effect of changes in an LP model's coefficients he/she is performing

A) a random analysis.
B) a coefficient analysis.
C) a sensitivity analysis.
D) a qualitative analysis.

Question 15

Meaningful Risk Solver Platform (RSP) sensitivity report headings can be defined

A) by adding cell notes to spreadsheet cells.
B) by using the Guess button in the Risk Solver Platform (RSP) dialog box.
C) by carefully labeling rows and columns in the spreadsheet model.
D) naming cells in the spreadsheet model.

Question 16

The coefficients in an LP model (c_j, a_ij, b_j) represent

A) random variables.
B) numeric constants.
C) random constants.
D) numeric variables.

Question 17

Which of the following statements is false concerning either of the Allowable Increase and Allowable Decrease columns in the Sensitivity Report?

A) The values equate the decision variable profit to the cost of resources expended.
B) The values give the range over which a shadow price is accurate.
C) The values give the range over which an objective function coefficient can change without changing the optimal solution.
D) The values provide a means to recognize when alternate optimal solution exist.

Question 18

The difference between the right-hand side (RHS) values of the constraints and the final (optimal) value assumed by the left-hand side (LHS) formula for each constraint is called the slack and is found in the .

A) Status report
B) Slack report
C) Results report
D) Cell Value report

Question 19

A binding less than or equal to ( $\le$ ) constraint in a maximization problem means

A) that all of the resource represented by the constraint is consumed in the solution.
B) it is not a constraint that the level curve contacts.
C) another constraint is limiting the solution.
D) the requirement for the constraint has been exceeded.

Question 20

A manager should consider how sensitive the model is to changes in all of the following except

A) differential coefficients.
B) objective function coefficients.
C) constraint coefficients.
D) right-hand side values for constraints.

Question 21

If the shadow price for a resource is 0 and 150 units of the resource are added what happens to the optimal solution?

A) increases by an unknown amount
B) increases more than 0 but less than 150
C) no change
D) decreases by an unknown amount

Question 22

When a solution is degenerate the reduced costs for the changing cells

A) is always equal to zero.
B) may not be unique.
C) may be set to any value the manager needs.
D) is equal to infinity.

Question 23

Given an objective function value of 150 and a shadow price for resource 1 of 5, if 10 more units of resource 1 are added (assuming the allowable increase is greater than 10), what is the impact on the objective function value?

A) increase of 50
B) increase of unknown amount
C) decrease of 50
D) increase of 10

Question 24

To convert=constraints into = constraints the Simplex method adds what type of variable to the constraint?

A) slack
B) dummy
C) redundant
D) spreading

Question 25

A change in the right hand side of a binding constraint may change all of the following except

A) optimal value of the decision variables
B) slack values
C) other right hand sides
D) objective function value

Question 26

The solution to an LP problem is degenerate if

A) the right hand sides of any of the constraints have an allowable increase or allowable decrease of zero.
B) the shadow prices of any of the constraints have an allowable increase or allowable decrease of infinity.
C) the objective coefficients of any of the variables have an allowable increase or allowable decrease of zero.
D) the shadow prices of any of the constraints have an allowable increase or allowable decrease of zero.

Question 27

When a solution is degenerate the shadow prices and their ranges

A) may be interpreted in the usual way but they may not be unique.
B) must be disregarded.
C) are always valid and unique.
D) are always understated

Question 28

What is the value of the objective function if X₁ is set to 0 in the following Limits Report? $\begin{array}{llc}&\text { Target }\\\text { Cell } & \text { Name } & \text { Value } \\\hline \$ \mathrm{E} \$ 5 & \text { Unit profit: Total Profit: } & 3200\end{array}$

$\begin{array}{llrrrrr}&\text {Adiustable}&&\text {Lower }&\text {Target }&\text {Upper}&\text { Targel}\\\text { Cell } & \text { Name } & \text { Value } & \text { Limit } & \text { Result } & \text { Limit } & \text { Result } \\\hline \$ \text { B } \$ 4 & \text { Number to make: } \mathrm{X} 1 & 80 & 0 & 800 & 79.9999999 & 3200 \\\$ \text { C } \$ 4 & \text { Number to make } \mathrm{X} 2 & 20 & 0 & 24 & 20 & 3200\end{array}$

A) 80
B) 800
C) 2400
D) 3200

Question 29

The reduced cost for a changing cell (decision variable) is

A) the amount by which the objective function value changes if the variable is increased by one unit.
B) how many more units to product to maximize profits.
C) the per unit profits minus the per unit costs for that variable.
D) equal to zero for variables at their optimal values.

Question 30

What is the value of the slack variable in the following constraint when X₁ and X₂ are nonbasic and only non-negativity is used as simple bounds? X₁ + X₂ + S₁ = 100

A) 0
B) 50
C) 100
D) can't be determined from the given information

Question 31

The shadow price of a nonbinding constraint is

A) positive
B) zero
C) negative
D) indeterminate

Question 32

The absolute value of the shadow price indicates the amount by which the objective function will be

A) improved if the corresponding constraint is loosened.
B) improved if the corresponding constraint is tightened.
C) made worse if the corresponding constraint is loosened.
D) improved if the corresponding constraint is unchanged.

Question 33

A variable with a final value equal to its simple lower or upper bound and a reduced cost of zero indicates that

A) an alternate optimal solution exists.
B) an error in formulation has been made.
C) the right hand sides should be increased.
D) the objective function needs new coefficients.

Question 34

How many basic variables are there in a linear programming model which has n variables and m constraints?

A) n
B) m
C) n + m
D) n- m

Question 35

For a minimization problem, if a decision variable's final value is 0, and its reduced cost is negative, which of the following is true?

A) Alternate optimal solutions exist.
B) There is evidence of degeneracy.
C) No feasible solution was found.
D) The variable has a non-negativity constraint.

Question 36

If the allowable increase for a constraint is 100 and we add 110 units of the resource what happens to the objective function value?

A) increase of 100
B) increase of 110
C) decrease of 100
D) increases but by unknown amount

Question 37

All of the following are true about a variable with a negative reduced cost in a maximization problem except

A) its objective function coefficient must increase by that amount in order to enter the basis.
B) it is at its simple lower bound.
C) it has surplus resources.
D) the objective function value will decrease by that value if the variable is increased by one unit.

Question 38

If the shadow price for a resource is 0 and 150 units of the resource are added what happens to the objective function value?

A) increase of 150
B) increases more than 0 but less than 150
C) no increase
D) increases but by an unknown amount

Question 39

A change in the right hand side of a constraint changes

A) the slope of the objective function
B) objective function coefficients
C) other right hand sides
D) the feasible region

Question 40

When the allowable increase or allowable decrease for the objective function coefficient of one or more variables is zero it indicates (in the absence of degeneracy) that

A) the problem is infeasible.
B) alternate optimal solutions exist.
C) there is only one optimal solution.
D) no optimal solution can be found.

Question 41

Is the optimal solution to this problem unique, or is there an alternate optimal solution? Explain your reasoning.

Question 42

The optimization technique that locates solutions in the interior of the feasible region is known as _____?

A) sub-optimal optimization
B) sensitivity analysis
C) robust optimization
D) USET optimization

Question 43

A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water both of which are in short supply. The following table summarizes the data for the problem.
Based on the following Risk Solver Platform (RSP) sensitivity output, how much can the price of Corn drop before it is no longer profitable to plant corn?

Question 44

Given the following Risk Solver Platform (RSP) sensitivity output how much does the objective function coefficient for X₂ have to increase before it enters the optimal solution at a strictly positive value?

Question 45

What is the smallest value of the objective function coefficient X₁ can assume without changing the optimal solution?
$$\begin{array} { l l } \operatorname { MAX } : & 7 \mathbf { X } _ { 1 } + 4 \mathbf { X } _ { \mathbf { 2 } } \\\text { Subject ta: } & 2 \mathbf { X } _ { 1 } + \mathbf { X } _ { \mathbf { 2 } } \leq 16 \\& \mathbf { X } _ { 1 } + \mathbf { X } _ { \mathbf { 2 } } \leq 10 \\& 2 \mathbf { X } _ { 1 } + 5 \mathbf { X } _ { \mathbf { 2 } } \leq 40 \\& \mathbf { X } _ { 1 } \mathbf { X } _ { \mathbf { 2 } } \geq \mathbf {0 }\end{array}$$ $\text {Changing Calls}$
$\begin{array}{llrrrrr}&&\text { Final} &\text {Reduced }&\text {Objective }&\text {Allowable}&\text { Allowable }\\\text { Cell } & \text { Name } & \text { Value } & \text { Cost } & \text { Coefficient } & \text { Increase } & \text { Decrease } \\\hline \$ \mathrm{~B} \$ 4 & \text { Number to make: } \mathrm{X} 1 &6& 0 & 7& 1& 3\\\$ \mathrm{C} \$ 4 & \text { Number to make: } \mathrm{X} 2 & 4 & 0 &4 &3 & 0.5\\\end{array}$

$\text {Constraints}$
$\begin{array}{llrrrrr}&&\text { Final } &\text {Shadow} &\text { Constraint } &\text {Allowable} &\text { Allowable }\\\text { Cell } & \text { Name } & \text { Value } & \text { Price } & \text { R.H. Side } & \text { Increase } & \text { Decrease } \\\hline \$ \mathrm{D} \$ 8 & \text { Used } & 16&3 & 16&4&2.67 \\\$ \mathrm{D} \$ 9 & \text { Used } & 10 & 1&10 & 1 & 2 \\\$ \mathrm{D} \$ 10 & \text { Used } & 32& 0 & 40 & 1 E+30 & 8\end{array}$

Question 46

Which of the constraints are binding at the optimal solution for the following problem and Risk Solver Platform (RSP) sensitivity output?

Question 47

Given the following Risk Solver Platform (RSP) sensitivity output what range of values can the objective function coefficient for variable X1 assume without changing the optimal solution?
$\text {Changing Calls}$
$\begin{array}{llrrrrr}&&\text { Final} &\text {Reduced }&\text {Objective }&\text {Allowable}&\text { Allowable }\\\text { Cell } & \text { Name } & \text { Value } & \text { Cost } & \text { Coefficient } & \text { Increase } & \text { Decrease } \\\hline \$ \mathrm{~B} \$ 4 & \text { Number to make: } \mathrm{X} 1 & 9.49 & 0 & 5 & 1.54 & 1 \\\$ \mathrm{C} \$ 4 & \text { Number to make: } \mathrm{X} 2 & 1.74 & 0 & 6 & 1.5 & 1.47\end{array}$

$\text {Constraints}$
$\begin{array}{llrrrrr}&&\text { Final } &\text {Shadow} &\text { Constraint } &\text {Allowable} &\text { Allowable }\\\text { Cell } & \text { Name } & \text { Value } & \text { Price } & \text { R.H. Side } & \text { Increase } & \text { Decrease } \\\hline \$ \mathrm{D} \$ 8 & \text { Used } & 42 & 0 & 48 & 1 \mathrm{E}+30 & 6 \\\$ \mathrm{D} \$ 9 & \text { Used } & 132 & 0.24 & 132 & 12 & 12 \\\$ \mathrm{D} \$ 10 & \text { Used } & 24 & 124 & 24 & 133 & 2\end{array}$

Question 48

Constraint 3 is a non-binding constraint in the final solution to a maximization problem. Complete the following entry for the Risk Solver Platform (RSP) sensitivity report. Cell labels are included to ease of reference.

Question 49

A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are in short supply. The following table summarizes the data for the problem.
Suppose the farmer can purchase more fertilizer for $2.50 per pound, should he purchase it and how much can he buy and still be sure of the value of the additional fertilizer? Base your response on the following Risk Solver Platform (RSP) sensitivity output.

Question 50

When automatically running multiple optimizations in Risk Solver Platform (RSP), what spreadsheet function indicates which optimization is being run?

A) =PsiOptNum()
B) =PsiOptValue()
C) =PsiOptIndex()
D) =PsiCurrentOpt()

Question 51

Consider the following linear programming model and Risk Solver Platform (RSP) sensitivity output. What is the optimal objective function value if the RHS of the first constraint increases to 18?

Question 52

Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited so at most 8 will be produced.
How much can the price of Desks drop before it is no longer profitable to produce them? Base your response on the following Risk Solver Platform (RSP) sensitivity output.
The LP model for the problem is

Question 53

A solution to the system of equations using a set of basic variables is called

A) a feasible solution.
B) basic feasible solution.
C) a nonbasic solution.
D) a nonbasic feasible solution

Question 54

What are the objective function coefficients for X₁ and X₂ based on the following Risk Solver Platform (RSP) sensitivity output?

Question 55

Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited so at most 8 will be produced.
Suppose the company can purchase more varnishing time for $3.00, should it be purchased and how much can be bought before the value of the additional time is uncertain? Base your response on the following Risk Solver Platform (RSP) sensitivity output.
The LP model for the problem is

Question 56

Exhibit 4.1
The following questions are based on the problem below and accompanying Risk Solver Platform (RSP) sensitivity report.
Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
$$\begin{array} { c c c c } \text { Week } & \text { Trucking Lanrits } & \text { Railway Limits } & \text { Air Carga Lavits } \\\hline 1 & 45 & 60 & 15 \\2 & 50 & 55 & 10 \\3 & 55 & 45 & 5 \\\hline \text { Costs } ( \$ \text { per } 1000 \text { tors) } & \$ 200 & \$ 140 & \$ 400\end{array}$$ The following is the LP model for this logistics problem.


-Refer to Exhibit 4.1. The Week 1 by Truck and Week 1 by Rail constraints each have a shadow price of -360. What do these values imply?

Question 57

Why might a decision maker prefer a solution in the interior of the feasible region of a linear programming problem?

A) Such a solution has a better objective function value than any other solution
B) Such a solution is likely to remain feasible if some of the coefficients in the problem change
C) The decision maker is not sure if he/she wants to maximize or minimize the objective
D) Such a solution has more binding constraints

Question 58

The Simplex method works by first

A) identifying any basic feasible solution.
B) choosing the largest value for X₁.
C) setting X₁ at one-half of the its maximum value.
D) going directly to the optimal solution.

Question 59

What is the optimal objective function value if X₁ is at its lower limit in the following Risk Solver Platform (RSP) sensitivity output?

Question 60

The Simplex method uses which of the following values to determine if the objective function value can be improved?

A) shadow price
B) target value
C) reduced cost
D) basic cost

Question 61

Use slack variables to rewrite this problem so that all its constraints are equality constraints.

Question 62

Identify the different sets of basic variables that might be used to obtain a solution to this problem.

Question 63

Exhibit 4.2
The following questions correspond to the problem below and associated Risk Solver Platform (RSP) sensitivity report.
Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations:
The following LP model allows Robert to maximize his numerical grade.

Refer to Exhibit 4.2. Constraint cell F9 corresponds to the constraint, W₁ + W₂ + W₃ + W₄ = 1, and has a shadow price of 75. Armed with this information, what can Robert request of his instructor regarding this constraint?

Question 64

Exhibit 4.2
The following questions correspond to the problem below and associated Risk Solver Platform (RSP) sensitivity report.
Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations:
The following LP model allows Robert to maximize his numerical grade.

Refer to Exhibit 4.2. Based on the Risk Solver Platform (RSP) sensitivity report information, Robert has been approved by his instructor to increase the total weight allowed for the project and final exam to 0.50 plus the allowable increase. When Robert re-solves his model, what will his new final grade score be?

Question 65

Exhibit 4.2
The following questions correspond to the problem below and associated Risk Solver Platform (RSP) sensitivity report.
Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations:
The following LP model allows Robert to maximize his numerical grade.

Refer to Exhibit 4.2. Based on the Risk Solver Platform (RSP) sensitivity report information, is there anything Robert can request of his instructor to improve his final grade?

Question 66

Solve this problem graphically. What is the optimal solution and what constraints are binding at the optimal solution?

Question 67

Exhibit 4.1
The following questions are based on the problem below and accompanying Risk Solver Platform (RSP) sensitivity report.
Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
The following is the LP model for this logistics problem.

Refer to Exhibit 4.1. Of the three percentage of effort constraints, Shipped by Truck, Shipped by Rail, and Shipped by Air, which should be examined for potential cost reduction?

Question 68

Solve this problem graphically. What is the optimal solution and what constraints are binding at the optimal solution?

Question 69

Exhibit 4.1
The following questions are based on the problem below and accompanying Risk Solver Platform (RSP) sensitivity report.
Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
The following is the LP model for this logistics problem.

Refer to Exhibit 4.1. Are there alternate optimal solutions to this problem?

Question 70

Identify the different sets of basic variables that might be used to obtain a solution to this problem.

Question 71

Exhibit 4.1
The following questions are based on the problem below and accompanying Risk Solver Platform (RSP) sensitivity report.
Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
The following is the LP model for this logistics problem.

Refer to Exhibit 4.1. Should the company negotiate for additional air delivery capacity?

Question 72

Use slack variables to rewrite this problem so that all its constraints are equality constraints.