Midterm 2 Review Operations Management (Lectures 4 - 6)

Mixed Integer

*Some* (BUT not all) decision variables required to have integer values

3. Conditional Constraint

*Tricky One* The selection of one is *conditional upon* the selection of another ex. x2 ≤ x1; x2 - x1 ≤ 0 (x2 is conditional on x1) *Test 0-1 and see if it makes sense*

Nonlinear Programming Function, Z

- ____p^2 + _____p +/- ______ Ex. -24.6p^2 +1696.8p - 22,000

________ variables are best suited to be the decision variables when dealing with yes-or-no decisions.

0-1

In a ________ linear programming model, the solution values of the decision variables are zero or one.

0-1 integer

4 Constraints of 0-1 Model

1. Contingency or Mutually Exclusive Constraint 2. Multiple Choice Constraint 3. Conditional Constraint 4. Co-requisite Constraint

If a maximization linear programming problem consists of all *less-than-or-equal-to* constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then *rounding down* the linear programming optimal solution values of the decision variables will ________ result in a feasible solution to the integer linear programming problem. A) always B) sometimes C) optimally D) never

A) always

If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a ________ constraint. A) multiple-choice B) mutually exclusive C) conditional D) corequisite

A) multiple choice

In a ________ integer model, all decision variables have integer solution values. A) total B) 0-1 C) mixed D) all of the above

A) total

Total Integer

All decision variables required to have integer values

0-1

All decision variables required to have integer values of 0 or 1

In formulating a mixed integer programming problem, the constraint x1 + x2 ≤ 500y1 where y1 is a 0-1 variable and x1 and x2 are continuous variables, then x1 + x2 = 500 if y1 is: A) 0. B) 1. C) 0 or 1. D) none of the above

B) 1.

Which of the following are assumptions or requirements of the transportation problem? A) There must be multiple sources. B) Goods are the same, regardless of source. C) There must be multiple destinations. D) There must be multiple routes between each source and each destination.

B) Goods are the same, regardless of source.

In a transportation problem, items are allocated from sources to destinations: A) at a maximum cost. B) at a minimum cost. C) at a minimum profit. D) at a minimum revenue.

B) at a minimum cost.

In a 0-1 integer programming model, if the constraint x1 - x2 ≤ 0, it means when project 2 is selected, project 1 ________ be selected. A) must always B) can sometimes C) can never D) is already

B) can sometimes *b/c this is a conditional constraint*

Assume that we are using 0-1 integer programming model to solve a capital budgeting problem and xj = 1 if project j is selected and xj = 0, otherwise. The constraint (x1 + x2 + x3 + x4 = 2) means that ________ out of the ________ projects must be selected. A) exactly 1, 2 B) exactly 2, 4 C) at least 2, 4 D) at most 1, 2

B) exactly 2, 4

If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a ________ constraint. A) multiple-choice B) mutually exclusive C) conditional D) corequisite

B) mutually exclusive

If the solution values of a linear program are rounded in order to obtain an integer solution, the solution is: A) always optimal and feasible. B) sometimes optimal and feasible. C) always feasible. D) never optimal and feasible.

B) sometimes optimal and feasible

If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a ________ constraint. A) multiple-choice B) mutually exclusive C) conditional D) corequisite

C) conditional

For a maximization/minimization integer linear programming problem, a feasible solution is ensured by rounding ________ non-integer solution values if all of the constraints are the *less-than-or-equal-to* type. A) up and down B) up C) down D) up or down

C) down

In a ________ integer model, some solution values for decision variables are integers and others can be non-integer. A) total B) 0-1 C) mixed D) all of the above

C) mixed

In the process of evaluating location alternatives, the transportation model method minimizes the: A) total demand. B) total supply. C) total shipping cost. D) number of destinations.

C) total shipping cost.

The problem that deals with the distribution of goods from several sources to several destinations is the: A) network problem. B) assignment problem. C) transportation problem . D) transshipment problem.

C) transportation problem .

Types of integer programming models are: A) total. B) 0-1. C) mixed. D) all of the above

D) all of the above

The branch and bound method of solving linear integer programming problems is: A) an integer method. B) a relaxation method. C) a graphical solution. D) an enumeration method.

D) an enumeration method

Which of the following is not an integer linear programming problem? A) total integer B) mixed integer C) 0-1 integer D) continuous

D) continuous

If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a ________ constraint. A) multiple-choice B) mutually exclusive C) conditional D) corequisite

D) corequisite

If a maximization linear programming problem consists of all less-than-or-equal-to constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then rounding down the linear programming optimal solution values of the decision variables will ________ result in a(n) ________ solution to the integer linear programming problem. A) always, optimal B) always, non-optimal C) never, non-optimal D) sometimes, optimal

D) sometimes, optimal

The linear programming model for a transportation problem has constraints for supply at each ________ and ________ at each destination. A) destination, source B) source, destination C) demand, source D) source, demand

D) source, demand

If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a conditional constraint.

False

In a 0-1 integer programming problem involving a capital budgeting application (where xj = 1, if project j is selected, xj = 0, otherwise) the constraint x1 - x2 ≤ 0 implies that if project 2 is selected, project 1 cannot be selected.

False

In a mixed integer model, all decision variables have integer solution values.

False

In a mixed integer model, the solution values of the decision variables are 0 or 1.

False

In a transportation problem, items are allocated from sources to destinations at a maximum value.

False

In an unbalanced transportation model, all constraints are equalities

False

In the classic game show Password, the suave, silver-haired host informed the contestants, "you can choose to pass or to play." This expression suggests a mixed integer model is most appropriate.

False

The "certainty" linear programming hypothesis is violated by integer programming.

False

The management scientist's fiance informed him that if they were to be married, he would also have to welcome her mother into their home. The management scientist should model this decision as a contingency constraint.

False

4. Co-requisite Constraint

If one is selected, the other will also and vice versa. ex. x1 = x2; x1 - x2 = 0 (If x1 is selected, so will x2)

Non-Linear Programming Break-Even Analysis

Max Z = vp - Cf - vCv Where: v = sales volume (demand) p = price Cf = fixed cost Cv= variable cost

How to find the Optimum point of the Nonlinear Programming Function

Take the Derivative of the Quadratic Function and set it equal to zero * The slope of the tangent line, i.e., a curve at any point, is equal to the derivative of the curve's function. * The slope of a curve at its highest point (*OPTIMAL POINT*) equals zero.

For most real-world applications, an unbalanced transportation model is a more likely occurrence than a balanced transportation model

True

If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a mutually exclusive constraint.

True

If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional constraint.

True

In a 0-1 integer model, the solution values of the decision variables are 0 or 1.

True

In a balanced transportation model where supply equals demand, all constraints are equalities

True

In a transportation problem, items are allocated from sources to destinations at a minimum cost.

True

One type of constraint in an integer program is a multiple-choice constraint.

True

Rounding non-integer solution values up to the nearest integer value *can* result in an infeasible solution to an integer programming problem.

True

The linear programming model for a transportation problem has constraints for supply at each source and demand at each destination

True

The three types of integer programming models are total, 0-1, and mixed.

True

In a ________ transportation model where supply equals demand, all constraints are equalities.

balanced

The ________ method is based on the principle that the total set of feasible solutions can be partitioned into smaller subsets of solutions.

branch and bound

If one location for a warehouse can be selected only if a specific location for a manufacturing facility is also selected, this decision can be represented by a ________ constraint.

conditional

If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a ________ constraint.

conditional

"It's me or the cat!" the exasperated husband bellowed to his well-educated wife. "Hmmmm," she thought, "I could model this decision with a ________ constraint."

contingency or mutually exclusive

In Max/Min Integer Linear Programming Problems, feasible solutions are ensured when rounding *UP* non-integer solutions on constraints that are:

greater-than-or-equal-to type

A ________ integer model allows for the possibility that some decision variables are not integers.

mixed

The cost to send a unit of product from supply source A to demand location B would be represented in the ________ of the linear programming statements.

objective function

In an integer program, if we were choosing between two locations to build a facility, this would be written as: ________.

x1 + x2 = 1

In an unbalanced transportation problem, if supply exceeds demand, the shadow price for at least one of the supply constraints will be equal to ________.

zero

2. Multiple Choice Constraint

A situation in which one some specific number out of total *must be selected* ex. a) x1 + x2 = 1 (Out of x1 and x2, 1 *has* to be selected) b) x1 + x2 + x3 + x4 = 2 (Out of 4 total variables (x1, x2, x3, and x4), 2 *has* to be selected) c) x1 + x2 + x3 + x4 ≤ 2 (No more than 2 selected out of a total of 4) d) x3 + x4 ≤ 1 (Out of the 2 variables, x3 and x4, only one *has* to be selected)

Binary variables are: A) 0 or 1 only. B) any integer value. C) any continuous value. D) any negative integer value

A) 0 or 1 only

In a balanced transportation model where supply equals demand: A) all constraints are equalities. B) none of the constraints are equalities. C) all constraints are inequalities. D) none of the constraints are inequalities.

A) all constraints are equalities.

Lagrange Multiplier

Analogous (equal to) the dual variable or shadow price

In a 0-1 integer programming model, if the constraint x1 - x2 = 0, it means when project 1 is selected, project 2 ________ be selected. A) can also B) can sometimes C) can never D) must also

Answer: D

In a ________ integer model, the solution values of the decision variables are 0 or 1. A) total B) 0-1 C) mixed D) all of the above

B) 0-1

For a maximization/minimization integer linear programming problem, a feasible solution is ensured by rounding ________ non-integer solution values if all of the constraints are the *greater-than-or-equal-to* type. A) up and down B) up C) down D) up or down

B) up

In a capital budgeting problem, if either project 1 or project 2 is selected, then project 5 cannot be selected. Which of the alternatives listed below correctly models this situation? A) x1 + x2 + x5 ≤ 1 B) x1 + x2 + x5 ≥ 1 C) x1 + x5 ≤ 1, x2 + x5 ≤ 1 D) x1 - x5 ≤ 1, x2 - x5 ≤ 1

C) x1 + x5 ≤ 1, x2 + x5 ≤ 1

Which of the following assumptions is not an assumption of the transportation model? A) Shipping costs per unit are constant. B) There is one transportation route between each source and destination. C) There is one transportation mode between each source and destination. D) Actual total supply and actual total demand must be equal.

D) Actual total supply and actual total demand must be equal.

In a transportation problem, only one of the following options *cannot* be correct - please indicate it: A) The objective function of this problem is: Min Z = 3x1A+ 3.5x1B+2.8x1C+ 3.5x2A+ 3x2B+ 4x2C+ 4.5x3A+ 3x3B+ 3.8x3C B) As part of the optimal solution: x1a= 210, x1b= 0, x1c= 110 C) Some of it constraints are: x1A+ x1B+x1C≤ 350; x2A+ x2B+x2C≤ 150 D) As part of the optimal solution: x1a= 138.56, x1b= 0.78, x1c= 80.67

D) As part of the optimal solution: x1a= 138.56, x1b= 0.78, x1c= 80.67

You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, the constraint for the first restriction is : A) S1 + S3 + S7 ≥ 1. B) S1 + S3 + S7 ≤1. C) S1 + S3 + S7 = 2. D) S1 + S3 + S7 ≤ 2.

D) S1 + S3 + S7 ≤ 2.

Compared to blending and product mix problems, transportation problems are unique because: A) they maximize profit B) the constraints are all equality constraints with no "≤" or "≥" constraints C) they contain fewer variables D) the solution values are always integers

D) the solution values are *always* integers

Max Z = 5x1 + 6x2 Subject to: 17x1 + 8x2 ≤ 136 3x1 + 4x2 ≤ 36 x1, x2 ≥ 0 and integer What is the optimal solution? A) x1 = 6, x2 = 4, Z = 54 B) x1 = 3, x2 = 6, Z = 51 C) x1 = 2, x2 = 6, Z = 46 D) x1 = 4, x2 = 6, Z = 56

D) x1 = 4, x2 = 6, Z = 56

1. Contingency or Mutually Exclusive Constraint

Either one or other can be selected, *NOT* both ex. x1 + x2 ≤ 1 ∙ 0 + 1 ≤ 1 (Good) ∙ 1 + 0 ≤ 1 (Good) ∙ 1 + 1 ≤ 1 (Bad)

A conditional constraint specifies the conditions under which variables are integers or real variables.

False

If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a mutually exclusive constraint.

False

If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a mutually exclusive constraint.

False

In an unbalanced transportation problem, if demand exceeds supply, the optimal solution will be infeasible.

False

Rounding non-integer solution values up to the nearest integer value *will* result in an infeasible solution to an integer linear programming problem.

False

The branch and bound solution method cannot be applied to 0-1 integer programming problems.

False

Unconstrained Optimization Model

Includes a nonlinear objective function and NO constraints

Constrained Optimization Model

Includes a nonlinear objective function and one or more nonlinear constraints ex. p<=20

LP Modeling

Linear programming examples: 1) A product mix example ** 2) A diet example 3) An investment example ** 4) A marketing example** 5) A transportation example ** concepts only 6) A blend example 7) A multi-period scheduling example 8) A data envelopment analysis example ** = Pay Attention Review online examples

In a mixed integer model, some solution values for decision variables are integer and others can be non-integer.

True

In a problem involving capital budgeting applications, the 0-1 variables designate the acceptance or rejection of the different projects.

True

In a total integer model, all decision variables have integer solution values.

True

The college dean is deciding among three equally qualified (in their eyes, at least) candidates for his associate dean position. If this situation could be modeled as an integer program, the decision variables would be cast as 0-1 integer variables.

True

The divisibility assumption is violated by integer programming.

True

The production planner for Airbus showed his boss the latest product mix suggestion from their slick new linear programming model: 12.5 model 320s and 17.4 model 340s. The boss looked over his glasses at the production planner and reminded him that they had several half airplanes from last year's production rusting in the parking lot. No one, it seems, is interested in half of an airplane. The production planner whipped out his red pen and crossed out the .5 and .4, turning the new plan into 12 model 320s and 17 model 340s. This production plan is definitely feasible.

True

Break-Even Analysis

Z = vp - Cf - vCv Where: v = sales volume (demand) p = price Cf = fixed cost Cv = variable cost *THERE IS A DEPENDENCY BETWEEN PRICE AND DEMAND*

If a firm's profit is z = -8p2 + 40p +150, then what is the level of price (p) that maximizes profit is: a) $2.00 b) $2.50 c) $2.75 d) $3.00

b) $2.50

Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

In a linear programming formulation of a transportation model, each of the possible combinations of supply and demand locations is a(n) ________.

decision variable

In order to model a "prohibited route" in a transportation or transshipment problem, the cost assigned to the route should be ________.

high

In Max/Min Integer Linear Programming Problems, feasible solutions are ensured when rounding *DOWN* non-integer solutions on constraints that are:

less-than-or-equal-to type

In a ________ linear programming model, some of the solution values for the decision variables are required to assume integer values and others can be integer or noninteger.

mixed integer

If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a ________ constraint.

multiple-choice

In choosing four electives from the dazzling array offered by the Decision Sciences Department next semester, the students that had already taken the management science class were able to craft a model using a ________ constraint.

multiple-choice

If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a ________ constraint.

mutually exclusive

In a ________ problem, items are allocated from sources to destinations at a minimum cost

transportation

In most real-world cases, the supply capacity and demanded amounts result in a(n) ________ transportation model.

unbalanced

In an integer program, if building one facility required the construction of another type of facility, this would be written as: ________.

x1 = x2

What is the point of maximization of profit using this information: z = vp- Cf - vCv v = 1500 - 24.6p Cv = 8, Cf = 10,000

z = (1500 - 24.6p (p)) - Cf - (1500 - 24.6p)(Cv) = 1500 - 24.6p2 -Cf - 1500Cv + 24.6pCv = 1500 - 24.6p2 - 10,000 - 1500(8) + 24.6p(8) = -24.6p^2 +1696.8p - 22,000 Take the Derivative of the function z, and set z = 0: -49.2p + 1696.8 = z -49.2p + 1696.8 = 0 p = 34.39 << Answer Plug p into the original equation: v = 1500 - 24.6(34.39) = 651.6 << Answer