56:171 Fall 2002 Operations Research Quizzes with Solutions

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1 56:7 Fall Operations Research Quizzes with Solutions Instructor: D. L. Bricker University of Iowa Dept. of Mechanical & Industrial Engineering Note: In most cases, each quiz is available in several versions!

2 Section A or B (circle) Name Version A Solutions 56:7 Operations Research Quiz # 6 September 56:7 Operations Research Quiz # Solutions 6 September Consider the following LP: Consider the following LP: Minimize X+ X subject to () X+ X () X+ X 6 () X+ X 6 X & X Minimize X+ X subject to () X+ X () X+ X 6 () X+ X 6 X & X. The feasible region has corner points, namely.. At point F, the slack (or surplus) variable for constraint # is positive. (If more than one such variable is positive, only one is required.). The optimal solution is at point Note: For your convenience, the (X, X ) coordinates of the points labeled above are: Point A B C D E F G H X. 5 6 X Which of the three matrices below (each of which are row-equivalent to A) is the result of a pivot in matrix A? (If more than one answer is correct, only one answer is required.) A=, B, C, D = = = 5. Which method of solving a system of linear equations requires more row operations? a. Gauss elimination b. Gauss-Jordan elimination c. Both require same number. The feasible region has corner points, namely _D, F, & H.. At point F, the slack (or surplus) variable for constraint # is positive. (If more than one such variable is positive, only one is required.). The optimal solution is at point F Note: For your convenience, the (X, X ) coordinates of the points labeled above are: Point A B C D E F G H X. 5 6 X Obj Which of the three matrices below (each of which are row-equivalent to A) is the result of a pivot in matrix A? (If more than one answer is correct, only one answer is required.) B A=, B, C, D = = = _b_5. Which method of solving a system of linear equations requires more row operations? a. Gauss elimination b. Gauss-Jordan elimination c. Both require same number O.R. Quiz # page of O.R. Quiz # page of

3 Section A or B (circle) Name Version B Solutions 56:7 Operations Research Quiz # 6 September 56:7 Operations Research Quiz # Solutions 6 September Consider the following LP: Consider the following LP: Maximize X+ X subject to () X+ X () X+ X 6 () X+ X 6 X & X Maximize X+ X subject to () X+ X () X+ X 6 () X+ X 6 X & X. The feasible region has corner points, namely.. At point F, the slack (or surplus) variable for constraint # is positive. (If more than one such variable is positive, only one is required.). The optimal solution is at point Note: For your convenience, the (X, X ) coordinates of the points labeled above are: Point A B C D E F G H X. 5 6 X Which of the three matrices below (each of which are row-equivalent to A) is the result of a pivot in matrix A? (If more than one answer is correct, only one answer is required.) A=, B, C, D = = = 5. Which method of solving a system of linear equations requires more row operations? a. Gauss elimination b. Gauss-Jordan elimination c. Both require same number. The feasible region has corner points, namely C, D, & F.. At point F, the slack (or surplus) variable for constraint # is positive. (If more than one such variable is positive, only one is required.). The optimal solution is at point D Note: For your convenience, the (X, X ) coordinates of the points labeled above are: Point A B C D E F G H X. 5 6 X Obj Which of the three matrices below (each of which are row-equivalent to A) is the result of a pivot in matrix A? (If more than one answer is correct, only one answer is required.) C A=, B, C, D = = = _b_5. Which method of solving a system of linear equations requires more row operations? a. Gauss elimination b. Gauss-Jordan elimination c. Both require same number O.R. Quiz # page of O.R. Quiz # page of

4 Version C Solutions 56:7 Operations Research Quiz # 6 September 56:7 Operations Research Quiz # Solutions 6 September Consider the following LP: Consider the following LP: Maximize X+ X subject to () X+ X () X+ X 6 () X+ X 6 X & X Maximize X+ X subject to () X+ X () X+ X 6 () X+ X 6 X & X. The feasible region has corner points, namely.. At point C, the slack (or surplus) variable for constraint # is positive. (If more than one such variable is positive, only one is required.). The optimal solution is at point Note: For your convenience, the (X, X ) coordinates of the points labeled above are: Point A B C D E F G H X. 5 6 X Which of the three matrices below (each of which are row-equivalent to A) is the result of a pivot in matrix A? (If more than one answer is correct, only one answer is required.) 5 A=, B, C, D = = = 5. Which method of solving a system of linear equations requires more row operations? a. Gauss elimination b. Gauss-Jordan elimination c. Both require same number. The feasible region has _5_ corner points, namely _B, C, E, F & G. At point C, the slack (or surplus) variable for constraint # is positive. (If more than one such variable is positive, only one is required.). The optimal solution is at point C Note: For your convenience, the (X, X ) coordinates of the points labeled above are: Point A B C D E F G H X. 5 6 X Obj Which of the three matrices below (each of which are row-equivalent to A) is the result of a pivot in matrix A? (If more than one answer is correct, only one answer is required.) D 5 A=, B, C, D = = = _b_5. Which method of solving a system of linear equations requires more row operations? a. Gauss elimination b. Gauss-Jordan elimination c. Both require same number

5 Version A Solution 56:7 Operations Research Quiz # (version A) Solution Fall Part I. For each statement, indicate "+"=true or "o"=false. +. A "pivot" in a nonbasic column of a tableau will make it a basic column. O. If a zero appears on the right-hand-side of row i of an LP tableau, then at the next iteration you cannot pivot in row i. O. A "pivot" in row i of the column for variable Xj will increase the number of basic variables. +. A basic solution of the problem "maximize cx subject to Ax b, x " corresponds to a corner of the feasible region In a basic LP solution, the nonbasic variables equal zero. Part II. Below are several simplex tableaus. Assume that the objective in each case is to be maximized. Classify each tableau by writing to the right of the tableau a letter A through F, according to the descriptions below. Also circle the pivot element when specified. (A) Nonoptimal, nondegenerate tableau with bounded solution. Circle a pivot element which would improve the objective. (B) Nonoptimal, degenerate tableau with bounded solution. Circle an appropriate pivot element. (C) Unique optimum. (D) Optimal tableau, with alternate optimum. Circle a pivot element which would lead to another optimal basic solution. (E) Objective unbounded (above). (F) Tableau with infeasible basic solution. () -z X X X X X 5 X 6 X 7 X 8 RHS max -6 9 B ( correct pivots) () -z X X X X X 5 X 6 X 7 X 8 RHS max D () -z X X X X X 5 X 6 X 7 X 8 RHS max E (variable X ) () -z X X X X X 5 X 6 X 7 X 8 RHS max A ( correct pivots) (5) -z X X X X X 5 X 6 X 7 X 8 RHS max F :7 O.R. Quiz # page of

6 Version B Solution 56:7 Operations Research Quiz # (version B) Solution Fall Part I. For each statement, indicate "+"=true or "o"=false. O. It may happen that an LP problem has (exactly) two optimal solutions. O. If a zero appears on the right-hand-side of row i of an LP tableau, then at the next iteration you must pivot in row i. +. A "pivot" in the simplex method corresponds to a move from one corner point of the feasible region to another. +. In the simplex method, every variable of the LP is either basic or nonbasic If there is a tie in the "minimum-ratio test" of the simplex method, the next basic solution will be degenerate, i.e., one of the basic variables will be zero. Part II. Below are several simplex tableaus. Assume that the objective in each case is to be maximized. Classify each tableau by writing to the right of the tableau a letter A through F, according to the descriptions below. Also circle the pivot element when specified. (A) Optimal tableau, with alternate optimum. Circle a pivot element which would lead to another optimal basic solution. (B) Objective unbounded (above). (C) Nonoptimal, nondegenerate tableau with bounded solution. Circle a pivot element which would improve the objective. (D) Nonoptimal, degenerate tableau with bounded solution. Circle an appropriate pivot element. (E) Unique optimum. (F) Tableau with infeasible basic solution. () -z X X X X X 5 X 6 X 7 X 8 RHS max F () -z X X X X X 5 X 6 X 7 X 8 RHS max -6 9 D () -z X X X X X 5 X 6 X 7 X 8 RHS max A () -z X X X X X 5 X 6 X 7 X 8 RHS max B (X ) (5) -z X X X X X 5 X 6 X 7 X 8 RHS max C :7 O.R. Quiz # page of

7 Version C Solution 56:7 Operations Research Quiz # (version C) Solution Fall Part I. For each statement, indicate "+"=true or "o"=false. O. If a zero appears on the right-hand-side of row i of an LP tableau, then at the next iteration you cannot pivot in row i. +. A basic solution of the problem "maximize cx subject to Ax b, x " corresponds to a corner of the feasible region. +. In a basic LP solution, the nonbasic variables equal zero. O. The "minimum ratio test" is used to select the pivot column in the simplex method for linear programming In the simplex tableau, all rows, including the objective row, are written in the form of equations. Part II. Below are several simplex tableaus. Assume that the objective in each case is to be maximized. Classify each tableau by writing to the right of the tableau a letter A through F, according to the descriptions below. Also circle the pivot element when specified. (A) Nonoptimal, nondegenerate tableau with bounded solution. Circle a pivot element which would improve the objective. (B) Nonoptimal, degenerate tableau with bounded solution. Circle an appropriate pivot element. (C) Unique optimum. (D) Optimal tableau, with alternate optimum. Circle a pivot element which would lead to another optimal basic solution. (E) Objective unbounded (above). (F) Tableau with infeasible basic solution. () -z X X X X X 5 X 6 X 7 X 8 RHS max _E (variable X ) () -z X X X X X 5 X 6 X 7 X 8 RHS max _A ( correct pivots) () -z X X X X X 5 X 6 X 7 X 8 RHS max -6 9 _B ( correct pivots) () -z X X X X X 5 X 6 X 7 X 8 RHS max _F (5) -z X X X X X 5 X 6 X 7 X 8 RHS max _D :7 O.R. Quiz# page of

8 Solutions 56:7 Operations Research Quiz # Solutoins -- September Part I. For each statement, indicate "+"=true or "o"=false. o a. When you enter an LP formulation into LINDO, you must first convert all inequalities to equations. o b. Unlike the ordinary simplex method, the "Revised Simplex Method" never requires the use of artificial variables. + c. Whether an LP is a minimization or a maximization problem, the first phase of the twophase method is exactly the same. o d. At the beginning of the first phase of the two-phase simplex method, the phase-one objective function will have the value. + e. At the end of the first phase of the two-phase simplex method, the phase-one objective function must be zero if the LP is feasible. o f. If a zero appears on the right-hand-side of row i of an LP tableau, then at the next iteration you must pivot in row i. + g. If an LP model has constraints of the form Ax b, x, and b is nonnegative, then there is no need for artificial variables. o h. If a zero appears on the right-hand-side of row i of an LP tableau, then at the next iteration you cannot pivot in row i. o i. Every variable in the primal problem has a corresponding dual variable. o j. The primal LP is a minimization problem, whereas the dual problem is a maximization problem. + k. If the slack or surplus variable in a constraint is positive, then the corresponding dual variable must be zero. + l. If the right-hand-side of constraint i in the LP problem Minimize cx st Ax b, x increases, then the optimal value must either decrease or remain unchanged. o m. If the right-hand-side of constraint i in the LP problem Maximize cx st A b, x increases, then the optimal value must either decrease or remain unchanged. o n. The revised simplex method usually requires fewer iterations than the ordinary simplex method. + o. The simplex multipliers at the termination of the revised simplex method are always feasible in the dual LP of the problem being solved. + p. In the two-phase method, the first phase finds a basic feasible solution to the LP being solved, while the second phase finds the optimal solution. + q. The original objective function is ignored during phase one of the two-phase method. + r. If a zero appears in row i of the column of substitution rates in the pivot column, then then row i cannot be the pivot row. Part II. Sensitivity analysis using LINDO. Ken and Larry, Inc., supplies its ice cream parlors with three flavors of ice cream: chocolate, vanilla, and banana. Because of extremely hot weather and a high demand for its products, the company has run short of its supply of ingredients: milk, sugar, & cream. Hence, they will not be able to fill all the orders received from their retail outlets, the ice cream parlors. Owing to these circumstances, the company has decided to choose the amount of each product to produce that will maximize total profit, given the constraints on supply of the basic ingredients. The chocolate, vanilla, and banana flavors generate, respectively, $., $.9, and $.95 per profit per gallon sold. The company has only gallons of milk, 5 pounds of sugar, and 6 gallons of cream left in its inventory. The LP formulation for this problem has variables C, V, and B representing gallons of chocolate, vanilla, and banana ice cream produced, respectively. 56:7 O.R. Quiz# Solutions page of

9 Solutions MAXIMIZE C+.9V+.95B ST.5C +.5V +.B <=! milk resource.5c +.V +.B <= 5! sugar resource.c +.5V +.B <= 6! cream resource END OBJECTIVE FUNCTION VALUE ).5 VARIABLE VALUE REDUCED COST C..75 V.. B 75.. ROW SLACK OR SURPLUS DUAL PRICES ).. )..875 ).. RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE C..75 INFINITY V B RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE. INFINITY True/False (+ or O): +. If the profit per gallon of chocolate increases to $., the basis and the values of the basic variables will be unchanged. o. If the profit per gallon of vanilla drops to $.88, the basis and the values of the basic variables will be unchanged. Multiple choice: (NSI = not sufficient information ) _d_. If the amount of cream available were to increase to 65 gallons, the increase in profit will be (choose nearest value): a. $. b. $.5 c. $ d. $5 e. $ f. NSI _a_. If the amount of milk available were to increase to 5 gallons, the increase in profit will be (choose nearest value): a. $. b. $.5 c. $ d. $5 e. $ f. NSI _e_ 5. If the profit per gallon of banana ice cream were to drop to $.9 per gallon, the loss in total profit would be (choose nearest value): a. $. b. $.5 c. $ d. $5 e. $ ($5) f. NSI 56:7 O.R. Quiz# Solutions page of

10 Solutions 56:7 Operations Research Quiz # -- 7 September "A manufacturer produces two types of plastic cladding. These have the trade names Ankalor and Beslite. One yard of Ankalor requires 8 lb of polyamine,.5 lb of diurethane and lb of monomer. A yard of Beslite needs lb of polyamine, lb of diurethane, and lb of monomer. The company has in stock 8, lb of polyamine,, lb of diurethane, and, lb of monomer. Both plastics can be produced by alternate parameter settings of the production plant, which is able to produce cladding at the rate of yards per hour. A total of 75 production plant hours are available for the next planning period. The contribution to profit on Ankalor is $/yard and on Beslite is $/yard. The company has a contract to deliver at least, yards of Ankalor. What production plan should be implemented in order to maximize the contribution to the firm's profit from this product division." Definition of variables: A = Number of yards of Ankalor produced LP model: B = Number of yards of Beslite produced ) Maximize A + B subject to ) 8 A + B 8, (lbs. Polyamine available) ).5 A + B, (lbs. Diurethane available) ) A + B, (lbs. Monomer available) 5) A + B 9, (lbs. Plant capacity) 6) A, (Contract) A, B The LINDO solution is: OBJECTIVE FUNCTION VALUE ). VARIABLE VALUE REDUCED COST A.. B 56.. ROW SLACK OR SURPLUS DUAL PRICES ).. ) 69.. ) 6.. 5).. 6). -6. RANGES IN WHCH THE BASIS IS UNCHANGED OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE A. 6. INFINITY B. INFINITY 7.5 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE INFINITY 69.. INFINITY INFINITY :7 O.R. Quiz # Solutions Fall page of

11 Solutions THE TABLEAU ROW (BASIS) A B SLK SLK SLK SLK 5 SLK 6 RHS ART E+6 B SLK SLK SLK A Consult the LINDO output above to answer the following questions. If there is not sufficient information in the LINDO output, answer "NSI". _b_. Suppose that the company can purchase pounds of additional polyamine for $.5 per pound. Should they make the purchase? a. yes b. no c. NSI (since the dual variable for row [] is only $./lb.) _a_. Regardless of your answer in (), suppose that they do purchase pounds of additional polyamine. Increasing the quantity of polyamine used in the model above is equivalent to a. decreasing the slack in row by d. decreasing surplus in row by b. increasing the surplus in row by e. none of the above c. increasing the slack in row by f. NSI (since 8A+B+SLK = 8 & 8A+B=8 SLK =.) _c_. If the company purchases pounds of additional polyamine, what is the total amount of Ankelor that they should deliver? (Choose nearest value!) a. 8 yards c. yards unchanged! e. yards b. 9 yards d. yards f. NSI (substitution rate of SLK for A is, so A is unchanged as SLK decreases, up to ALLOWABLE INCREASE for RHS of row#, i.e.,.) _d_. If the company purchases pounds of additional polyamine, what is the total amount of Beslite that they should deliver? (Choose nearest value!) a. 55 yards c. 57 yards e. 59 yards b. 56 yards d. 58 yards f. NSI (substitution rate of SLK for B is., so B increases by. for each pound decrease in SLK) _c_5. How will the decision to purchase pounds of additional polyamine change the quantity of diurethane used during the next planning period? a. increase by pounds c. increase by pounds e. none of the above b. decrease by pounds d. decrease by pounds f. NSI (substitution rate of SLK for SLK is -., so SLK decreases by.lb for each pound decrease in SLK.5A+B (i.e., amount of diurethane used) increases by pounds.) _b_6. If the profit contribution from Beslite were to decrease from $ to $/yard, will the optimal values of A &/or B change? a. yes b. no c. NSI (Since decrease of $7 is less than ALLOWABLE DECREASE which is $7.5.) _a_7. If the profit contribution from Ankelor were to increase from $ to $7/yard, will the optimal values of A &/or B change? a. yes b. no c. NSI (Since increase of $7 is greater than ALLOWABLE INCREASE which is $6.) _a_8. Suppose that the company could deliver yards less than the contracted amount of Ankalor if they were to pay a penalty of $5/yard shortage. Should they do so? a. yes b. no c. NSI (Since dual variable for row [6] is $6., profit would decrease by $6 for every pound increase, or decrease by $6 for every pound decrease in requirements.) _b_9. Is the optimal solution of this LP degenerate? a. yes b. no c. NSI (No zero appears on RHS of optimal tableau.) _b_. Are there multiple optimal solutions of this LP? a. yes b. no c. NSI (No zero appears in objective row () of any nonbasic column of tableau.) 56:7 O.R. Quiz # Solutions Fall page of

12 Solution--version A Part. Transportation Simplex Method. 56:7 Operations Research Quiz #5 Version A Solution -- Fall Consider the feasible solution of the transportation problem below: dstn source 5 Supply A 8 9 & basic 7 9 B nonbasic 7 7 C nonbasic 5 D nonbasic 5 7 E nonbasic Demand= _a_. Which additional variable (=) of those below might be made a basic variable in order to complete the (degenerate) basis? a. XA c. XB e. XE b. XD d. XC f. None of above _d_. If we choose to assign dual variable U D =5, what must be the value of dual variable V? a. c. 6 e. b. d. 8 f. None of above _e_. If, in addition to U D =5, we determine that V =, then the reduced cost of X D is a. c. +5 e. +9 b. d. 8 f. None of above _a_. If we initially assign U D = (rather than U D =5) and then compute the remaining dual variables, a different value of the reduced cost of X D will be obtained, although its sign will remain the same. a. True b. False c. Cannot be determined _e_5. Suppose XE enters the basis (ignoring whether it would improve the solution or not). Which variable must leave the basis? a. XA c. XC e. XE b. XC d. XD f. None of above 56:7 O.R. Quiz #5 Solution Fall page of

13 Solution--version A Part : Assignment Problem Consider the problem of assigning machines (rows) to jobs (columns), with cost matrix By drawing lines in rows &5 and in columns &, all the nine zeroes are covered. _c_. After the next step of the Hungarian method, the number of zeroes in the cost matrix will be a. 7 c. 9 e. b. 8 d. f. None of the above Consider the reduced assignment problem shown below: \ _a_. A zero-cost solution of the problem with this cost matrix must have a. X= c. X= e. X5= b. X= d. X5= f. None of the above Part. True(+) or False(o)? _+_. The transportation problem is a special case of a linear programming problem. _o_. The Hungarian algorithm can be used to solve a transportation problem. _+_. Every basic feasible solution of an assignment problem must be degenerate. _o_. If the current basic solution of a transportation problem is degenerate, no improvement of the objective function will occur in the next iteration of the (transportation) simplex method. _o_ 5. The Hungarian algorithm is the simplex method specialized to the assignment problem. _+_6. At each iteration of the Hungarian method, the original cost matrix is replaced with a new cost matrix having the same optimal assignment. 56:7 O.R. Quiz #5 Solution Fall page of

14 Solution -- Version B Part. Transportation Simplex Method. 56:7 Operations Research Quiz #5 Version B -- Fall Consider the feasible solution of the transportation problem below: dstn source 5 Supply A 8 9 & basic 7 9 B nonbasic 7 7 C nonbasic 5 D nonbasic 5 7 E nonbasic Demand= _c_. Which additional variable (=) of those below might be made a basic variable in order to complete the (degenerate) basis? a. XB c. XA e. XE b. XC d. XD f. None of above _d_. If we choose to assign dual variable U D =5, what must be the value of dual variable V? a. c. 6 e. b. d. 8 f. None of above _e_. If, in addition to U D =5, we determine that V =, then the reduced cost of X D is a. c. +5 e. +9 b. d. 8 f. None of above _a_. If we initially assign U D = (rather than U D =5) and then compute the remaining dual variables, a different value of the reduced cost of X D will be obtained, although its sign will remain the same. a. True b. False c. Cannot be determined _e_5. Suppose X E enters the basis (ignoring whether it would improve the solution or not). Which variable must leave the basis? a. XC c. XA e. XE b. XC d. XD f. None of above 56:7 O.R. Quiz #5 Solution Fall page of

15 Solution -- Version B Part : Assignment Problem Consider the problem of assigning machines (rows) to jobs (columns), with cost matrix By drawing lines in rows &5 and in columns &, all the nine zeroes are covered. _c_. After the next step of the Hungarian method, the number of zeroes in the cost matrix will be a. 7 c. 9 e. b. 8 d. f. None of the above Consider the reduced assignment problem shown below: \ _a_. A zero-cost solution of the problem with this cost matrix must have a. X= c. X= e. X5= b. X= d. X5= f. None of the above Part. True(+) or False(o)? _+_. The assignment problem is a special case of an transportation problem. _o_. The number of basic variables in a n n assignment problem is n. _o_. The dual variables of the transportation problem are uniquely determined at each iteration of the simplex method. _+_. The transportation simplex method can be applied to solution of an assignment problem. _o_ 5. The optimal dual variables of the transportation problem obtained at the final iteration must be nonnegative. _o_ 6. A balanced transportation problem has an equal number of sources and destinations. 56:7 O.R. Quiz #5 Solution Fall page of

16 Part. Transportation Simplex Method. 56:7 Operations Research Quiz #5 Version C -- Fall Consider the feasible solution of the transportation problem below: dstn source 5 Supply A 8 9 & basic 7 9 B nonbasic 7 7 C nonbasic 5 D nonbasic 5 7 E nonbasic Demand= _d_. Which additional variable (=) of those below might be made a basic variable in order to complete the (degenerate) basis? a. XD b. XC c. XE d XA e. XB f. None of above _e_. If we choose to assign dual variable U D =5, what must be the value of dual variable V? a. b. 6 c. d. e. 8 f. None of above _c_. If, in addition to U D =5, we determine that V =, then the reduced cost of XD is a. b. +5 c. +9 d. e. 8 f. None of above _a_. If we initially assign U D = (rather than U D =5) and then compute the remaining dual variables, a different value of the reduced cost of X D will be obtained, although its sign will remain the same. a. True b. False c. Cannot be determined _c_5. Suppose X E enters the basis (ignoring whether it would improve the solution or not). Which variable must leave the basis? a. XA b. XC c. XE d. XC e. XD f. None of above

17 Part : Assignment Problem Consider the problem of assigning machines (rows) to jobs (columns), with cost matrix By drawing lines in rows &5 and in columns &, all the eight zeroes are covered. _b_. After the next step of the Hungarian method, the number of zeroes in the cost matrix will be a. 7 b. 9 c. d. 8 e. f. None of the above Consider the reduced assignment problem shown below: \ _a_. A zero-cost solution of the problem with this cost matrix must have a. X= b. X= c. X5= d. X= e. X5= f. None of the above Part. True(+) or False(o)? _o_. The Hungarian algorithm can be used to solve a transportation problem. _o_. The number of basic variables in a n n assignment problem is n. _o_. If the current basic solution of a transportation problem is degenerate, no improvement of the objective function will occur in the next iteration of the (transportation) simplex method. _o_. The Hungarian algorithm is the simplex method specialized to the assignment problem. _+_ 5. At each iteration of the Hungarian method, the original cost matrix is replaced with a new cost matrix having the same optimal assignment. _+_ 6. A balanced transportation problem has an equal number of supply and demand.

18 SOLUTIONS 56:7 Operations Research Quiz #6 Solutions Fall Part I: True(+) or False(o)? _+. The critical path in a project network is the longest path from a specified source node (beginning of project) to a specified destination node (end of project). _o. There is at most one critical path in a project network. _o. The latest times of the events in a project schedule must be computed before the earliest times of those events. _+. In PERT, the total completion time of the project is assumed to have a normal distribution. _+ 5. All tasks on the critical path of a project schedule have their latest finish time equal to their earliest finish time. _+ 6. In the LP formulation of the project scheduling problem, the constraints include YB YA da if activity A must precede activity B, where da = duration of activity A. _o 7. In CPM, the "forward pass" is used to determine the latest time (LT) for each event (node). _+ 8. The A-O-N project network does not require any "dummy" activities (except for the "begin" and "end" activities). _o 9. The PERT method assumes that the completion time of the project has a beta distribution. _o. A "dummy" activity in an A-O-A project network always has duration zero and cannot be a critical activity. _o(?).the Hungarian algorithm can be used to solve a transportation problem. (If the supplies & demands are integers, one could, however, replace each source i with Si rows and each destination j with Dj columns, and apply the Hungarian algorithm.) _o.the number of basic variables in a n n assignment problem is n. _+. At each iteration of the Hungarian method, the original cost matrix is replaced with a new cost matrix having the same optimal assignment. _+. Every basic feasible solution of an assignment problem must be degenerate. _+ 5. In order to apply the Hungarian algorithm, the assignment cost matrix must be square. _+ 6. The transportation problem is a special case of a linear programming problem. _o 7. The PERT method assumes that the completion time of the project has a beta distribution. _o 8. If the current basic solution of a transportation problem is degenerate, no improvement of the objective function will occur in the next iteration of the (transportation) simplex method. _o 9. The PERT method assumes that the duration of each activity has a normal distribution. _+. A "dummy" activity in an A-O-A project network always has duration zero. _o. At each iteration of the Hungarian method, the number of zeroes in the cost matrix will increase. _+. If at some iteration of the Hungarian method, the zeroes of a n n assignment cost matrix cannot be covered with fewer than n lines, this cost matrix must have a zero-cost assignment. _+. Every A-O-A project network has at least one critical path. 56:7 O.R. Quiz #5 Solutions Fall page of

19 SOLUTIONS Part II: Project Scheduling. Consider the project with the A-O-A (activity-on-arrow) network below. The activity durations are given on the arrows. The Earliest (event) Times (ET) and Latest (event) Times (LT) for each node are written in the box beside each node. Note: There are three different versions, each having different durations of activity A: a D() 7 7 A() F() G() H() 6 b B() C() I() 8 8 ET LT E() c 7 _c d d f d_. Complete the labeling of the nodes on the network. Note: The labeling above is one of several such that an arrow always goes from lower-numbered node to a higher-numbered node.. The number of activities (i.e., tasks), not including "dummies", which are required to complete this project is a. six c. eight e. ten g. b. seven d. nine f. eleven h. NOTA. The latest event time (LT) indicated by a in the network above is: a. one c. three e. five g. seven b. two d. four f. six h. NOTA. The latest event time (LT) indicated by b in the network above is: a. one c. three e. five g. seven b. two d. four f. six h. NOTA 5. The earliest event time (ET) indicated by c in the network above is: a. one c. three e. five g. seven b. two d. four f. six h. NOTA 6. The slack ("total float") for activity D is: solution: depends upon ET() which depends upon duration of A. For network shown above, answer is. a. zero c. two e. four g. six b. one d. three f. five h. NOTA 7. Which activities are critical? (circle: A B C D E F G H I ) Suppose that the non-zero durations are random, with each value in the above network being the expected values and each standard deviation equal to.. Then... _c_ 8. The expected earliest completion time for the project is a. six c. eight e. ten g. twelve b. seven d. nine f. eleven h. NOTA _e_ 9. The variance σ of the earliest completion time for the project is a. zero c. two e. four g. six b. one d. three f. five h. NOTA Note: variance of sum of durations of four activities along critical path is sum of variances of those activities. 56:7 O.R. Quiz #5 Solutions Fall page of

20 56:7 Operations Research Quiz #7 Version A Solution Fall (Version A) SOLUTION Part I: Consider a decision problem whose payoffs are given by the following payoff table: Decision A B one 8 5 two 5 three 6 7 Prior Probability..6 _c_. Which alternative should be chosen under the maximin payoff criterion? a. one b. two c. three d. NOTA _a_. Which alternative should be chosen under the maximax payoff criterion? a. one b. two c. three d. NOTA _c_. Which alternative should be chosen under the maximum expected payoff criterion? a. one b. two c. three d. NOTA _c_. What will be the entry in the regret table for decision three & State-of-Nature A? a. zero b. c. d. e. f. NOTA Suppose that you perform an experiment to predict the state of nature (A or B) above. The experiment has two possible outcomes which we label as positive and negative. If the state of nature is A, there is a 6% probability that the outcome will be positive, whereas if the state of nature is B, there is a % probability that the outcome will be positive. According to Bayes rule, { } P A positive P = { α β} P{ γ} P{} δ In this equation, _c_ 5. α = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _a_ 6. β = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _a_ 7. γ = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _c_ 8. δ = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _c_ 9. Suppose that the outcome of the experiment is positive. Then the probability that the state of nature is A is revised to (choose nearest value): a..5 b..6 c..7 / d..8 e..9 f. NOTA 56:7 O.R. Quiz #7 Solution Fall page of

21 (Version A) SOLUTION Part II. Consider the decision tree above. Fold back the branches and write the values of each node in the table below: Node # # # # Value _b_ 5. What is the optimal decision at node #? a. A b. A 56:7 O.R. Quiz #7 Solution Fall page of

22 56:7 Operations Research Quiz #7 Version B Solution Fall (Version B) SOLUTION Part I: Consider a decision problem whose payoffs are given by the following payoff table: Decision A B one 5 two 5 three 7 Prior Probability..6 _b_. Which alternative should be chosen under the maximin payoff criterion? a. one b. two c. three d. NOTA _c_. Which alternative should be chosen under the maximax payoff criterion? a. one b. two c. three d. NOTA _c_. Which alternative should be chosen under the maximum expected payoff criterion? a. one b. two c. three d. NOTA _c_. What will be the entry in the regret table for decision three & State-of-Nature A? a. zero b. c. d. e. f. NOTA Suppose that you perform an experiment to predict the state of nature (A or B) above. The experiment has two possible outcomes which we label as positive and negative. If the state of nature is A, there is a 6% probability that the outcome will be positive, whereas if the state of nature is B, there is a % probability that the outcome will be positive. According to Bayes rule, { } P A negative P = { α β} P{ γ} P{} δ In this equation, _d_ 5. α = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _a_ 6. β = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _a_ 7. γ = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _d_ 8. δ = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _d_ 9. Suppose that the outcome of the experiment is negative. Then the probability that the state of nature is A is revised to (choose nearest value): a.. b..5 c.. d..5 e.. f. NOTA 56:7 O.R. Quiz #7 Solution Fall page of

23 (Version B) SOLUTION Part II. Consider the decision tree above. Fold back the branches and write the values of each node in the table below: Node # # # # $ $ $5 $5 Value _a_ 5. What is the optimal decision at node #? a. A b. A 56:7 O.R. Quiz #7 Solution Fall page of

24 (Version C) SOLUTION 56:7 Operations Research Quiz #7 Version C Solution Fall Part I: Consider a decision problem whose payoffs are given by the following payoff table: Decision A B one 7 5 two 8 three 6 Prior Probability..6 _a_. Which alternative should be chosen under the maximin payoff criterion? a. one b. two c. three d. NOTA _b_. Which alternative should be chosen under the maximax payoff criterion? a. one b. two c. three d. NOTA _b_. Which alternative should be chosen under the maximum expected payoff criterion? a. one b. two c. three d. NOTA _b_. What will be the entry in the regret table for decision three & State-of-Nature A? a. zero b. c. d. e. f. NOTA Suppose that you perform an experiment to predict the state of nature (A or B) above. The experiment has two possible outcomes which we label as positive and negative. If the state of nature is A, there is a 6% probability that the outcome will be positive, whereas if the state of nature is B, there is a % probability that the outcome will be positive. According to Bayes rule, { } P B positive P = { α β} P{ γ} P{} δ In this equation, _c_ 5. α = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _b_ 6. β = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _b_ 7. γ = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _c_ 8. δ = a. state of nature is A c. experimental outcome is positive e. NOTA b. state of nature is B d. experimental outcome is negative _b_ 9. Suppose that the outcome of the experiment is positive. Then the probability that the state of nature is B is revised to (choose nearest value): a.. b.. / c.. d..5 e..9 f. NOTA 56:7 O.R. Quiz #7Solution Fall page of

25 (Version C) SOLUTION Part II. $5 A (5%) one B (5%) $ A (6%) $ two three B (%) $ four $5 Consider the decision tree above. Fold back the branches and write the values of each node in the table below: Node # # # # $ $ $6 $6 Value _a_ 5. What is the optimal decision at node #? a. A b. A 56:7 O.R. Quiz #7Solution Fall page of

26 Version A SOLUTION 56:7 Operations Research Quiz #8 Solution Fall Suppose that you are in the position of having to buy a used car, and you have narrowed down your choices to two possible models: one car (A) a private sale and the other (B) is from a dealer. You must now choose between them. The cars are similar, and the only criterion is to minimize expected cost. The dealer car is more expensive, but it comes with a one-year warranty which would cover all costs of repairs. You decide that, if car A will last for year, you can sell it again and recover a large part of your investment. If it falls apart, it will not be worth fixing. After test driving both cars and checking for obvious flaws, you make the following evaluation of probable resale value: Car Purchase price Probability of lasting one year Estimated resale price A: Private $8.6 $6 B: Dealer $. $6 Note that car B, because of the warranty, is guaranteed to last the year and bring full resale price! survives.6 $6 _a. Which car should you buy?. a. Car A b. Car B _d. What is the expected cost for the year, after resale of the car? This is the expected value without Car B information (about the future), denoted EVWOI. $ (choose nearest value) a. $ b. $ c. $ d. $ ( $) e. $5 survives Calculation of Expected Value of Perfect Information Suppose that a seer of the future will give you perfect information before you purchase the car-- namely, whether car A will survive the year. The outcome of consulting this seer is the initial (left-most) event node to the right. _c. What is the expected value with perfect information (denoted EVWPI), i.e., the Car A will fail. Car B Car A fails $ $8 $6 $6 Car A fails $ $8 expected value at node? Car A (nearest value) $ will survive a. $ b. $ c. $( $) d. $ d. $.6 survives _a. What is the expected value of perfect information (EVPI)? (nearest value) a. $ (= $) b. $ c. $ d. $ e. $5 $6 $6 Car A fails $ $8 Car B $ $6 56:7 O.R. Quiz #8 Version A Fall page of

27 Version A SOLUTION For $5, you may take car A to an independent mechanic, who will do a complete inspection and offer you an opinion as to whether the car will last year. For various subjective reasons, you assign the following probabilities to the accuracy of the mechanic s opinion: Given: Mechanic says Yes Mechanic says No A car that will survive year 9% % A car that will fail in next year % 7% That is, the mechanic is 9% likely to correctly identify a car that will survive the year, but only 7% likely to correctly identify a car that will fail. Let AS and AF be the states of nature, namely car A Survives and car A Fails during the next year, respectively. Let PR and NR be the outcomes of the mechanic s inspection, namely Positive Report and Negative Report, respectively. Bayes' Rule states that if Si are the states of nature and Oj are the outcomes of an experiment, P{ Oj Si} P{ Si} P{ Si Oj} = P O { j} { j} = { j i} { i} where P O P O S P S The probability that the mechanic will give a postive report, i.e., P{ PR} is P{ PR} P{ PR AS} P{ AS} P{ PR AF} P{ AF} i = + = (.9)(.6) + (.)(.) =.66 _e 5. According to Bayes theorem, the probability that car A will survive, given that the mechanic gives a positive report, i.e., P{ AS PR }, is (choose nearest value): a..6 b..65 c..7 d..75 e..8 (.88) f..85 g..9 h..95 The decision tree on the next page includes your decision as to whether or not to hire the mechanic. 6. Insert P{AS PR}, i.e., the probability that car A survives if the mechanic gives a positive report, and P{AF PR} on the appropriate branches of the tree. 56:7 O.R. Quiz #8 Version A Fall page of

28 Version A SOLUTION 7. "Fold back" nodes through 8, and write the value of each node below: Node Value Node Value Node Value $ $9 7 $ $7 5 $5 8 $6 $8 6 $8 8. Should you hire the mechanic? Circle: Yes No e_ 9. The expected value of the mechanic s opinion is (Choose nearest value): a. $ b. $5 c. $ d. $5 e. $6 ($66) f. $75 g. $9 h. $5 56:7 O.R. Quiz #8 Version A Fall page of

29 Version B SOLUTION 56:7 Operations Research Quiz #8 November Suppose that you are in the position of having to buy a used car, and you have narrowed down your choices to two possible models: one car (A) a private sale and the other (B) is from a dealer. You must now choose between them. The cars are similar, and the only criterion is to minimize expected cost. The dealer car is more expensive, but it comes with a one-year warranty which would cover all costs of repairs. You decide that, if car A will last for year, you can sell it again and recover a large part of your investment. If it falls apart, it will not be worth fixing. After test driving both cars and checking for obvious flaws, you make the following evaluation of probable resale value: Car Purchase price Probability of lasting one year Estimated resale price A: Private $9.5 $6 B: Dealer $. $6 Note that car B, because of the warranty, is guaranteed to last the year and bring full resale price! _b. Which car should you buy? a. Car A b. Car B _e. What is the expected cost for the year, after resale of the car? This is the expected value without information (about the future), denoted EVWOI. (choose nearest value) a. $ b. $ c. $ d. $ e. $5 Calculation of Expected Value of Perfect Information Suppose that a seer of the future will give you perfect information before you purchase the car-- namely, whether car A will survive the year. The outcome of consulting this seer is the initial event node to the right. _d. What is the expected value with perfect information (denoted EVWPI)? (nearest value) a. $ b. $ c. $ d. $ _a. What is the expected value of perfect information (EVPI)? a. $ b. $ c. $ d. $ e. $5 Car A will survive Car A will fail.5.5 Car B $ survives $6 Car A fails $ $9 survives $6 $6 Car A fails $ $9 Car B $ $6 56:7 O.R. Quiz #8 Version B Fall page of

30 Version B SOLUTION For $5, you may take car A to an independent mechanic, who will do a complete inspection and offer you an opinion as to whether the car will last year. For various subjective reasons, you assign the following probabilities to the accuracy of the mechanic s opinion: Given: Mechanic says Yes Mechanic says No A car that will survive year 9% % A car that will fail in next year % 7% That is, the mechanic is 9% likely to correctly identify a car that will survive the year, but only 7% likely to correctly identify a car that will fail. Let AS and AF be the states of nature, namely car A Survives and car A Fails during the next year, respectively. Let PR and NR be the outcomes of the mechanic s inspection, namely Positive Report and Negative Report, respectively. Bayes' Rule states that if Si are the states of nature and Oj are the outcomes of an experiment, P{ Oj Si} P{ Si} P{ Si Oj} = P O { j} { j} = { j i} { i} where P O P O S P S The probability that the mechanic will give a postive report, i.e., P{ PR} is P{ PR} P{ PR AS} P{ AS} P{ PR AF} P{ AF} i = + = (.9)(.5) + (.)(.5) =.6 _d 5. According to Bayes theorem, the probability that car A will survive, given that the mechanic gives a positive report, i.e., P{ AS PR }, is (choose nearest value): a..6 b..65 c..7 d..75 e..8 f..85 g The decision tree below includes your decision as to whether or not to hire the mechanic. Insert P{AS PR} and P{AF PR} on the appropriate branches. 56:7 O.R. Quiz #8 Version B Fall page of

31 Version B SOLUTION 7. "Fold back" nodes through 8, and write the value of each node below: Node Value Node Value Node Value $5 $5 7 $5 $7 5 $5 8 $ $5 6 $75 8. Should you hire the mechanic? Circle: Yes No 9. The expected value of the mechanic s opinion is (Choose nearest value): a. $ b. $5 c. $ d. $5 e. $6 f. $75 g. $9 h. $5 56:7 O.R. Quiz #8 Version B Fall page of

32 Version A SOLUTION 56:7 Operations Research Quiz #9 November 8, Markov Chains. Consider the discrete-time Markov chain diagrammed below: A E row sum P ( n) ( n) n f f _b_. Which is the matrix Q (used in computation of E)? a... b...5 c d. e. f. None of the above (NOTA) _a_. Which is the matrix R (used in computation of A)? a... b...5 c d. e. f. NOTA _e_. If the system begins in state #, what is the probability that it is absorbed into state #? (Choose nearest value) a. % or less b. 5% c. % d. 5% e. 5% f. 55% g. 6% h. 65% or more _c_. If the system begins in state #, what is the expected number of stages (including the initial stage) that the system exists before it is absorbed into one of the two absorbing states? (Choose nearest value) a. b. c. d. e. 5 f. 6 g. 7 h. 8 or more c_ 5. For a discrete-time Markov chain, let P be the matrix of transition probabilities. The sum of each... a. column is c. row is b. column is d. row is e. NOTA 56:7 O.R. Quiz#9 Fall page of

33 Version A SOLUTION a_ 6. An absorbing state of a Markov chain is one in which the probability of a. moving out of that state is zero c. moving out of that state is one. b. moving into that state is one. d. moving into that state is zero e. NOTA g_ 7. The minimal closed set(s) in the above Markov chain = a. {,,,} b. {,} c. {.} d. {,,,} & {,} e. {} & {} f. {,} & {,} g. {} & {} h. NOTA _h 8. The probability that the system reaches an absorbing state, beginning in state, is (choose nearest value): a.. b.. c..5 d..6 e..7 f..8 g..9 h.. b_ 9. The recurrent state(s) in the above Markov chain = a. & b. & c.,,, & d. NOTA a_. The transient state(s) in the above Markov chain = a. & b. & c.,,, & d. NOTA e_. The quantity f () is a. the probability that the system, beginning in state, is in state at stage n b. the probability that the system first visits state before state. c. the expected number of visits to state during the first stages, beginning in state d. the stage in which the system, beginning in state, visits state for the fourth time e. the probability that the system, beginning in state, first reaches state in stage f. NOTA _b. From which state is the system more likely to eventually reach state? a. b. c. equally likely d. NOTA _g_. What is the probability that the system is absorbed into state, starting in state, if the first transition is to state? (choose nearest value) a. % or less b. 5% c. % d. 5% e. 5% f. 55% g. 6% h. 65% or more g_. The probability that the system, starting in state, is in state after stages? (choose nearest value) a. 5% or less b. % c. 5% d. % e. 5% f. % g. 5% h. % or more c_ 5. The probability that the system, starting in state, is in state after stages? (choose nearest value) a. 5% or less b. % c. 5% d. % e. 5% f. % g. 5% h. % or more a.. b.. c.. d.. e..5 f..6 g..7 h..8 d_ 6. In general, the steadystate probability distribution π (if it exists) must satisfy a. Pπ= b. Pπ= c. πp= d. πp=π e. Pπ=π f. πp= g. NOTA 56:7 O.R. Quiz#9 Fall page of

34 Version B--SOLUTIONS 56:7 Operations Research Quiz #9 November 8, Markov Chains. Consider the discrete-time Markov chain diagrammed below: A E row sum ( n) ( n) n f f P _a_. Which is the matrix R (used in computation of A)? a... b...5 c d. e. f. None of the above (NOTA) _b_. Which is the matrix Q (used in computation of E)? a... b...5 c d. e. f. None of the above (NOTA) _d_. If the system begins in state #, what is the probability that it is absorbed into state #? (Choose nearest value) a. % or less b. 5% c. % d. 5% e. 5% f. 55% g. 6% h. 65% or more _c_. If the system begins in state #, what is the expected number of stages (including the initial stage) that the system exists before it is absorbed into one of the two absorbing states? (Choose nearest value) a. b. c. d. e. 5 f. 6 g. 7 h. 8 or more _b_ 5. For a discrete-time Markov chain, let P be the matrix of transition probabilities. The sum of each... a. column is b. row is c. column is d. row is e. NOTA 56:7 O.R. Quiz#9 Fall page of