UNIVERSITY of LIMERICK

Size: px

Start display at page:

Download "UNIVERSITY of LIMERICK"

Job Tyrone Long
5 years ago
Views:

1 UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Department of Mathematics & Statistics Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS4303 SEMESTER: Spring 2018 MODULE TITLE: Operations Research 1 LECTURER: Dr. J. Kinsella DURATION OF EXAMINATION: 2 1/2 hours GRADING SCHEME: EXTERNAL EXAMINER: Prof. J. King INSTRUCTIONS TO CANDIDATES: Answer four of the five questions correctly for full marks 70%. For convenience, each question is assigned 25 marks. Ruled paper for recording transportation tableaux is appended to this Examination paper. Your examination paper must be returned with your script.

2 1 (a) A restaurant is open seven days per week. Based on experience, the number of staff needed on a particular day is as follows: Day Mon Tue Wed Thu Fri Sat Sun Number Symbol n 1 n 2 n 3 n 4 n 5 n 6 n 7 Every worker works five consecutive days and then takes two days off, repeating this pattern indefinitely. Formulate a Linear Program to minimize the number of workers that staff the restaurant. Hint: A worker who starts a shift on (say) Tuesday (Day 2) is only available for work on Days 2 6. (i) Make a choice of intermediate variables, i.e. variables that are useful when tracking the day-to-day variation of the problem. 1 (ii) Make a choice of decision variables for the problem. 1 (iii) Express the intermediate variables in terms of the decision variables a simple 7-day table may be helpful. 5 (iv) What are the constraints expressed in terms of the intermediate variables? (Use the symbols n i, i = 1,..., 7.) 1 (v) Express the constraints in terms of the decision variables, just write out the first two constraints. 2 (vi) Express your first constraint as an equality constraint. What is the coefficient of the slack variable? Explain the reason for this choice. 2 (vii) Write the objective function z in terms of the decision variables only. 1 (viii) Explain whether z is to be maximised or minimised. 1 (ix) What other constraints must be applied to the decision variables for the problem to be a Standard Form L.P.? 1 (b) A pharmaceutical company, PharmCo, produces two drugs: A and B. These drugs are produced via two manufacturing processes, Process 1 & Process 2. Each Process can run as often as needed in the time period a week. (The abbreviation g. is used below for grams & kg. for kilograms.) Each run of Process 1 requires 2 hours of labour and 1 kg. of raw material to produce 20 g. of A and 10 g. of B. Each run of Process 2 requires 3 hours of labour and 2 kg. of raw material to produce 30 g. of A and 20 g. of B. A minimimum of 150 g. of A and 270 g. of B must be produced. Sixty hours of labour and 40 kg. of raw material are available. Chemical A sells for E16 per g. and B sells for E14 per g. Formulate a linear program that maximises Pharmco s revenue. (i) Make a choice of intermediate variables, i.e. variables that are useful when tracking the day-to-day variation of the problem. 1 (ii) Make a choice of the decision variables for the problem. 1 (iii) Express the intermediate variables in terms of the decision variables. 2 (iv) What are the constraints expressed in terms of the intermediate variables? 1 1

3 (v) Express the constraints in terms of the decision variables. 2 (vi) Express the constraint (expressed in terms of the decision variables) corresponding to product A as an equality constraint. What is the coefficient of the slack variable? Briefly explain the reason for this choice. 1 (vii) State the objective function z (in algebraic notation) in terms of the intermediate variables. 1/2 (viii) Re-phrase z in terms of the decision variables only.. 1/2 (ix) Explain whether z is to be maximised or minimised.. 1/2 (x) What other constraints must be applied to the decision variables for the problem to be a Standard Form L.P.? 1/2 2 (a) Consider the following LP: with x 1, x 2 0. max 6x 1 3x 2 subject to 2x 1 + 3x 2 3 6x 1 + 2x 2 4 (i) Reformulate the LP into a Standard Form (SF) LP. 1 (ii) Write a Simplex Tableau for your SF LP. 2 (iii) Explain why the tableau is not in Canonical Form (CF). 1 (iv) The Dual Simplex method (DSM) is needed to pivot to CF. A. Which row & column should you pivot on? Explain your choice. 1 B. Use the DSM to pivot to CF explain each row operation clearly. 4 (v) Explain why the tableau is not in Optimal Form (OF). 1 (vi) The Simplex method (SM) is needed to pivot to OF. A. Which row & column should you pivot on? Explain your choice. 1 B. Use the SM to pivot to OF explain each row operation clearly. 4 (vii) Explain why the tableau is now in OF. (viii) What are the the optimal values of x 1, x 2 and the objective function z for the original max problem? 1 (Q.2 is continued on the next page.) 2

4 (b) As part of the process of formulating an LP in SF, free variables must be expressed in terms of non-negative variables. (i) Starting with a LP that is otherwise in SF, explain carefully how free variables can be eliminated using the expression x = y we where e is a constant vector of ones, x is a vector of free variables, y is a non-negative vector and w is a non-negative scalar variable. 6 (ii) Consider your starting tableau in part (a)(ii) of this Question with the nonnegativity conditions dropped. Explain why an extra column must be added to the starting tableau. Write the extra column of the tableau. 3 3 (a) Start with the standard dual pair in Appendix C and show (using steps (i) (v) below) that the optimal solutions to both are equal. (I m is the m m identity matrix and 0 T m is a row of m zeros.) (i) Introduce a vector s of slacks s 1,..., s m (where m is the number of constraints) into the min problem and show that the corresponding Simplex Tableau takes the form x s P = 0 c T 0 T m. b A I m (ii) Let P be the simplex tableau found in part (i) and P the OF tableau, say 3 P = x s -d u T v T b D B with optimal vector x and d = c T x. Recall that the pivot matrix Q is the result of applying the same pivots to I m+1 as were applied to P in order to pivot it to OF and that P[ = QP. ] [ ] 1 v T v T Show that Q = where are the right-hand m columns of P 0 B B corresponding to the m slack variables. 6 (iii) Now use the fact that P = QP to show that v 0, A T v c and d = v T b. 5 (iv) Show (without using tableaux) for any primal feasible x and any dual feasible y that c T x b T y. 2 (v) Finally explain why you can conclude that c T x = b T v and that v is the optimal vector for the dual problem. 3 (b) Show that the dual of an LP in SF is 6 max b T y (1) subject to A T y c y free. 3

5 4 (a) Consider the optimal form (OF) tableau T 0 = x 1 x 2 x 3 x 4 x 5 x 6 x (i) Find (without re-solving the LP) the optimal vector and objective value when the additional requirement x 7 = 4 is added to the problem. 2 (ii) Suppose now that the condition x 7 = 10 must be satisfied. A. Explain why the tableau T 0 must first be pivoted so that x 7 is increased up to its minimum row ratio for the tableau. N.B. Do not perform the pivot. 1 B. In what row & column must the pivot be performed? 1 (iii) Given that the result of the pivot is (writing the tableau in two equivalent formats) T 1 = x 1 x 2 x 3 x 4 x 5 x 6 x /7 1/ / /7 1/ / /7 3/ / /7 4/ /7 0 x 1 x 2 x 3 x 4 x 5 x 6 x /7 1/ / /7 1/ / /7 3/ / /7 4/ /7 0 state which of the currently non-basic variables can be increased from zero (so that x 7 is increased from 8 to 10) noting the corresponding increase in z? 2 (iv) Determine which choice of variables to increase is cheaper and find the optimal vector and objective value corresponding to x 7 = (v) Explain carefully whether z has increased or decreased from its initial value (based on T 0 ) and why. 1 (Q.4 is continued on the next page.) 4

6 (b) The following tableau T is the initial canonical form tableau for a resource allocation problem of the form max c T x such that Ax b with x 0: with optimal form (OF) tableau x 1 x 2 x 3 x 4 s 1 s 2 s T = x 1 x 2 x 3 x 4 s 1 s 2 s (i) Write the matrix Q such that T = QT. (See Q.3(a)(ii) above.) 2 (ii) Explain using Q what the effect on T is of adding a (positive or negative) to the inital resources available for resource 1. 3 (iii) Find the new optimal vector and objective if 26 units of resource 1 are available instead of the original (c) Suppose, for a resource allocation problem as in part (b), that the price in the starting canonical form of a variable x i that is basic in the optimal tableau is changed by the addition of q (positive or negative) to the price. (i) Show that the effect on an optimal tableau is to add q times the row in T associated with x i to the top (objective) row of T so only z changes and not the optimal x. 3 (ii) What is the maximum amount by which the price associated with the variable x 2 in T in part (b) above may be increased or decreased while keeping the currently basic variables basic? 2 (iii) Suppose that the price associated with variable x 2 in the tableau T in part (b) is increased from E6 to E7. What is the effect on the optimal value of the objective function z? 2 5

7 5 The (balanced) Transportation Problem m n min c ij x ij i=1 j=1 subject to n x ij = a i, for i = 1,..., m. j=1 m x ij = b j, for j = 1,..., n. i=1 x ij 0, for each i and j. with m i=1 a i = n j=1 = b j can be written as an LP in Standard Form: min subject to c T x Ax = b x 0 referring to Appendix E for the definitions of A, b and c. (a) (i) Use the expression for the dual of an LP in SF in Eq. 1 in Q3 (b) above to show that the dual of the Transportation Problem is 7 max m a i u i + i=1 subject to n b j v j j=1 u i + v j c ij, i = 1,..., m, j = 1,..., n u & v free. (ii) Explain briefly how this result is used to determine whether a transportation tableau is optimal. You should use no further algebra but you should refer to the result (established in Q.3 (a)) that the optimal objective values for a primaldual pair are equal. 2 6

8 (b) A company has three manufacturing plants, all making the same product. There are four outlets that require the product. One of the plants can only ship to certain destinations. Plants 1, 2 and 3 produce 10, 15 & 20 items respectively per day. Plant 1 can ship to all four outlets with unit shipping costs of 1, 2, 3 & 4 respectively. Plant 2 can only ship to outlets 1, 2 & 3 with unit shipping costs of 5,1 & 6 respectively. Plant 3 can ship to all four outlets with unit shipping costs of 7, 3, 5 & 2 respectively. The four outlets require 5, 10, 15 and 20 items per day respectively. There is an shortfall in production capacity. (i) Formulate this problem as a transportation problem by writing a transportation tableau. (Note that supply does not equal demand so a dummy supply with zero shipping costs is needed. Assign a large positive cost N to the forbidden link.) 2 (ii) Use the NW Corner Method (NWCM) to find an initial feasible tableau. 2 (iii) Use the SCEM method to find an initial feasible tableau. 2 (iv) Calculate the costs of the two starting solutions. Which is better? 1 (c) Using the cheaper of your two starting tableau and referring to the statement of the Transportation Algorithm in App. D if you wish: (i) Calculate dual variables u and v. 1 (ii) Adjust costs replacing the costs in non-basic positions by c ij u i v j. 1 (iii) Is the tableau optimal? Explain why or why not. 1/2 (d) Now take the following tableau as your current tableau for the next part of the question N (i) Find the loop connecting a succession of basic positions, starting at the nonbasic position with most negative cost. 2 (ii) Determine the max allowable increase/decrease t in x ij for the basic positions in the loop and update your tableau changing the basic variables and updating the assignments as necessary. 1 (iii) What is the cost reduction associated with this update? 1/2 7

9 (e) Suppose that the following optimal transportation tableau results from calculating the dual variables u and v and adjusting the costs N (i) What is the cost associated with this optimal tableau using the starting costs? 1 (ii) Draw a network diagram (arrow diagram) displaying your solution. 1 (iii) Which consumer(s) have excess stock and how much? 1 8

10 Appendix of Results A Algorithm 1 (Simplex Method) begin (Start with a Canonical tableau s.t. b 0.) while NOT finished do if c j 0 for all j then STOP (Tableau is optimal.) else Select j s.t. c j < 0. fi if a ij 0 for all i = 1,..., m then STOP (Problem is unbounded.) fi Select k such { that: } b k b a kj = min i i a ij such that a ij > 0 (k attains the min.) Pivot on a kj. (Divide Row k across by a kj and add end... multiples of Row k to the rows above & below end introducing zeros into column j.) B Algorithm 2 (Dual Simplex Method) begin (Start with a tableau s.t. c 0.) while NOT finished do if b i 0 for all i then STOP (Tableau is optimal.) else Select i s.t. b i < 0. fi if a ij 0 for all j = 1,..., n then STOP (Dual unbounded Primal infeasible.) fi Select k such { that: } c k cj a ik = max j a ij such that a ij < 0 (k attains max.) Pivot on a ik. (Divide Row k across by a ik and add end... multiples of Row k to the rows above & below end introducing zeros into column i.) C The following pair of LP s are the standard dual pair: D Transportation Algorithm min c T x max b T y subject to Ax b subject to A T y c (a) Find an initial basic feasible point x. (b) x 0 y 0. For the current solution x and the current cost coefficients c ij, find a dual vector (u, v) such that u i + v j = c ij for all basic positions (i, j). Calculate the adjusted cost coefficients (a.c.c.) for all positions (i, j). 9

11 (c) If each a.c.c is 0, STOP. ELSE Pick the position with the most negative a.c.c and find the unique loop starting there (with all other positions basic). (d) Shift as much as possible around the loop to get a new basic feasible point x. GOTO Step 2 with the a.c.c s as the new costs. E Notation for the Transportation Problem written as a LP in matrix notation. I n is the n n identity matrix. e T n = [ ] (n ones). z T n = [ ] (n zeros). e T n z T n z T n... z T n z T n z T n e T n z T n... z T n z T n A is the (m + n) (mn) matrix A = z T n z T n z T n... e T n z T n z T n z T n z T n... z T n e T n I n I n I n... I n I n a 1 a 2. a b = m. b 1 b 2. b n The cost coefficients c are just the column vector of length mn: c = c 11 c 12. c 1n c 21 c 22. c 2n. c m1 c m2. c mn. 10

12 Ruled Paper for use with MS4303 Assessment 5 May

13 Ruled Paper for use with MS4303 Assessment

14 Ruled Paper for use with MS4303 Assessment 5 May

15 Ruled Paper for use with MS4303 Assessment 5 May

16 Ruled Paper for use with MS4303 Assessment 5 May

17 Ruled Paper for use with MS4303 Assessment 5 May

18 Ruled Paper for use with MS4303 Assessment 5 May

19 Ruled Paper for use with MS4303 Assessment 5 May

Part 1. The Review of Linear Programming

In the name of God Part 1. The Review of Linear Programming 1.5. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Formulation of the Dual Problem Primal-Dual Relationship Economic Interpretation