June 00 (adapted to include 00 1 flows and simplex questions) 1. igure 1 0 1 15 10 16 16 19 0 15 igure 1 shows a network of roads connecting six villages,,,, and. he lengths of the roads are given in km. (a) omplete the table in the answer booklet, in which the entries are the shortest distances between pairs of villages. You should do this by inspection. he table can now be taken to represent a complete network. (b) Use the nearest-neighbour algorithm, starting at, on your completed table in part (a). Obtain an upper bound to the length of a tour in this complete network, which starts and finishes at and visits every village exactly once. (c) Interpret your answer in part (b) in terms of the original network of roads connecting the six villages. (d) y choosing a different vertex as your starting point, use the nearest-neighbour algorithm to obtain a shorter tour than that found in part (b). tate the tour and its length.. two-person zero-sum game is represented by the following pay-off matrix for player. I II III IV I 4 5 4 II 1 1 1 III 0 5 4 IV 1 3 1 1 (a) etermine the play-safe strategy for each player. (b) Verify that there is a stable solution and determine the saddle points. (c) tate the value of the game to.
3. igure 7 9 8 8 4. ndrew () and arbara () play a zero-sum game. his game is represented by the following payoff matrix for ndrew. 3 1 6 5 4 3 4 7 7 9 10 6 6 3 6 9 (a) xplain why this matrix may be reduced to 3 5. 6 3 (b) Hence find the best strategy for each player and the value of the game. (8) he network in ig. shows possible routes that an aircraft can take from to. he numbers on the directed arcs give the amount of fuel used on that part of the route, in appropriate units. he airline wishes to choose the route for which the maximum amount of fuel used on any part of the route is as small as possible. his is the rninimax route. (a) omplete the table in the answer booklet. (b) Hence obtain the minimax route from to and state the maximum amount of fuel used on any part of this route. (8) 5. n engineering company has 4 machines available and 4 jobs to be completed. ach machine is to be assigned to one job. he time, in hours, required by each machine to complete each job is shown in the table below. Job 1 Job Job 3 Job 4 Machine 1 14 5 8 7 Machine 1 6 5 Machine 3 7 8 3 9 Machine 4 4 6 10 Use the Hungarian algorithm, reducing rows first, to obtain the allocation of machines to jobs which minimises the total time required. tate this minimum time. (11)
6. he table below shows the distances, in km, between six towns,,,, and. 85 110 175 108 100 85 38 175 160 93 110 38 148 156 73 175 175 148 110 84 108 160 156 110 9 100 93 73 84 9 (a) tarting from, use Prim s algorithm to find a minimum connector and draw the minimum spanning tree. You must make your method clear by stating the order in which the arcs are selected. (b) (i) Using your answer to part (a) obtain an initial upper hound for the solution of the travelling salesman problem. (ii) Use a short cut to reduce the upper bound to a value less than 680. (c) tarting by deleting, find a lower bound for the solution of the travelling salesman problem. 7. steel manufacturer has 3 factories 1, and 3 which can produce 35, 5 and 15 kilotonnes of steel per year, respectively. hree businesses 1, and 3 have annual requirements of 0, 5 and 30 kilotonnes respectively. he table below shows the cost ij in appropriate units, of transporting one kilotonne of steel from factory i to business j. actory usiness 1 3 1 10 4 11 1 5 8 3 9 6 7 he manufacturer wishes to transport the steel to the businesses at minimum total cost. (a) Write down the transportation pattern obtained by using the North-West corner rule. (b) alculate all of the improvement indices Iij, and hence show that this pattern is not optimal. (c) Use the stepping-stone method to obtain an improved solution. (d) how that the transportation pattern obtained in part (c) is optimal and find its cost. (5)
8. igure 4 3 8 4 5 1 7 6 16 9 7 10 11 3 G he network in ig. 4 models a drainage system. he number on each arc indicates the capacity of that arc, in litres per second. (a) Write down the source vertices. igure 5 6 4 3 1 7 3 16 7 5 10 11 3 G igure 5 shows a feasible flow through the same network. (b) tate the value of the feasible flow shown in ig. 5. aking the flow in ig. 5 as your initial flow pattern, (c) use the labelling procedure on iagram 1 to find a maximum flow through this network. You should list each flow-augmenting route you use, together with its flow. (6) (d) how the maximal flow on iagram and state its value. (e) Prove that your flow is maximal. H H 9. 4 o. Ltd produces three different blends of tea, Morning, fternoon and vening. he teas must be processed, blended and then packed for distribution. he table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne. Processing lending Packing Profit ( 100) Morning blend 3 1 4 fternoon blend 3 4 5 vening blend 4 3 3 he total times available each week for processing, blending and packing are 35, 0 and 4 hours respectively. 4 o. Ltd wishes to maximise the weekly profit. Let x, y and z be the number of tonnes of Morning, fternoon and vening blend produced each week. (a) ormulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities. n initial implex tableau for the above situation is asic variable x y z r s t Value r 3 4 1 0 0 35 s 1 3 0 1 0 0 t 4 3 0 0 1 4 P 4 5 3 0 0 0 0 (b) olve this linear programming problem using the implex algorithm. ake the most negative number in the profit row to indicate the pivot column at each stage. (11) 4 o. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three. (c) Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.
10. While solving a maximizing linear programming problem, the following tableau was obtained. asic x y z r s t Value variable 1 r 0 0 1 3 1 0 6 3 1 1 y 0 1 3 3 0 1 3 1 x 1 0 3 0 1 1 3 P 0 0 1 0 1 1 11 (a) xplain why this is an optimal tableau. (b) Write down the optimal solution of this problem, stating the value of every variable. (c) Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of P. 1 11. company wishes to transport its products from 3 factories 1, and 3 to a single retail outlet R. he capacities of the possible routes, in van loads per day, are shown in ig. 5. 1 3 8 6 3 6 6 8 4 igure 5 14 15 R (a) On iagram 1 in the answer booklet add a supersource to obtain a capacitated network with a single source and a single sink. tate the minimum capacity of each arc you have added. (b) (i) tate the maximum flow along 1R and 3R. (ii) how these maximum flows on iagram in the answer booklet, using numbers in circles. aking your answer to part (b)(ii) as the initial flow pattern, (c) (i) use the labelling procedure to find a maximum flow from to R. Your working should be shown on iagram 3. List each flow-augmenting route you find together with its flow. (ii) Prove that your final flow is maximal. (7)
003 (adapted for new spec) 1. two person zero-sum game is represented by the following pay-off matrix for player. 1. igure W1 W W3 1 3 8 6 10 8 11 9 16 6 R1 R plays I plays II plays III plays I 3 5 plays II 4 1 4 (a) Write down the pay off matrix for player. (b) ormulate the game as a linear programming problem for player, writing the constraints as equalities and stating your variables clearly. (otal 6 marks). (a) xplain the difference between the classical and practical travelling salesman problems. company has 3 warehouses W1, W, and W3. It needs to transport the goods stored there to retail outlets R1 and R. he capacities of the possible routes, in van loads per day, are shown in ig. Warehouses W1, W and W3 have 14, 1 and 14 van loads respectively available per day and retail outlets R1 and R can accept 6 and 5 van loads respectively per day. (a) On iagram 1 on the answer sheet add a supersource W, a supersink R and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. tate the minimum capacity of each arc you have added. (b) tate the maximum flow along (i) W W1 R1 R, (ii) WW3 R R. (c) aking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from W to R. how your working on iagram. List each flowaugmenting route you use, together with its flow. (5) (d) rom your final flow pattern, determine the number of van loads passing through each day. (b) 8 13 19 18 0 18 10 31 9 he network in the diagram above shows the distances, in kilometres, between eight Mcurger restaurants. n inspector from head office wishes to visit each restaurant. His route should start and finish at, visit each restaurant at least once and cover a minimum distance. Obtain a minimum spanning tree for the network using Kruskal s algorithm. You should draw your tree and state the order in which the arcs were added. 13 G 11 17 14 0 H (c) Use your answer to part (b) to determine an initial upper bound for the length of the route. (d) tarting from your initial upper bound and using an appropriate method, find an upper bound which is less than 135 km. tate your tour. (otal 10 marks)
3. alkalot ollege holds an induction meeting for new students. he meeting consists of four talks: I (Welcome), II (Options and acilities), III (tudy ips) and IV (Planning for uccess). he four department heads, live, Julie, Nicky and teve, deliver one of these talks each. he talks are delivered consecutively and there are no breaks between talks. he meeting starts at 10 a.m. and ends when all four talks have been delivered. he time, in minutes, each department head takes to deliver each talk is given in the table below. (a) alk I alk II alk III alk IV live 1 34 8 16 Julie 13 3 36 1 Nicky 15 3 3 14 teve 11 33 36 10 Use the Hungarian algorithm to find the earliest time that the meeting could end. You must make your method clear and show 5. he manager of a car hire firm has to arrange to move cars from three garages, and to three airports, and so that customers can collect them. he table below shows the transportation cost of moving one car from each garage to each airport. It also shows the number of cars available in each garage and the number of cars required at each airport. he total number of cars available is equal to the total number required. (a) irport irport irport ars available Garage 0 40 10 6 Garage 0 30 40 5 Garage 10 0 30 8 ars required 6 9 4 Use the North-West corner rule to obtain a possible pattern of distribution and find its cost. (b) (i) the state of the table after each stage in the algorithm, (ii) the final allocation. Modify the table so it could be used to find the latest time that the meeting could end. (10) (b) (c) alculate shadow costs for this pattern and hence obtain improvement indices for each route. Use the stepping-stone method to obtain an optimal solution and state its cost. (7) (otal 14 marks) 4. two person zero-sum game is represented by the following pay-off matrix for player. (otal 13 marks) 6. Kris produces custom made racing cycles. he can produce up to four cycles each month, but if she wishes to produce more than three in any one month she has to hire additional help at a cost of 350 for that month. In any month when cycles are produced, the overhead costs are 00. maximum of 3 cycles can be held in stock in any one month, at a cost of 40 per cycle per month. ycles must be delivered at the end of the month. he order book for cycles is plays I plays II plays III Month ugust eptember October November plays I 1 3 Number of cycles required 3 3 5 (a) plays II 1 3 0 plays III 0 1 3 Identify the play safe strategies for each player. isregarding the cost of parts and Kris time, (a) determine the total cost of storing cycles and producing 4 cycles in a given month, making your calculations clear. (b) (c) Verify that there is no stable solution to this game. xplain why the pay-off matrix above may be reduced to here is no stock at the beginning of ugust and Kris plans to have no stock after the November delivery. (b) Use dynamic programming to determine the production schedule which minimises the costs, showing your working in the table below. plays I plays II plays III tage emand tate ction estination Value plays I 1 3 plays II 1 3 0 1 (Nov) 0 (in stock) (make) 0 00 1 (in stock) (make) 1 0 40 (in stock) (make) 0 0 80 (d) ind the best strategy for player, and the value of the game. (7) (otal 14 marks) (Oct) 5 1 4 0 590 + 00 = 790 3 0 4 1 (13)
he fixed cost of parts is 600 per cycle and of Kris time is 500 per month. he sells the cycles for 000 each. (c) etermine her total profit for the four month period. (otal 18 marks) (c) ind the value of cuts 1 and. tarting with the given feasible flow of 68, (d) use the labelling procedure on iagram to find a maximal flow through this network. List each flow-augmenting route you use, together with its flow. 7. iagram igure 1 igure 1 shows a capacitated, directed network. he unbracketed number on each arc indicates the capacity of that arc, and the numbers in brackets show a feasible flow of value 68 through the network. (a) 1 45 (38) 35 (30) 4 (0) 0 (18) 1 (1) 15 (15) 0 (18) 10 (5) dd a supersource and a supersink, and arcs of appropriate capacity, to iagram 1 below. 8 (5) 1 8 ( y) 4 G 8 ( x) 5 (18) H 9 (9) 5 (5) 15 (10) 8 (6) 1 (e) 1 how your maximal flow on iagram 3 and state its value. iagram 3 G H 1 (6) iagram 1 1 1 45 (38) 35 (30) 4 (0) 0 (18) 1 (1) 15 (15) 0 (18) 10 (5) 8 (5) 8 ( y) 4 G 8 ( x) 5 (18) 9 (9) 15 (10) H 8 (6) 5 (5) 1 (f) 1 Prove that your flow is maximal. G H (otal 18 marks) (b) ind the values of x and y, explaining your method briefly.
8. he tableau below is the initial tableau for a maximising linear programming problem. 004 (adapted for new spec) (a) asic variable x y z r s Value r 3 4 1 0 8 s 3 3 1 0 1 10 P 8 9 5 0 0 0 or this problem x 0, y 0, z 0. Write down the other two inequalities and the objective function. 1. In game theory explain what is meant by (a) zero-sum game, (b) saddle point. (otal 4 marks). In a quiz there are four individual rounds, rt, Literature, Music and cience. team consists of four people, onna, Hannah, Kerwin and homas. ach of four rounds must be answered by a different team member. he table shows the number of points that each team member is likely to get on each individual round. rt Literature Music cience (b) olve this linear programming problem. You may not need to use all of these tableaux. onna 31 4 3 35 Hannah 16 10 19 Kerwin 19 14 0 1 b.v. x y z r s Value homas 18 15 1 3 P Use the Hungarian algorithm, reducing rows first, to obtain an allocation which maximises the total points likely to be scored in the four rounds. You must make your method clear and show the table after each stage. (otal 9 marks) - b.v. x y z r s Value P b.v. x y z r s Value 3. he table shows the least distances, in km, between five towns,,,, and. Nassim wishes to find an interval which contains the solution to the travelling salesman problem for this network. (a) Making your method clear, find an initial upper bound starting at and using (i) (ii) the minimum spanning tree method, the nearest neighbour algorithm. 153 98 14 115 153 74 131 149 98 74 8 103 14 131 8 134 115 149 103 134 (7) P (b) y deleting, find a lower bound. b.v. x y z r s Value (c) Using your answers to parts (a) and (b), state the smallest interval that Nassim could correctly write down. (otal 1 marks) P (8) 4. mma and reddie play a zero-sum game. his game is represented by the following pay-off matrix for 4 1 3 mma. 1 (c) tate the final value of P, the objective function, and of each of the variables. (otal 14 marks) (a) (b) how that there is no stable solution. ind the best strategy for mma and the value of the game to her. (8) (c) Write down the value of the game to reddie and his pay-off matrix. (otal 14 marks)
5. (a) escribe a practical problem that could be solved using the transportation algorithm. problem is to be solved using the transportation problem. he costs are shown in the table. he supply is from, and and the demand is at d and e. d e upply 5 3 45 4 6 35 4 40 emand 50 60 (b) xplain why it is necessary to add a third demand f. (c) Use the north-west corner rule to obtain a possible pattern of distribution and find its cost. able 3 ravel costs ( ) G H Home 70 80 150 80 90 70 180 150 140 10 00 10 00 160 10 170 100 110 It is decided to use dynamic programming to find a schedule that maximises the total expected profit, taking into account the travel costs. (a) efine suitable stage, state and action variables. d e f upply 5 3 45 4 6 35 4 40 emand 50 60 (5) 7. (b) (c) etermine the schedule that maximises the total profit. how your working in a table. (1) dvise Joan on the shows that she should visit and state her total expected profit. (otal 18 marks) igure 1 (d) alculate shadow costs and improvement indices for this pattern. 4 0 G (e) Use the stepping-stone method once to obtain an improved solution and its cost. (5) (5) (otal 16 marks) 6. Joan sells ice cream. he needs to decide which three shows to visit over a three-week period in the summer. he starts the three-week period at home and finishes at home. he will spend one week at each of the three shows she chooses travelling directly from one show to the next. 14 1 11 15 9 6 4 10 5 H 7 I 8 able 1 gives the week in which each show is held. able gives the expected profits from visiting each show. able 3 gives the cost of travel between shows. able 1 Week 1 3 hows,,,, G, H able how G H xpected Profit ( ) 900 800 1000 1500 1300 500 700 600 igure 1 shows a capacitated directed network. he number on each arc is its capacity. igure 0 G 0 x 7 0 14 0 y 14 I 0 6 4 4 H
igure shows a feasible initial flow through the same network. d) how your maximal flow pattern on iagram. (a) Write down the values of the flow x and the flow y. iagram (b) Obtain the value of the initial flow through the network, and explain how you know it is not maximal. G (c) Use this initial flow and the labelling procedure on iagram 1 below to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow. iagram 1 I I G H H (e) Prove that your flow is maximal. (otal 13 marks) 8. three-variable linear programming problem in x, y and z is to be solved. he objective is to maximise the profit P. he following tableau was obtained. asic variable x y Z r s t Value s 3 0 0 1 3 r 4 0 7 1 0 8 y 5 1 7 0 0 3 7 P 3 0 0 0 8 63 3 9 (a) (b) tate, giving your reason, whether this tableau represents the optimal solution. tate the values of every variable. (c) alculate the profit made on each unit of y. (otal 6 marks) (5)
9. 10. latland UK Ltd makes three types of carpet, the Lincoln, the Norfolk and the uffolk. he carpets all require units of black, green and red wool. 15 1 14 14 7 7 1 he diagram above shows a network of roads represented by arcs. he capacity of the road represented by that arc is shown on each arc. he numbers in circles represent a possible flow of 6 from to L. hree cuts 1, and 3 are shown on he diagram above. 5 5 14 14 1 7 7 7 H I 7 G 7 9 7 8 3 1 J 1 4 3 18 6 K 8 3 8 L or each roll of carpet, the Lincoln requires 1 unit of black, 1 of green and 3 of red, the Norfolk requires 1 unit of black, of green and of red, and the uffolk requires units of black, 1 of green and 1 of red. here are up to 30 units of black, 40 units of green and 50 units of red available each day. Profits of 50, 80 and 60 are made on each roll of Lincoln, Norfolk and uffolk respectively. latland UK Ltd wishes to maximise its profit. Let the number of rolls of the Lincoln, Norfolk and uffolk made daily be x, y and z respectively. (a) ormulate the above situation as a linear programming problem, listing clearly the constraints as inequalities in their simplest form, and stating the objective function. his problem is to be solved using the implex algorithm. he most negative number in the profit row is taken to indicate the pivot column at each stage. (b) tating your row operations, show that after one complete iteration the tableau becomes asic variable x y z r s t Value r 1 0 1 1 1 1 0 10 y 1 1 1 0 1 0 0 t 0 0 0 1 1 10 P 10 0 0 0 40 0 1600 (a) ind the capacity of each of the three cuts. (b) Verify that the flow of 6 is maximal. he government aims to maximise the possible flow from to L by using one of two options. Option 1: uild a new road from to J with capacity 5. You may not need to use all of the tableaux. asic variable x y z r s t Value r s t Row operations or Option : uild a new road from to H with capacity 3. P (c) y considering both options, explain which one meets the government s aim - asic variable x y z r s t Value Row operations -
(c) xplain the practical meaning of the value 10 in the top row. (d) (i) Perform one further complete iteration of the implex algorithm. asic variable x y z r s t Value Row operations 005 (adapted for new spec) 1. reezy o. has three factories, and. It supplies freezers to three shops, and. he table shows the transportation cost in pounds of moving one freezer from each factory to each outlet. It also shows the number of freezers available for delivery at each factory and the number of freezers required at each shop. he total number of freezers required is equal to the total number of freezers available. vailable 1 4 16 4 18 3 17 3 15 19 5 14 - asic variable x y z r s t Value Row operations (a) Required 0 30 0 Use the north-west corner rule to find an initial solution. (b) (c) Obtain improvement indices for each unused route. Use the stepping-stone method once to obtain a better solution and state its cost. (5) (otal 11 marks) (ii) tate whether your current answer to part (d)(i) is optimal. Give a reason for your answer.. (iii) Interpret your current tableau, giving the value of each variable. (8) (otal 18 marks) 1 15 18 14 0 16 19 8 G 7 11 8 31 he network in the diagram shows the distances, in km, of the cables between seven electricity relay stations,,,,, and G. n inspector needs to visit each relay station. He wishes to travel a minimum distance, and his route must start and finish at the same station. y deleting, a lower bound for the length of the route is found to be 19 km. (a) (b) ind another lower bound for the length of the route by deleting. tate which is the better lower bound of the two. y inspection, complete the table of least distances. (5) he table can now be taken to represent a complete network. (c) Using the nearest-neighbour algorithm, starting at, obtain an upper bound to the length of the route. tate your route. (otal 11 marks)
3. hree warehouses W, X and Y supply televisions to three supermarkets J, K and L. he table gives the cost, in pounds, of transporting a television from each warehouse to each supermarket. he warehouses have stocks of 34, 57 and 5 televisions respectively, and the supermarkets require 0, 56 and 40 televisions respectively. he total cost of transporting the televisions is to be minimised. J K L W 3 6 3 X 5 8 4 Y 5 7 ormulate this transportation problem as a linear programming problem. Make clear your decision variables, objective function and constraints. (otal 7 marks) 4. (a) xplain what is meant by a maximin route in dynamic programming, and give an example of a situation that would require a maximin solution. 5. our salesperson,, and are to be sent to visit four companies 1,, 3 and 4. ach salesperson will visit exactly one company, and all companies will be visited. Previous sales figures show that each salesperson will make sales of different values, depending on the company that they visit. hese values (in 10 000s) are shown in the table below. (a) (b) 1 3 4 nn 6 30 30 30 renda 30 3 6 9 onnor 30 5 7 4 ave 30 7 5 1 Use the Hungarian algorithm to obtain an allocation that maximises the sales. You must make your method clear and show the table after each stage. tate the value of the maximum sales. (11) 4 5 7 3 18 5 16 19 14 H 16 3 I 17 0 15 G 8 J maximin route is to be found through the network shown in the diagram. (b) omplete the table in the answer book, and hence find a maximin route. 19 18 1 K (9) 6. (c) how that there is a second allocation that maximises the sales. 1 1 1 35 17 6 6 8 6 6 34 0 3 3 4 0 0 9 1 43 15 3 0 17 17 10 0 40 33 1 3 1 (otal 15 marks) (c) List all other maximin routes through the network. (otal 14 marks) his figure shows a capacitated directed network. he number on each arc is its capacity. he numbers in circles show a feasible flow through the network. ake this as the initial flow. (a) On iagram 1 and iagram in the answer book, add a supersource and a supersink. On iagram 1 show the minimum capacities of the arcs you have added. iagram in the answer book shows the first stage of the labelling procedure for the given initial flow. (b) omplete the initial labelling procedure in iagram. (c) ind the maximum flow through the network. You must list each flow-augmenting route you use together with its flow, and state the maximal flow. (6)
(d) how a maximal flow pattern on iagram 3. (c) ind the best strategy for player and the value of the game to her. (8) (e) (f) Prove that your flow is maximal. escribe briefly a situation for which this network could be a suitable model. (otal 16 marks) (d) ormulate the game as a linear programming problem for player. Write the constraints as inequalities and define your variables clearly. (5) (otal 17 marks) 8. Polly has a bird food stall at the local market. ach week she makes and sells three types of packs, and. 1 0 18 1 17 0 6 0 3 4 0 8 1 6 11 3 0 17 10 0 8 6 19 1 1 3 1 Pack contains 4 kg of bird seed, suet blocks and 1 kg of peanuts. Pack contains 5 kg of bird seed, 1 suet block and kg of peanuts. Pack contains 10 kg of bird seed, 4 suet blocks and 3 kg of peanuts. ach week Polly has 140 kg of bird seed, 60 suet blocks and 60 kg of peanuts available for the packs. he profit made on each pack of, and sold is 3.50, 3.50 and 6.50 respectively. Polly sells every pack on her stall and wishes to maximise her profit, P pence. Let x, y and z be the numbers of packs, and sold each week. n initial implex tableau for the above situation is 0 0 15 0 asic variable x y z r s t Value r 4 5 1 0 1 0 0 140 s 1 4 0 1 0 60 t 1 3 0 0 1 60 P 350 350 650 0 0 0 0 (a) (b) xplain the meaning of the variables r, s and t in the context of this question. Perform one complete iteration of the implex algorithm to form a new tableau. ake the most negative number in the profit row to indicate the pivotal column. (5) (c) tate the value of every variable as given by tableau. (d) Write down the profit equation given by tableau. 7. (a) xplain briefly what is meant by a zero-sum game. (e) Use your profit equation to explain why tableau is not optimal. aking the most negative number in the profit row to indicate the pivotal column, two person zero-sum game is represented by the following pay-off matrix for player. I II III (f) identify clearly the location of the next pivotal element. (otal 15 marks) I 5 3 II 3 5 4 (b) Verify that there is no stable solution to this game.
9. iagram 8 5 9 0 1 14 15 7 3 3 15 11 1 10 his diagram shows a capacitated directed network. he number on each arc is its capacity. (a) tate the maximum flow along (i), (ii), (iii). (b) how these maximum flows on iagram 1 below. (ii) raw your final flow pattern on iagram 3 below. iagram 3 (5) iagram 1 (iii) Prove that your flow is maximal. ake your answer to part (b) as the initial flow pattern. (c) (i) Use the labelling procedure to find a maximum flow from to. Your working should be shown on iagram below. List each flow-augmenting route you use, together with its flow. (d) Give an example of a practical situation that could have been modelled by the original network. (otal 14 marks)
Jan 006 (adapted for new spec) 1. theme park has four sites,,, and, on which to put kiosks. ach kiosk will sell a different type of refreshment. he income from each kiosk depends upon what it sells and where it is located. he table below shows the expected daily income, in pounds, from each kiosk at each site. Hot dogs and beef burgers (H) Ice cream (I) Popcorn, candyfloss and drinks (P) nacks and hot drinks () ite 67 7 76 61 ite 64 71 78 63 ite 67 73 75 63 ite 61 69 74 57 Reducing rows first, use the Hungarian algorithm to determine a site for each kiosk in order to maximise the total income. tate the site for each kiosk and the total expected income. You must make your method clear and show the table after each stage. (otal 13 marks). n engineering firm makes motors. hey can make up to five in any one month, but if they make more than four they have to hire additional premises at a cost of 500 per month. hey can store up to two motors for 100 per motor per month. he overhead costs are 00 in any month in which work is done. Motors are delivered to buyers at the end of each month. here are no motors in stock at the beginning of May and there should be none in stock after the eptember delivery. he order book for motors is: Month May June July ugust eptember Number of motors 3 3 7 5 4 Use dynamic programming to determine the production schedule that minimises the costs, showing your working in the table provided below. tage (month) tate (Number in store at start of month) ction (Number made in month) estinatio n (Number in store at end of month) Value (cost) 3. hree depots,, G and H, supply petrol to three service stations,, and U. he table gives the cost, in pounds, of transporting 1000 litres of petrol from each depot to each service station., G and H have stocks of 540 000, 789 000 and 673 000 litres respectively., and U require 57 000, 348 000 and 41 000 litres respectively. he total cost of transporting the petrol is to be minimised. G H 3 35 41 31 38 50 U 46 51 63 ormulate this problem as a linear programming problem. Make clear your decision variables, objective function and constraints. 4. he following minimising transportation problem is to be solved. (a) (b) J K upply 1 15 9 8 17 13 4 9 1 emand 9 11 omplete the table below. J K L upply 1 15 9 8 17 13 4 9 1 emand 9 11 34 xplain why an extra demand column was added to the table above. possible north-west corner solution is: J K L 9 0 11 1 (otal 8 marks) (c) xplain why it was necessary to place a zero in the first row of the second column. fter three iterations of the stepping-stone method the table becomes: J K L Production schedule 8 1 Month May June July ugust eptember Number to be made otal cost:... (otal 1 marks) (d) 13 9 3 aking the most negative improvement index as the entering square for the stepping stone method, solve the transportation problem. You must make your shadow costs and improvement indices clear and demonstrate that your solution is optimal. (11) (otal 16 marks)
5. two-person zero-sum game is represented by the following pay-off matrix for player. (a) (b) (c) plays 1 plays plays 3 plays 4 plays 1 1 3 1 plays 1 3 1 plays 3 4 0 1 plays 4 1 1 3 Verify that there is no stable solution to this game. xplain why the 4 4 game above may be reduced to the following 3 3 game. ormulate the 3 3 game as a linear programming problem for player. Write the constraints as inequalities. efine your variables clearly. 6. he network in the figure above, shows the distances in km, along the roads between eight towns,,,,,,, G and H. Keith has a shop in each town and needs to visit each one. He wishes to travel a minimum distance and his route should start and finish at. y deleting, a lower bound for the length of the route was found to be 586 km. y deleting, a lower bound for the length of the route was found to be 590 km. (a) (b) y deleting, find another lower bound for the length of the route. tate which is the best lower bound of the three, giving a reason for your answer. (5) y inspection complete the table of least distances. he table can now be taken to represent a complete network. he nearest neighbour algorithm was used to obtain upper bounds for the length of the route: tarting at, an upper bound for the length of the route was found to be 838 km. tarting at, an upper bound for the length of the route was found to be 707 km. (c) tarting at, use the nearest neighbour algorithm to obtain another upper bound for the length of the route. tate which is the best upper bound of the three, giving a reason for your answer. 5 97 G 84 77 115 14 85 163 140 9 1 3 1 3 1 1 (8) (otal 13 marks) G H - 84 85 138 173 149 5 84-130 77 16 13 136 85 130-53 88 83 9 138 77 53-49 190 173 16 88 49-100 180 15 13 83 100-163 115 G 149 9 180 163-97 H 5 136 190 15 115 97-83 53 88 100 49 (otal 13 marks) 7. (a) efine the terms (i) cut, (ii) minimum cut, as applied to a directed network flow. he figure above shows a capacitated directed network and two cuts 1 and. he number on each arc is its capacity. (b) tate the values of the cuts 1 and. Given that one of these two cuts is a minimum cut, (c) (d) find a maximum flow pattern by inspection, and show it on the diagram below. ind a second minimum cut for this network. In order to increase the flow through the network it is decided to add an arc of capacity 100 joining either to or to G. (e) 408 39 1 14 153 164 85 1 307 G 19 tate, with a reason, which of these arcs should be added, and the value of the increased flow. (otal 11 marks) 08 51 G 3 79 36
June 006 (adapted for new spec) 1. (a) tate ellman s principle of optimality. (b) (c) xplain what is meant by a minimax route. escribe a practical problem that would require a minimax route as its solution. (otal 4 marks). hree workers, P, Q and R, are to be assigned to three tasks, 1, and 3. ach worker is to be assigned to one task and each task must be assigned to one worker. he cost, in hundreds of pounds, of using each worker for each task is given in the table below. he cost is to be minimised. ost (in 100s) ask 1 ask ask 3 Worker P 8 7 3 Worker Q 9 5 6 Worker R 10 4 4 ormulate the above situation as a linear programming problem, defining the decision variables and making the objective and constraints clear. (otal 7 marks) 3. college wants to offer five full-day activities with a different activity each day from Monday to riday. he sports hall will only be used for these activities. ach evening the caretaker will prepare the hall by putting away the equipment from the previous activity and setting up the hall for the activity next day. On riday evening he will put away the equipment used that day and set up the hall for the following Monday. he 5 activities offered are adminton (), ricket nets (), ancing (), ootball coaching () and ennis (). ach will be on the same day from week to week. he college decides to offer the activities in the order that minimises the total time the caretaker has to spend preparing the hall each week. he hall is initially set up for adminton on Monday. he table below shows the time, in minutes, it will take the caretaker to put away the equipment from one activity and set up the hall for the next. o ime 108 150 64 100 rom 108 54 104 60 150 54 150 10 64 104 150 68 100 60 10 68 possible ordering of activities is (b) (c) (d) Monday uesday Wednesday hursday riday ind the total time taken by the caretaker each week using this ordering. tarting with adminton on Monday, use a suitable algorithm to find an ordering that reduces the total time spent each week to less than 7 hours. y deleting, use a suitable algorithm to find a lower bound for the time taken each week. Make your method clear. (otal 11 marks) 4. uring the school holidays four building tasks, rebuilding a wall (W), repairing the roof (R), repainting the hall (H) and relaying the playground (P), need to be carried out at a Junior chool. our builders,,, and will be hired for these tasks. ach builder must be assigned to one task. uilder is not able to rebuild the wall and therefore cannot be assigned to this task. he cost, in thousands of pounds, of using each builder for each task is given in the table below. (a) (b) ost H P R W 3 5 11 9 3 7 8 5 10 7 8 3 7 6 Use the Hungarian algorithm, reducing rows first, to obtain an allocation that minimises the total cost. tate the allocation and its total cost. You must make your method clear and show the table after each stage. tate, with a reason, whether this allocation is unique. (9) (otal 11 marks) (a) xplain why this problem is equivalent to the travelling salesman problem.
5. Victor owns some kiosks selling ice cream, hot dogs and soft drinks. he network below shows the choices of action and the profits, in thousands of pounds, they generate over the next four years. he negative numbers indicate losses due to the purchases of new kiosks. Use a suitable algorithm to determine the sequence of actions so that the profit over the four years is maximised and state this maximum profit. (otal 1 marks) 6. (a) xplain briefly the circumstances under which a degenerate feasible solution may occur to a transportation problem. (b) xplain why a dummy location may be needed when solving a transportation problem. he table below shows the cost of transporting one unit of stock from each of three supply points, and to each of two demand points 1 and. It also shows the stock held at each supply point and the stock required at each demand point. (c) 10 3 5 3 6 15 3 17 1 upply 6 47 15 61 48 1 68 58 17 emand 16 11 omplete the table below to show a possible initial feasible solution generated by the north-west corner method. 4 3 8 10 G 8 10 4 J K I H 1 0 3 4 (d) Use the stepping-stone method to obtain an optimal solution and state its cost. You should make your method clear by stating shadow costs, improvement indices, stepping-stone route, and the entering and exiting squares at each stage. (10) (otal 14 marks) 7. two person zero-sum game is represented by the following pay-off matrix for player. (a) (b) (c) (d) plays 1 plays plays 3 plays 1 5 7 plays 3 8 4 plays 3 6 4 9 ormulate the game as a linear programming problem for player, writing the constraints as equalities and clearly defining your variables. xplain why it is necessary to use the simplex algorithm to solve this game theory problem. Write down an initial simplex tableau making your variables clear. Perform two complete iterations of the simplex algorithm, indicating your pivots and stating the row operations that you use. (8) (otal 16 marks) 8. he tableau below is the initial tableau for a maximising linear programming problem. (a) (b) (c) asic variable x y z r s t Value r 7 10 10 1 0 0 3600 s 6 9 1 0 1 0 3600 t 3 4 0 0 1 400 P 35 55 60 0 0 0 0 Write down the four equations represented in the initial tableau above. aking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. tate the row operations that you use. tate the values of the objective function and each variable. (5) (9) (otal 16 marks) 1 3 0
b.v. x y z r s t Value Row Operations 9. 1 5 5 31 31 1 6 6 40 55 49 b.v. x y z r s t Value Row Operations 0 0 18 4 18 75 45 18 5 18 8 7 7 7 1 b.v. x y z r s t Value Row Operations he figure above shows a capacitated, directed network. he capacity of each arc is shown on each arc. he numbers in circles represent an initial flow from to. wo cuts 1 and are shown on the figure. (a) Write down the capacity of each of the two cuts and the value of the initial flow. (b) omplete the initialisation of the labelling procedure on the diagram below by entering values along arcs,, and. b.v. x y z r s t Value Row Operations b.v. x y z r s t Value Row Operations
0 5 0 31 0 0 30 45 0 55 7 7 18 18 0 7 (d) how your maximal flow pattern on the diagram below. (e) Prove that your flow is maximal. (otal 14 marks) (c) Hence use the labelling procedure to find a maximal flow through the network. You must list each flow-augmenting path you use, together with its flow. (5)
G G G cutting stitching fillingdressing cutting stitching filling dressing G G
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G H I J KL xyz P x y z r s t r s t P
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008 (adapted for new spec) 1. 1 1 17 x 19 5 10 5 8 1 1 18 16 3 (c) Use the north-west corner rule to obtain a possible pattern of distribution. (d) aking the most negative improvement index to indicate the entering square, use the stepping-stone method to obtain an optimal solution. You must make your shadow costs and improvement indices clear and demonstrate that your solution is optimal. (7) (e) tate the cost of your optimal solution. (otal 13 marks) 4. (a) xplain the difference between a maximin route and a minimax route in dynamic programming. 8 7 11 y 4 0 8 8 9 8 1 3 he diagram above shows a capacitated, directed network of pipes. he number on each arc represents the capacity of that pipe. he numbers in circles represent a feasible flow. (a) tate the values of x and y. (b) (c) List the saturated arcs. tate the value of the feasible flow. (d) tate the capacities of the cuts 1,, and 3. (e) (f) y inspection, find a flow-augmenting route to increase the flow by one unit. You must state your route. Prove that the new flow is maximal.. xplain what is meant, in a network, by (a) a walk, (b) a tour. (otal 11 marks) (otal 4 marks) 3. Jameson cars are made in two factories and. ales have been made at the two main showrooms in London and dinburgh. ars are to be transported from the factories to the showrooms. he table below shows the cost, in pounds, of transporting one car from each factory to each showroom. It also shows the number of cars available at each factory and the number required at each showroom. London (L) dinburgh () upply 80 70 55 60 50 45 emand 35 60 maximin route from L to R is to be found through the staged network shown above. (b) Use dynamic programming to complete a table below and hence find a maximin route. (10) (otal 1 marks) 5. (a) In game theory, explain the circumstances under which column (x) dominates column (y) in a two-person zero-sum game. Liz and Mark play a zero-sum game. his game is represented by the following pay-off matrix for Liz. Mark plays 1 Mark plays Mark plays 3 Liz plays 1 5 3 Liz plays 4 5 6 Liz plays 3 6 4 3 (b) Verify that there is no stable solution to this game. (c) ind the best strategy for Liz and the value of the game to her. (9) he game now changes so that when Liz plays 1 and Mark plays 3 the pay-off to Liz changes from to 4. ll other pay-offs for this zero-sum game remain the same. (d) xplain why a graphical approach is no longer possible and briefly describe the method Liz should use to determine her best strategy. (otal 16 marks) It is decided to use the transportation algorithm to obtain a minimal cost solution. (a) (b) xplain why it is necessary to add a dummy demand point. omplete the table below. L ummy upply 80 70 55 60 50 45 emand 35 60 100 6. our salespersons, Joe, Min-eong, Olivia and Robert, are to attend four business fairs,,, and. ach salesperson must attend just one fair and each fair must be attended by just one salesperson. he expected sales, in thousands of pounds, that each salesperson would make at each fair is shown in the table below. Joe 48 49 4 4 Min-eong 53 49 51 50 Olivia 51 53 48 48 Robert 47 50 46 43
7. (a) Use the Hungarian algorithm, reducing rows first, to obtain an allocation that maximises the total expected sales from the four salespersons. You must make your method clear and show the table after each stage. (b) tate all possible optimal allocations and the optimal total value. (10) (otal 14 marks) b.v. x y z r s t Value Row operations P b.v. x y z r s t Value Row operations he network in the diagram above shows the distances, in km, between eight weather data collection points. tarting and finishing at, lice needs to visit each collection point at least once, in a minimum distance. (a) Obtain a minimum spanning tree for the network using Kruskal s algorithm, stating the order in which you select the arcs. (b) Use your answer to part (a) to determine an initial upper bound for the length of the route. (c) tarting from your initial upper bound use short cuts to find an upper bound, which is below 630km. tate the corresponding route. (d) Use the nearest neighbour algorithm starting at to find a second upper bound for the length of the route. (e) y deleting, and all of its arcs, find a lower bound for the length of the route. (f) Use your results to write down the smallest interval which you are confident contains the optimal length of the route. (otal 16 marks) 8. he tableau below is the initial tableau for a maximising linear programming problem in x, y and z. asic variable x y z r s t Value 7 5 r 4 1 0 0 64 3 s 1 3 0 0 1 0 16 t 4 0 0 1 60 7 P 5 4 0 0 0 0 P b.v. x y z r s t Value Row operations P b.v. x y z r s t Value Row operations P (a) aking the most negative number in the profit row to indicate the pivot column at each stage, perform two complete iterations of the simplex algorithm. tate the row operations you use. You may not need to use all of these tableaux. (b) xplain how you know that your solution is not optimal. (9) (otal 10 marks)
Write your answers in the answer book for this paper. Paper Reference(s) 6690/01 dexcel G ecision Mathematics dvanced/dvanced ubsidiary Monday 1 June 009 Morning ime: 1 hour 30 minutes Materials required for examination Nil Items included with question papers nswer ook andidates may use any calculator allowed by the regulations of the Joint ouncil for Qualifications. alculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. 1. company, Kleenitquick, has developed a new stain remover. o promote sales, three salespersons, Jess, Matt and Rachel, will be assigned to three of four department stores 1,, 3 and 4, to demonstrate the stain remover. ach salesperson can only be assigned to one department store. he table below shows the cost, in pounds, of assigning each salesperson to each department store. 1 3 4 Jess 15 11 14 1 Matt 13 8 17 13 Rachel 14 9 13 15 (a) xplain why a dummy row needs to be added to the table. (b) omplete able 1 in the answer book. (c) Reducing rows first, use the Hungarian algorithm to obtain an allocation that minimises the cost of assigning salespersons to department stores. You must make your method clear and show the table after each iteration. (6) Instructions to andidates Write your answers for this paper in the answer book provided. In the boxes on the answer book, write your centre number, candidate number, your surname, initials and signature. heck that you have the correct question paper. nswer LL the questions. When a calculator is used, the answer should be given to an appropriate degree of accuracy. o not return the question paper with the answer book. Information for andidates ull marks may be obtained for answers to LL questions. he marks for individual questions and the parts of questions are shown in round brackets: e.g.. here are 8 questions in this question paper. he total mark for this question paper is 75. here are 8 pages in this question paper. he answer book has 16 pages. ny blank pages are indicated. dvice to andidates You must ensure that your answers to parts of questions are clearly labelled. You should show sufficient working to make your methods clear to the xaminer. nswers without working may not gain full credit. (d) ind the minimum cost. (otal 9 marks) Printer s Log. No. M3484 W850/R6690/57570 5/5/4/3 *M3484* his publication may be reproduced only in accordance with dexcel Limited copyright policy. 009 dexcel Limited. urn over M3484
. (a) xplain the difference between the classical and the practical travelling salesperson problems. he table below shows the distances, in km, between six data collection points,,,,,, and. - 77 34 56 67 1 77-58 58 36 74 34 58-73 70 4 56 58 73-68 38 67 36 70 68-71 1 74 4 38 71 - Rachel must visit each collection point. he will start and finish at and wishes to minimise the total distance travelled. 3. two-person zero-sum game is represented by the following pay-off matrix for player. plays 1 plays plays 3 plays 1 5 6 3 plays 1 4 13 plays 3 3 1 (a) Verify that there is no stable solution to this game. (b) Reduce the game so that player has a choice of only two actions. (c) Write down the reduced pay-off matrix for player. (d) ind the best strategy for player and the value of the game to player. (7) (b) tarting at, use the nearest neighbour algorithm to obtain an upper bound. Make your method clear. (otal 13 marks) tarting at, a second upper bound of 93 km was found. (c) tate the better upper bound of these two, giving a reason for your answer. y deleting, a lower bound was found to be 45 km. (d) y deleting, find a second lower bound. Make your method clear. (e) tate the better lower bound of these two, giving a reason for your answer. (f) aking your answers to (c) and (e), use inequalities to write down an interval that must contain the length of Rachel s optimal route. (otal 1 marks) M3484 3 urn over M3484 4
4. 5. While solving a maximising linear programming problem, the following tableau was obtained. 7 7 1 8 10 11 11 1 6 3 10 6 8 4 9 9 4 5 0 8 5 5 14 G 7 4 5 asic Variable x y z r s t value z 1 4 1 4 1 1 4 0 0 s 5 4 7 4 0 3 4 1 0 4 t 3 5 0 1 0 1 P 4 0 5 4 0 0 10 (a) Write down the values of x, y and z as indicated by this tableau. (b) Write down the profit equation from the tableau. 1 igure 1 (otal 4 marks) igure 1 shows a capacitated network. he capacity of each arc is shown on the arc. he numbers in circles represent an initial flow from to. wo cuts 1 and are shown in igure 1. (a) ind the capacity of each of the two cuts. (b) ind the maximum flow through the network. You must list each flow-augmenting route you use together with its flow. (otal 5 marks) M3484 5 urn over M3484 6