CONCEPT QUIZState True or False. 1. A linear programming problem is infeasible if there are any slack variables in the solution. 2. A solution becomes degenerate whenever there is a tie in the row to be replaced. 3. An optimal solution to a maximisation problem is arrived at when all values in the net evaluation row are positive or zero. 4. An optimal solution to a minimisation problem is arrived at when all values hi the net evaluation row are positive or zero. 5. The number of constraints in the primal is the same as the number of decision variables in the dual. 6. Each constraint, excluding non-negativity constraints, in the mathematical formulation of a linear programme generates one row in the simplex tableau. 7. Every inequality constraint in a linear programme adds exactly one variable to the problem. 8. All the rules and procedures of the simplex method are identical whether solving maximization or a minimization problem. 9. There should be no artificial variables in the final solution. 10. A maximization problem can be easily solved with the Two-phase method, if all constraints are of "less than equal to" type. Tick the correct answer/answers. 1. If the solution contains a variable that has a value of zero the problem is (a) Infeasible. (b) Unbounded. (c) Degenerate. (d) None of the above. 2. Shadow price is: (a) The cost of a unit of a resource. (b) The contribution of one unit of a resource towards the objective function. (c) The amount by which a resource can be increased without changing the solution. (d) The contribution of one unit of a scarce resource towards the objective function. 3. In the case of a maximization problem, the incoming variable has: (a) A value of zero in the net evaluation row. (b) A positive value in the Zj row. (c) The least negative value in the net evaluation row. (d) The highest positive value in the net evaluation row. 4. The outgoing variable in a simplex pivot operation is the variable with: (a) The least replacement ratio. (b) A negative replacement ratio. (c) The maximum positive replacement ratio. (d) The least positive replacement ratio. 5. In a maximization problem, if a constraint is of the 'greater than or equal to' type, the artificial variable is assigned: (a) A very large value. (b) A very small value. (c) A very large positive value, i.e. M. (d) A very large negative value, i.e. -M. 6. Which of the following statements is true about a primal linear programme and its corresponding dual: (a) The optimal value for the primal is greater than that of the dual. (b) The shadow prices of the primal at optimality are the values of the structural variables of the dual at optimality. (b) The number of tableaus is determined by the number of variables. (d) 10. Determine the . Define slack and surplus variables in a linear programming 3. Explain various steps of the simplex method involved in the computation of an optimal solution to a linear programming problem. 6000. 6.Zj row with the Zj row. 8000 and 10000. (c) Dividing the quantity column with the column of the incoming variable. (d) The range of the resource availability over which the shadow prices remain constant. (d) To convert inequalities of the 'less than or equal to' type into equalities. Right-hand side ranging is used to determine: (a) The availability of resources. be determined by: (a) Dividing the Cj . Artificial variables are required: (a) Only in a minimisation problem. Explain the primal dual relationship. (c) To convert inequalities of the 'greater than or equal to' type into equalities. Explain the meaning of basic feasible solution and degenerate solution in a linear programming problem. 7.(c) Each problem does not have a dual. In the simplex procedure: (a) Each successive tableau presents a solution superior to the one preceding it. They have limited their cosmetic and neurosurgery practice to a total of 150 hours a week. Health Centre is a clinic specializing in four types of patient care: dermatology cosmetic surgery. (c) The range of the resource availability over which the solution remains the same. (b) The requirement of resources. 9. 4. Right-hand side ranging can 8. problem. None of (c) the above. QUESTIONS 1. respectively to profits of the clinic. (b) Dividing the quantity column by the column representing the slack of the variable whose range we are determining. (b) Only in a maximisation problem. The number of tableaus is determined by the number of constraints. 5. (d) None of the above. Past data reveals the following time requirements limitations: Hours required per patient Speciality Dermatology Cosmetics Obstetrics Neurosurgery Hours available per week Lab 5 5 3 2 200 XRay 8 2 1 4 140 Therapy 10 1 0 8 110 Surgery 8 4 Doctonl 14 j 10 8 12 320 6 16 240 The doctors have access to as many patients as they wish. (d) Both the primal and the dual are either maximisation or minimisation problems. What do you understand by shadow prices? What is the managerial implication of shadow prices? 7. A patient in each of these specialties contributes Rs 4000. obstetrics and neurosurgery. Explain the use of artificial variable in linear programming. 2. Each chair must receive work in each class and the time in hours required for each chair in each class is given below: Chair I Year 2 3 2 // Year /// Year 3 2 4 A B 4 3 1 . There are three wood working classes—I year. II year and III year. The Annual Handmade Furniture Show and Sale occurs next month and the School for Vocational Training is planning to make furniture for the sale. 8.optimal patient mix on a weekly basis. B and C. at the school and they have decided to make three styles of chairs—A. Rs 350 and C. B. determine the number of each type of chair to be produced for profit maximisation. Assuming abundant supply of raw materials and unlimited demand for finished products. Each chair is first made in the carpentry shop and then varnished. waxed and polished in the finishing shop. assuming that whatever is produced can be sold. A furniture company can produce 4 types of chairs. for which the profits per unit are Rs 10. using the simplex method. Determine the most profitable mix. 9. The teacher has determined that the profit from each type of chair will be: A. 4 and 2 and 1.During the next month there will be 120 hours in the I Year class. respectively. (c) Show that the total available hours of X and Y have been fully utilized and there is surplus hours of Z. respectively. Three products A. Corresponding requirements for rings and valves are 1. packing and allied formalities is 100. 5 and 6 hours. 160 hours in the II Year class and 100 hours in the III Year class. . B and C are produced at three machining centres X. wants to decide the most profitable mix. A factory engaged in the manufacture of pistons. Y and Z Each product involves operations at each of the machine centres. The time required for each operation on various products is given in the table below: Machine centres Product X 10 2 1 100 Y 7 3 2 77 Z 2 4 1 80 Profit per unit 12 3 1 A B C Available hours (a) Formulate a linear programming problem on the basis of the above information. Man hours required in each shop are: Chair type Shop Carpentry Finishing Contribution 7 4 1 120 2 9 1 200 3 7 3 180 4 10 40 400 Total number of man hours available per month in Carpentry and Finishing shops are 6000 and 4000. respectively. 600 and 300. The teacher of the wood working classes feels that a maximum of 40 chairs can be sold at the show. (b) Find a suitable product mix so as to maximise profit. It takes 1 hour of preparatory work. Rs 400. 10 hours of machining and 2 hours of packing and allied formalities for a piston. 11. Rs 300. Determine the optimal mix of chairs using the simplex method. 6 and 4 respectively. 10. The total number of hours available for preparatory work. rings and valves. If there is any other optimal product mix. 13. Vehicles B and C require a crew of two men each. With this capacity determine the company's optimal product-mix and total maximum profit. semi finished— Rs 250 per unit and finished—Rs 350 per unit. Vehicle C is a modified form of Vehicle B. The manufacturer can sell all the items that he can produce. 4 hours of machining. 240 and 100 worker-days. The firm has a permanent contract to supply at least 2 tons of X and at least 3 tons of Y per day to another company. and 4 hours of plating. it carries sleeping quarters for one driver and that reduces its carrying capacity to 18 tons and raises the cost to Rs 150000. 1 hour for machining and 1 hour for plating. the daily maximum possible number of machine hours is 360. could be run for an average of 18 hours per day. The company has 150 drivers available each day and would find it difficult to obtain further crews. but whereas B would be driven 18 hours per day with three shifts. How many vehicles of each type should be purchased if the company wishes to maximise its capacity in ton-km per day? 15. and if driven on three shifts per day. Vehicle A has a 10 ton payload and is expected to average 35 km per hour. respectively in the three production processes. The machining department has a capacity of 88 hours and the plating department has a capacity of 40 hours per week. A company manufactures three models of cars. It costs Rs 130000. Model A requires 60. (c) Finished—5 hours of casting. It costs Rs 80000. Each ton of X requires 20 machine hours of production time and each ton of Y requires 50 machine hours of production time. All the firms output can be sold. 4 hours of machining and 2 hours of plating. Model C requires 200. Rs 15000 and Rs 30000 respectively. identify that too. There is a backlog of orders with the company. He may sell his product as: (a) Raw castings—5 hours for casting. The profit per unit of sale is: Raw casting—Rs 150 per unit. Model B requires 100. (b) Semi finished—5 hours of casting. and the profit made is Rs 80 per ton of X and Rs 120 . A trucking company with Rs 4000000 to spend on new equipment is contemplating three types of vehicles. Maintenance facilities are such that the total number of vehicles must not exceed 30. A manufacturer has to decide how much finishing to perform on his product prior to sale. X and Y requiring the same production capacity. and an average worker is on the job for 200 working days a year. Vehicle A requires a crew of one man. Vehicle B has a 20 ton payload and is expected to average 30 km per hour. A firm makes two products X and Y and has total production capacity of 9 tons per day. 100 and 80 worker-days in three production processes. 30 and 15 respectively.12. The expected profit for each model is Rs 7500. 360 and 160 worker-days. The weekly production capacity of the casting department is 130 hours. C would average 21 hours per day. The number of workers employed in the three production processes is 15. What should the product mix be to maximise profits? 14. The profit margin per typewriter is Rs 150. The ABC Candy Company makes three different types of candy bars. in grams. Also because of lack of company cars. 1. respectively. The ABC Company sells two types of porch furniture. 2. It makes a profit of Rs 100 on each glider and Rs 40 on each chair. For the coming week three of the salesmen will be on leave. 36 and 25 typewriters per week. Jain. leaving only 12 men for duty. The selling expenses for salesmen per week for salesmen in each area are Rs 800 for Market 1 and Rs 700 and Rs 500 per week for Markets 2 and 3. ABC has 900 square feet space and 30 hours of labour available for assembly. The ingredients. 16. The budget for the next week is Rs 7000. Each glider requires 40 square feet of display space and each chair requires 25 square feet of display space. His market research data indicated that two mixtures of three screw types (1. Mr. for each candy bar are as follows: Candy bar A B C Availability Per unit contribution Chocolate 12 6 10 25000 Nuts 4 10 2 15000 Caramel 15 8 15 30000 Weekly demand Rs. It is required to determine the production schedule for maximum profit and to calculate this profit. 2 and 3) properly priced would be most acceptable to the buyer.50 900 Very large Very large What should be the product mix to maximise the profits? 17. The relevant data is: Mixture Specifications > 50% Type 1 A < 30% Type 2 any quantity of Type 3 B > 35% of Type 1 < 45% of Type 2 any quantity of Type 3 Rs 4 per kg Rs 5 per kg Selling price . Determine how many salesmen should be assigned to each area so as to maximise the profits.per ton of Y. 19. Salesmen in the other two markets can sell on the average. gliders and chairs. Formulate and solve the LP model.00 Rs. Market 1 is in an urban area and salesmen can sell on the average 40 typewriters per week. maximum of 5 salesmen can be allocated to Market 1. is trying to decide how to allocate his salesmen to the company's three primary markets.50 Rs. A certain manufacturer of screw fastenings found that there was a market for packages of mixed screw sizes. respectively. It takes 1% hours to assemble a glider and 2/3 hours to assemble a chair. 2. 18. The sales manager wants at least two chairs displayed for every glider displayed. the marketing manager of ABC Typewriter Company. 50 2. the total available capacity and the demand are given in the following table: Product Production line A B C Demand 1 150 200 160 2000 2 100 100 80 3000 3 500 760 890 3000 IVj 4 400 400 600 6000 tax. Line B Rs 5000 per day and Line C Rs 4000 per day. A piano manufacturer manufactures three types of pianos. There are three production lines on which the products could be processed. Formulate and solve the dual of Question 21.70 What production should the manufacturer schedule for greatest profits assuming that he can sell all that he manufactures? 20. . Solve using suitable software package. The data below give the production hours per unit in each of the three operations. if Line A requires Rs 3000 per day. Four products have to be processed through the plant.For these screws. 21. The rates of production in units per day. line capacity (days) 20 20 18 Formulate the above as a linear programming problem to minimise the cost of operations. 22.50 3. Operations (hrs) Piano types A B C Max time available 1 2 4 2 600 2 5 2 3 400 3 5 2 10 900 Profit per unit (OOs of Rs) 30 40 20 How many units of each type of piano should be produced to maximise the total profit? Write the dual and use it to check the optimal solution. maximum time available and profit per unit. plant capacity and manufacturing costs are given below: Screw type 1 2 3 Plant capacity (kg /day Manufacturing cost x 100) (Rs/kg) 10 10 6 4. correspondence boxes and lunch boxes are Rs 20. filing cabinets. leather.2 0. and yields a profit of Rs. wood and glue are required in the amount as shown below: Resources required for one unit Product Leather (kg) Wood (sq m) 4 7 28000 Glue (litres) 0.25 2200 (a) Formulate the LP model and find the optimal solution. each requiring a different manufacturing technique. The finishing department can handle either 1500 trousers or 2000 shirts (or a combination of the two) each day. 25. 400.2 1400 A B Availability 0. The Deluxe machine requires 18 hours of labour. Its inputs are sheet metals of two different thicknesses. filing cabinets. (b) Which resources are fully consumed? How much of each resource remains unused? (c) What are the shadow prices of the resources? 24.50 0. Formulate and solve this as a linear programming problem. 200. Waste cans Sheet metal A Sheet metal B Labour 6 0 4 Filing cabinets Correspondence boxes Lunch boxes 0 10 8 2 0 2 3 0 3 The sales revenue per unit of waste cans. The company's objective is profit maximisation. A knitting machine can produce 1000 trousers or 3000 shirts (or a combination of the two) each day. and 190 units of labour. The marketing department requires that at least 400 trousers be produced each day.23. A marketing forecast has shown that the monthly demand for the Standard machine to be no more than 250. Management wants to know the number of each model to produce monthly that will maximise total profits. Product A offers a profit of Rs 25 per unit and product B yields a profit of Rs 40 per unit. Input output relationships for the company are shown in the table. Rs 90 and Rs 20. called A and B and manual labour. and yields a profit of Rs. What is the company's optimal sales revenue? 26. file boxes for correspondence and lunch boxes. The Standard machine requires 3 hours of labour. A metal product company produces waste cans. 9 hours of testing. If the profit from the trousers is Rs 40 and that from a shirt is Rs 15. A company manufactures two different kinds of machines. 4 hours of testing. To manufacture the products. respectively. There are 225 units of Sheet metal A available in the company's inventory. Rs 400. how many of each type should be produced? . There are 800 hours of labour and 600 hours of testing available each month. 300 of Sheet metal B. 0 minutes of punch press time and 2. 29.0 minutes of punch press time and 1. B and C are to be manufactured.0 minutes of assembly time. 28. each of which requires three operations as part of the manufacturing process. Find the quantities of A and B to be produced in the next week so as to maximise profits.50. The capacity of the punch press is 20 hours per week.118 Quantitative Techniques for Decision Making 27. Each product passes through three processes. Product C requires 2. Product A gives a profit of Rs 18 per unit and one unit of Product B gives Rs 15.5 minutes of welding time and 2. A chemical manufacturing company produces two products A and B. Three products A. . 24 and 20. B and C is Re 0. A manufacturing company makes three products. of welding 10 hours per week and of assembly section 25 hours per week.4 minutes of punch press time and 5. Find the optimal feasible solution and the maximum profit. The processing time in hours for each of the two products in each process is given below: Process Product A Product B 1 2 3 2 7 4 5 2 3 The total hours available for each process in a week are 30.60. Product B requires 3. Also determine the shadow prices of the three resources.70 and Re 0. Product A requires 2. The company can sell all of the products it can manufacture but its production capacity is limited by the capacity of its operations centres. Additional data concerning the company is as follows: Manufacturing requirements hour/unit Product Centre 1 Centre 2 Cost (Rs) Centre 3 2 1 2 80 Selling price (Rs) A B C Hours available 1 3 2 160 3 4 2 120 11 12 10 - 15 20 16 What should the product mix be? Write the dual of the given problem. Profit per unit of A.5 minutes of welding time. respectively.5 minutes of assembly time. respectively. Re 0. Standard and Economy tour packages to offer for this charter. (c) At least 30 per cent must be of the Economy type.A timber merchant manufactures three types of plywood. The travel agent has hired an aircraft for a flat fee of Rs 200000 for the entire trip. The problem for the travel agent is to determine the number of Deluxe. board and lodging and selected tour options. Tour plan Deluxe Standard Economy Price (Rs) 10000 7000 6500 Hotel costs (Rs) 3000 2200 1900 Meals and other expenses (Rs) 4750 2500 2200 In planning the trip the following considerations must be taken into account: (a) At least 10 per cent of the packages must be of the deluxe type. The data given in the following table shows the production hours per unit in each of three production operations. meal plans and tour options. Operations (hrs) Plywood Grade A Grade B Grade C Max time available / 2 5 10 900 II 2 5 3 400 III 4 2 2 600 Profit per unit (Rs) 40 30 20 How many units of each grade of plywood should be produced to maximise the total profit? 31. (d) The maximum number of Deluxe packages available in any aircraft is restricted to 60. . surface transportation. The charter trip is restricted to 200 persons and past experience indicates that there will be no problem in getting 200 persons. These three plans each differ according to seating and service on the flight. Determine the number of packages to offer in each type so as to maximise profits. (e) The hotel desires that at least 120 tourists should be on the Deluxe and Standard packages together. The following table summarises the estimated price for the three packages and the corresponding expenses for the travel agent per person. The eight day seven night package includes the fare for the round trip travel. (b) At least 35 per cent but not more than 70 per cent must be of the Standard type. maximum time available and profits per unit. quality of accommodation. A local travel agent is planning a charter trip to a major sea resort. A. X. 20 minutes and 1 hour.32. A company makes three products X. 35. A manufacturer produces four products. (b) If XYZ Company decides to rent out its capacity to another almirah manufacturer—ABC Company. C and D. painting and packing departments is 600 hours. each of which is processed on three machines. respectively. The Type III requires 40 minutes each of assembly. The time required for manufacturing one unit of each of the four products and capacity of each machine is indicated as follows: . XYZ Company Y\as three departments—assembly.The total time available at assembly. and Can make three types of almirahs. An almirah of Type 1 requires one hour assembly. 40 minutes of painting and 20 minutes of packing time. 1500 and 2000 as profit respectively. Y and Z. Lathe and Assembly. (a) Determine the number of almirahs of each type that should be produced in order to maximise the profits. Type II and Type III almirahs yield Rs 1000. what should it charge as rental rates? 34. B. the profit contribution of the products and the total time available in each department are shown in the table below: Time required per unit Product Drill 3 6 7 210 Lathe 3 5 4 240 Assembly 8 10 12 260 Profit per unit X Y Z Hours available 9 15 20 The marketing department indicates that the sales potential for Products X and Y is unlimited but for the Product Z it is only 30 units. painting and packing. which flow through three departments: Drill. Type I. respectively Similarly almirah of Type II 80 minutes. Determine the optimal j product mix. Y and Z. The hours required by each product in each department. 400 hours and 800 hours. respectively. painting and packing time . Product Machine X A B C D Capacity 1. protein and vitamin content of the three basic foods are given in the following table: Nutritional elements Food Calories Proteins Vitamin A Vitamin B Cost per serving (Rs) 350 250 100 75 1. The specifications for Hi-Pro have been established by a panel of medical experts. ABC Foods Company is developing a low calorie high protein diet supplement called Hi-Pro.50 Units of nutritional elements (per 100 gm basic food) 1 250 300 150 125 2.5 2 4 3 550 Processing time in hrs Machine Y 4 1 2 1 700 Machine Z Profit per unit 2 3 1 2 200 4 6 3 1 (a) What is the optimal product mix? What is the maximum profit? (b) Which machine (s) has(have) excess capacity? How much? (c) If the profit contribution from Product B increases by Rs 2 per unit. will the optimal product mix change? (d) If machine Y is to be shut down for 50 hours due to repairs. will the product mix change? (e) What are the shadow prices of the machine hours on the three machines? 36.00 Food 2 Food 3 200 150 75 150 1. Food 2 and Food 3 should be used? . These specifications along with the calorie.20 Hi-Pro 300 200 100 100 What quantities of Food 1.