1. MBA@IICMR

2. Linear Programming Formulation 2 Components / Model Structure Objective FunctionDecision VariableConstraintsParameters Assumptions / Model Validity LinearityDivisiblityCertainityNon - Negativity

3. MBA@IICMR Linear Programming Formulation 3 Maximize: 4X1 + 7X2 + 5X 3 (Profit) - objective function Subject to: System Constraints 2X1 + 3X 2 + 6X 3 < 300 labor hrs 5X1 + X2 + 2X 3  200 lb raw material 3X 1 + 5X 2 + 2X 3  360 Individual Constraints X1 = 3 0 X 2  40 Non Negativity Constrain ts X 1, X 2, X 3  0

4. MBA@IICMR Formulating linear programming models involves the following steps: 4 Step 1 Define the problem/problem definition Step 2 Identify the decision variables or represent unknown quantities Step 3 Determine the objective function Step 4 Identify the constraints

5. MBA@IICMR Question 1 – LPP Formulation A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time. The profit earned on a racket and on a bat is Rs 20 and Rs 10 respectively. You are the operation manager of the company, Formulate the LPP so that profit is maximum 5

6. MBA@IICMR What are you are requested to find out? You are the manager of a company, whether you want to maximize the profit or minimize it?. Yes I know all of you unanimously want to maximize it. Hence we want to maximize our objective function. But it is already given in the problem. If it is not specified, you should decide whether to maximize or minimize the objective function based on the situation. 6

7. MBA@IICMR Steps to Formulate LPP 7 Step 1 The operations manager is interested to know how many units of Tennis Rackets and Crickets Bats must be manufactured so that the profit is maximum? Step 2 The decision variable of unknown quantities are No of Tennis rackets and No of Cricket Bats Let x denotes the number of Tennis Rackets and y denotes the number of Cricket Bats to be manufactured. Step 3 Determine the objective function Here objective function is to maximize Z= 20 x + 10 y with respect to the given constraints Step 4 Constraints 1.5 x + 3 y ≤ 42 ( Refer Working Note 1) 3 x + y ≤ 24 ( Refer Working Note 2) x, y ≥ 0 ( Non Negativity Constraint)

8. MBA@IICMR Working Note 1: 8 Machine Time required to manufacture 1 tennis racket is 1.5 hours. Here we manufacture x units of Tennis racket, hence Machine time taken to manufacture x units is (1.5) x hours. Machine Time required for tennis racket Similarly Machine time taken to manufacture 1 cricket bat is 3 hours. To manufacture y units of cricket bats, Machine time required is (3)y hours. Machine time required for cricket bat But they have total machine hours of 42. Availability This can be mathematically represented as 1.5 x + 3 y ≤ 42 – Constraint No 1 ( ≤ denotes machine time cannot exceed 42 hours) Machine Time Constraint

9. MBA@IICMR Working Note 2: 9 Craft man time for Tennis Racket Craft’s man Time required to manufacture 1 tennis racket is 3 hours Here we manufacture x units of Tennis racket, hence crafts’ man time taken to manufacture x units is 3 x hours. Craft man time for Cricket Bat Similarly Crafts man time taken to manufacture 1 cricket bat is 1 hour. To manufacture y units of cricket bats, Machine time required is y hours. Availability But they have total crafts man time of 24 hours. This can be mathematically represented as 3 x + y ≤ 24 – Constraint No 2 (≤ denotes Craft Man hours cannot exceed 24 hours )

10. MBA@IICMR Solution 10 Tennis Racket X Cricket Bat Y Availability Particulars Machine Time (h) 1.5342Constraint 1 Craftsman’s Time (h) 3124Constraint 2 Profit 2010 Objective Function objective function is to maximize Z= 20 x + 10 y with respect to the given constraints Constraints 1.5 x + 3 y ≤ 42 ( Refer Working Note 1 – Machine hour Constraint) 3 x + y ≤ 24 ( Refer Working Note 2 – Craft Man hours Constraint ) x, y ≥ 0 ( Non Negativity Constraint)

11. MBA@IICMR Question No 2 A merchant plans to sell two types of personal computers − a desktop model and a portable Laptop model that will cost Rs 30000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 300 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000. 11 X Y

12. MBA@IICMR 12 Desktop X Laptop Y AvailabilityParticulars Cost 300004000070 LakhsConstraint 1 Total Demand 300 unitsConstraint 2 Profit 45005000 Objective Function 30000 X +40000Y <= 70 Lakhs X +Y <=300 units Let X be the number of units of Desktop Let y be the units of Laptop Objective function is to maximise Z=4500 X +5000 y with respect to

13. MBA@IICMR Solution 13 Step 1 The merchant is interested to how many units of desktop and laptop should be sold so that his profit is maximum Step 2 The decision variable of unknown quantities are No of units of desktop and laptop to be Purchased. Let x denotes the number of desktop and y denotes the number of laptop to be Purchased Step 3: Determine the objective function Here objective function is to maximize Z= 4500 x + 5000 y with respect to the given constraints Step 4: Constraints 30000 x + 40000 y ≤ 70,00,000 – Constraint No 1 ( ≤Investment can't exceed 70 Lakhs) x + y ≤ 300 -Constraint No 2 (≤ denotes Total Demand won't exceed 300 ) x, y ≥ 0 ( Non Negativity Constraint)

14. MBA@IICMR Question No 3 The Manager of an oil refinery has to decide the optimum mix of two possible blending processes of which inputs and outputs per production run are given in the table. 14 ProcessInput ( Units)Output (units) Crude ACrude B Gasoline X Gasoline Y I5358 II4544

15. MBA@IICMR The maximum amounts of Crude A and B available are 200 units and 150 units respectively. Market requirements show that at least 100 units of Gasoline X and 80 units of Gasoline Y must be produced. Profits per production run from process I and Process II are Rs.300 and Rs.400 respectively. Formulate the LPP. ( SPPU May 2009) 15

16. MBA@IICMR 16 X Y

17. MBA@IICMR Steps for solving LPP 17 Step 1 The manager is interested to find out how many production of run of process I and process II should be carried out Step 2 Decision variables are Let X be the number of production run of Process I Let Y be the number of production run of Process II Step 3 The objective function is to maximize the profit i.e to maximize Z= 300 X + 400 Y with respect to the following constraints Step 4 Constraint 5 X + 4 Y ≤ 200 -Constraint1 (≤ Crude A can’t exceed 200 units- Crude A availability) 3 X + 5 Y ≤ 150 -Constraint2 (≤ Crude B can’t exceed 150 units- Crude B availability) 5 X + 4 Y ≥ 100 -Constraint3 (≥GasolineX should exceed 100units- market requirement) 8 X+ 4Y≥ 80 -Constraint4 (≥Gasoline Y should exceed 80 - market requirement) X, Y ≥ 0 (Non Negativity Constraint)

18. MBA@IICMR 18 ProcessInput ( Units)Output (units) Crude ACrude BGasoline XGasoline Y I5358 II4544 Availability / Demand 200150More than 100More than 80

19. MBA@IICMR SPPU April 2014 – MBA 2013 Pattern 19

20. MBA@IICMR 20 MACHINETIME PER UNIT (MINUTES) MACHINE CAPACITY MINUTES PER DAY PRODUCT 1PRODUCT 2PRODUCT 3 M1M1 232440 M2M2 4-3470 M3M3 25-430 PROFIT436 P Q R

21. MBA@IICMR 21 Step 1 To determine the no of units of Product 1, Product 2 and Product 3 to be manufactured so that profit is maximum Step 2 Decision variables are how many units of Product 1, Product 2 and Product 3 to be manufactured Let P be the number of units of Product 1 to be manufactured Let Q be the number of units of Product 2 to be manufactured Let R be the number of units of Product 3 to be manufactured Step 3 The objective function is to maximize the profit i.e to maximize Z= 4 P + 3 Q + 6 R with respect to the following constraints Step 4: Constrai nt 2P + 3Q + 2R ≤ 440- Constraint1 ( ≤denotes Machine M1 Capacity can't exceed 440 ) 4 P + 3 R ≤ 470 -Constraint2 ( ≤ denotes Machine M 2 Capacity can't exceed 470) 2P + 5 Q ≤ 430 -Constraint3 ( ≤ denotes Machine M 3 Capacity can't exceed 430) P, Q, R ≥ 0 (Non Negativity Constraint)

22. MBA@IICMR Question No 5 A garment manufacturer has a production line making two styles of shirts. Style I requires 200 grams of cotton thread, 300 grams of dacron thread, and 300 grams of linen thread. Style II requires 200 grams of cotton thread, 200 grams of dacron thread and 100 grams of linen thread. The manufacturer makes a net profit of Rs. 19.50 on Style 1, Rs. 15.90 on Style II. He has in hand an inventory of 24 kg of cotton thread, 26 kg of dacron thread and 22 kg of linen thread. His immediate problem is to determine a production schedule, given the current inventory to make a maximum profit. Formulate the LPP model. 22

23. MBA@IICMR 23 Style IStyle IIAvailabilityParticulars Cotton Thread 200 gm 24 Kg /24000gmConstraint 1 Dacron Thread 300gm200gm26 Kg/ 26000 gmConstraint 2 Linen Thread 300gm100gm22 Kg/ 22000 gmConstraint 3 ProfitRs.19.5Rs.15.9 Objective Function

24. MBA@IICMR Solution 24 Step 1 The manager has to determine the production schedule so that profit is maximum Step 2 Decision variables are how many units of Style I and Style II to be manufactured Let X be the number of units Style I to be manufactured Let Y be the number of units of Style II to be manufactured Step 3: The objective function is to maximize the profit i.e Z= 19.5 X + 15.9 Y with respect to the following constraints Step 4: Constrai nt 200 X + 200 Y ≤ 24000-Constraint1 ( Cotton Thread can't exceed 24000 units ) 300 X + 200 Y ≤ 26000-Constraint2 ( Dacron Thread cannot exceed 26000 units ) 300 X + 100 Y ≤ 22000- Constraint3 (Linen Thread cannot exceed -22000 units ) X,Y ≥ 0 (Non Negativity Constraint)

25. MBA@IICMR Question 6 An animal feed company must produce 2000 kg of a mixture consisting of ingredients X1, and X2 daily. X1 cost Rs. 30 per kg and X2Rs. 80 per kg. Not more than 800 kg of X1 can be used, and at least 600 kg of X2 must be used. Find how much of each ingredient should be used if the company wants to minimize cost. 25

26. MBA@IICMR Solution 26 ParticularsX1X1 X2X2 Required Quantity of Mixture 2000 Cost per Kg3080 Availability in Kg ≤ 800≥600 XY

27. MBA@IICMR 27 Step 1 The manager has to determine the how ingredients must be used to minimise the cost Step 2 Decision variables are how many units of X1 and X2 to be used in the mixture Let X be the number of units X 1 to be used Let Y be the number of units of X 2 to be used Step 3: The objective function is to minimize the cost i.e to minimize Z= 30 X + 80 Y with respect to the following constraints Step 4: X+ Y =2000 -Constraint 1 (Quantity of Mixture to be produced ) X ≤ 800- Constraint 2 (Quantity of X 1 should be used - X 1 Availability) Y ≥600 -Constraint 3 (Quantity of X 2 should be used - X 2 Availability) X,Y ≥ 0 ( Non Negativity Constraint)

28. MBA@IICMR LPP formulation SPPU OCT 2012 28

29. MBA@IICMR LPP Formulation – April 2013 29

30. MBA@IICMR April 2014 - SPPU 30

31. MBA@IICMR OCT 2015 SPPU 31