Shah 2
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Linear programming problem • A firm is engaged in the mfring of 2 products P1 and P2. Each unit of P1 requires 2 kg of RM & 4 labour hrs for processing whereas each unit of P2 requires 5 kg of RM & 3 LH of the same type . Every week the frim has the availability of 50 kg of RM & 60 LH . One unit of P1 sold gets profit Rs 20 & one unit of P2 sold gives Rs 30 as profit . Formulate this problem as linear programming problem to determine as to how many units of each product should be produced per week so that
Step 1 : Objective Function Goal is to maximise profit by selling P1 and P2. Let P1 and P2 be represented by x1 and x 2. Therefore Total profit = Z ( objective function ) is given by 20x1 + 30x 2 Total profit represented by Z = 20x1 + 30x 2
Step 2 : Establishing the constraints • Resources are in a limited supply • So we represent this by inequality signs ≤ or ≥ • The Raw Material RM constraints are • Each Unit of P1 requires 2 kg of RM • Each Unit of P2 requires 5 kg of RM • Total consumption cant exceed 50 kg / week So 2x + 5x ≤ 50 is the material
Step 2 Cont… Labour Constraints • P1 requires 4 labour hrs for production • P2 requires 3 labour hrs for production • Total available labour hours =60 Hence Labour constraint will be 4x1 + 3x 2 ≤ 60 Non Negativity Constraints are that x1 & x 2 ≥ 0
Complete LPP formulation Now we can write the problem in complete LPP form as follows Z = 20x1 + 30x 2 Profit Function 2x1 + 5x 2 ≤ 50 Material Constraint 4x1 + 3x 2 ≤ 60 Labour Constraint x1 & x 2 ≥ 0 Non
Terminology of LPP 1. Decision Variables 2. Well Defined Objective Function 3. Presence of constraints or restrictions 4. Non negativity constraints 5. Linearity i.e proportional relationship
Media Selection LPP Problem Lintas Advt Agency has been asked to help a client develop an advt budget for the introduction of a improved product . Advt funds are to be spent upon both television & magazine announcements. The client has specified the following requirements ii.No more than Rs 2 lakh has to be spent on TV ads iii.At least one lakh must be spent upon magazine ads
Media Selection LPP Problem cont… i. Advt costs Rs 5000 in TV & Rs 2000 in magazine respectively . Total exp should not exceed Rs 5 Lacs . ii. From previous Mkting Res results agency has determined that 300 people are exposed to message for each rupee spent on TV., and 200 people are exposed to message for each rupee spent on magazine .
Media Selection LPP Problem cont… • Determine the amounts to be spent on TV and magazine ads to maximise the total nb of people exposed to the new product Advt _______________________________________ Sol: Max ( Total exposure) Z = 300 x1 + 200 x2 Subject to 5000 x1 ≤ 200,000 TV cost constraint 2000 x2 ≥ 100,000 Mag Cost constraint
Z = 300 x1 + 200 x2 Subject to 5000 x1 ≤ 200,000 TV cost constraint 2000 x2 ≥ 100,000 Mag Cost constraint 5000 x1 + 2000 x2 ≤ 500,000 Expenditure const Where x1 & x 2 ≥ 0 Non negativity constraints
Diet Problem
Step 1 : Objective Function Goal is to maximise profit by selling P1 and P2. Let P1 and P2 be represented by x1 and x 2. Therefore Total profit = Z ( objective function ) is given by 20x1 + 30x 2 Total profit represented by Z = 20x1 + 30x 2
Step 2 : Establishing the constraints • Resources are in a limited supply • So we represent this by inequality signs ≤ or ≥ • The Raw Material RM constraints are • Each Unit of P1 requires 2 kg of RM • Each Unit of P2 requires 5 kg of RM • Total consumption cant exceed 50 kg / week So 2x + 5x ≤ 50 is the material
Step 2 Cont… Labour Constraints • P1 requires 4 labour hrs for production • P2 requires 3 labour hrs for production • Total available labour hours =60 Hence Labour constraint will be 4x1 + 3x 2 ≤ 60 Non Negativity Constraints are that x1 & x 2 ≥ 0
Complete LPP formulation Now we can write the problem in complete LPP form as follows Z = 20x1 + 30x 2 Profit Function 2x1 + 5x 2 ≤ 50 Material Constraint 4x1 + 3x 2 ≤ 60 Labour Constraint x1 & x 2 ≥ 0 Non
Terminology of LPP 1. Decision Variables 2. Well Defined Objective Function 3. Presence of constraints or restrictions 4. Non negativity constraints 5. Linearity i.e proportional relationship
Media Selection LPP Problem Lintas Advt Agency has been asked to help a client develop an advt budget for the introduction of a improved product . Advt funds are to be spent upon both television & magazine announcements. The client has specified the following requirements ii.No more than Rs 2 lakh has to be spent on TV ads iii.At least one lakh must be spent upon magazine ads
Media Selection LPP Problem cont… i. Advt costs Rs 5000 in TV & Rs 2000 in magazine respectively . Total exp should not exceed Rs 5 Lacs . ii. From previous Mkting Res results agency has determined that 300 people are exposed to message for each rupee spent on TV., and 200 people are exposed to message for each rupee spent on magazine .
Media Selection LPP Problem cont… • Determine the amounts to be spent on TV and magazine ads to maximise the total nb of people exposed to the new product Advt _______________________________________ Sol: Max ( Total exposure) Z = 300 x1 + 200 x2 Subject to 5000 x1 ≤ 200,000 TV cost constraint 2000 x2 ≥ 100,000 Mag Cost constraint
Z = 300 x1 + 200 x2 Subject to 5000 x1 ≤ 200,000 TV cost constraint 2000 x2 ≥ 100,000 Mag Cost constraint 5000 x1 + 2000 x2 ≤ 500,000 Expenditure const Where x1 & x 2 ≥ 0 Non negativity constraints
Diet Problem
