Linear Programming Problem with Example
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Defining The Problem Of Beyond Era (Pvt) LtdThe beyond era pvt ltd is company they produce and selling water bottle to market. Last month they imported new 2 water bottle product machine from china. Water bottles filing machineWater bottles packing machine Those days very hot and have high demand for water bottle. They produce two size wetter bottles.500 ML bottle1000 ML bottleThe marketing team has concluded that the company could sell as much of either product per day as could be produced by these machine. However, those tow machine hours are limiting factors. According to machine manufacturing company those tow machine only can work per day as follows Filing Machine  12 HPacking machine 10 H So beyond era pvt ltd management thinking about how many no of water bottle pack both tow size should be produced to maximize their profit .this is product mix type linear programming problem. so we have to select best product mix using linear programmer model .Identified Some Date To Solved This ProblemHow many bottles in one pack? 100 PCSNumber of time of produce time use in each machine to produce each 2 bottle pack? it’s as follows  MACHINE TIME REQUIRED TO PRODUCE  100 PCS BOTTLE PACKAVAILABLE HOURS PER DAY500 ML BOTTLE PACK1000 ML BOTTLE PACKFILING2M4M12*60 = 720MPACKING3M2M10*60 = 600MHow much each bottle pack profit? 500 ml 100 Pcs pack Rs 3000/=1000 ml 100 Pcs pack Rs 5000/=According to above data beyond era pvt ltd management want to know that what is the best product mix to produce to reach the maximum profit .Mathematical Formulation Beyond era management is looking best product mix of both products to maximize their profit. So, we identified this is product mix linear programming problem . using graphical and simplex method we will advice to management to select their best production mix to maximize their profit .  Data we have collect as followsMACHINE TIME REQUIRED TO PRODUCE  100PCS BOTTLE PACKAVAILABLE HOURS 500 ML BOTTLES PACK(X)1000 ML BOTTLES PACK(Y)PER DAYFILING2M4M12*60 = 720MPACKING3M2M10*60 = 600MPROFIT PER UNITRS 3000RS 5000

The objective is to maximize profit.The constraints areThe hours of filing machine time use cannot exceed 12 hours per day.The hours of packing machine time use cannot exceed 10 hours per dayThe decision variables representing the actual decision we will make areX = number of 500 ml bottles pack to be produced per dayY = number of 1000 ml bottles pack to be produced per day.So we can develop mathematical function as follows Profit maximization = Rs3000X  +  Rs5000YFiling machine time using =  2X + 4Y  <  720Packing machine time using = 3X + 2Y < 600Both of these constraints restrict production capacity and affect to profit  The number of production X and Y greater than or equal to 0 so X,Y > 0Develop A Solution 3.1) Graphical solution To use above data we solved linear programmed using graphical method as follows.We can short collect data like this XYZ30005000  FILING 24<720PACKING32<600The fist step in solving the problem is to identify a set of region of feasible solutionTo do this we plot each constraint equation on a graph.When 1000ml bottles production is 0 500ml bottles production by fully using filling machine as follows 2x + 4y    <= 720 2x + 4(0) <= 720         2x   <= 720            X   = 720/2            X   = 360Similarly no 500ml bottle2x + 4y    <= 720 2(0) + 4Y <= 720         4Y  <= 720            Y   = 720/4            Y  =  180After solving above tow equations we can make flowing table for filing machineFILINGXY36000180Same as we can make table for packing machine as follows 3x + 2y    <= 600 3x + 2(0) <= 600         3x   <= 600