Graphical Solution of Linear Programming Problems CBSE Class 12 Maths

[{"selector":"#anim-b08829f1-b2ff-47b6-8657-f88667b7e21d","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-39396505-8485-45ed-95f8-28c110582946","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] By Team Aakash Byju's

[{"selector":"#anim-c291b1fa-f411-4d8b-a591-4e74f0a449dc","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-08da62d7-bb4e-43aa-94e6-c5d0b7e70cd4","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] ‘Linear programming’ is a method to attain the best outcome in a mathematical model whose requirements are expressed by linear relationships.

Click Here To Attend Free Trail Class

[{"selector":"#anim-2e188d9d-1d0f-4e63-9d9d-634842d27335","keyframes":[{"offset":0,"transform":"translate3d(0, -707.778%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":0.29,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"},{"offset":0.45,"transform":"translate3d(0, -199.0271736%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":0.61,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"},{"offset":0.71,"transform":"translate3d(0, -67.6635768%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":0.8,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"},{"offset":0.85,"transform":"translate3d(0, -25.409230200000003%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":0.92,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"},{"offset":0.96,"transform":"translate3d(0, -11.0413368%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":1,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"}],"delay":0,"duration":1300,"fill":"both"}] Download Aakash APP

Let us see how the solutions for different problems in linear programming are expressed.

[{"selector":"#anim-9aae573f-7e2a-4407-8788-7444f451a1f1","keyframes":{"transform":["translate3d(114.92064%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(0.4, 0.4, 0.0, 1)","fill":"both"}] [{"selector":"#anim-8c4a418c-a1d9-4ef4-ba15-45979cd2eef8","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(0.4, 0.4, 0.0, 1)","fill":"both"}] [{"selector":"#anim-4df0ee56-35de-46dc-9c8c-ce4a47b52365","keyframes":{"transform":["scale(0.15)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(0.4, 0.4, 0.0, 1)","fill":"forwards"}]

Step 1: List out the given constraints i.e., the general constraints and non-negative constraints.

[{"selector":"#anim-9dbe649d-3160-460a-a87b-386baf548534","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-03b40cfc-9de4-46d5-a06e-27c9b05f9a19","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}]

[{"selector":"#anim-0372d5c6-0119-44c0-9273-ed088919fdc7","keyframes":{"opacity":[1,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-71729d7c-d8cd-4453-8a2e-e0d3d07c4515","keyframes":{"transform":["scale(3)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] Step 2: Plot the graph for the given constraints and find the feasible region. Let us see how the graph looks for given constraints.

[{"selector":"#anim-96c9ca5a-cc2e-41cd-bd73-f8a5fe2d175c","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-562dc1ef-601a-41a0-9702-d7e2075c93a0","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] Step 3: This step is to identify the coordinates of the feasible region that we get from step 2. For the given example, we see that (1, 2) is the coordinate for the feasible region.

[{"selector":"#anim-0a557ad6-5d44-4e29-8cdc-400177b76fed","keyframes":{"opacity":[1,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-724fc05d-00de-4a8f-8bac-5aba4f5bd37c","keyframes":{"transform":["scale(3)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] Step 4: See what is being asked in the problem. Is it to maximize the given linear program or to minimize it?

[{"selector":"#anim-241ed00b-bd84-4d1b-a304-26206c483a80","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-ad9e86d9-7d4a-4756-839c-8e755018cf17","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] Substitute all intersecting points on the graph to the given linear program to find the maximum or minimum values.

[{"selector":"#anim-e19c3059-cffa-4384-be69-c4ee2dcff400","keyframes":{"opacity":[1,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-5e579e19-4dee-440b-93c9-8cc83836b595","keyframes":{"transform":["scale(3)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] Hence the given function is maximum at (0,3) and minimum at (0,0).

[{"selector":"#anim-39e98450-63fa-4694-89c8-167bd16fbedd","keyframes":{"opacity":[1,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-4c706e42-b131-43db-819b-766ffc68a364","keyframes":{"transform":["scale(3)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] Let us see another example of graphical solutions. Minimize B = -3x+4y subject to x+2y≤8 and 3x+2y≤12, x, y ≥0

[{"selector":"#anim-f3f9fe03-5fc1-4f1c-93af-eb231a0801ec","keyframes":{"opacity":[1,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-e48ac307-b677-45f0-8086-305c496b8307","keyframes":{"transform":["scale(3)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] Hope you have got the idea of finding graphical solutions for given linear programming. Now, share these with your friends and help them to score well.