Find solution using 0-1 Integer programming problem method
MAX Z = 6500x1 + 7000x2 + 2250x3 + 2500x4
subject to
700x1 + 850x2 + 300x3 + 350x4 <= 1200
550x1 + 550x2 + 150x3 + 200x4 <= 700
400x1 + 350x2 + 100x3 <= 400
x1 + x2 >= 1
-x3 + x4 <= 1
and x1,x2,x3,x4 >= 0

Solution:
To apply branch and bound method, the following 4 constraints have to be added to the LP model.
`x_1<=1`

`x_2<=1`

`x_3<=1`

`x_4<=1`

Solution steps by BigM method

Max `Z` `=` `` `6500` `x_1` ` + ` `7000` `x_2` ` + ` `2250` `x_3` ` + ` `2500` `x_4`

subject to

`` `700` `x_1` ` + ` `850` `x_2` ` + ` `300` `x_3` ` + ` `350` `x_4` ≤ `1200` `` `550` `x_1` ` + ` `550` `x_2` ` + ` `150` `x_3` ` + ` `200` `x_4` ≤ `700` `` `400` `x_1` ` + ` `350` `x_2` ` + ` `100` `x_3` ≤ `400` `` `` `x_1` ` + ` `` `x_2` ≥ `1` ` - ` `` `x_3` ` + ` `` `x_4` ≤ `1` `` `` `x_1` ≤ `1` `` `` `x_2` ≤ `1` `` `` `x_3` ≤ `1` `` `` `x_4` ≤ `1`

and `x_1,x_2,x_3,x_4 >= 0; `

Solution is
Max `Z_(A)=9062.5` `(x_1=0,x_2=1,x_3=0.5,x_4=0.375)`

and `Z_L=7000` `(x_1=0,x_2=1,x_3=0,x_4=0)` obtainted by the rounded off solution values.

The branch and bound diagram

A `x_1=0,x_2=1,x_3=0.5,x_4=0.375` `Z_A=9062.5` `Z_L=7000` Solution steps by BigM method

In Sub-problem A, `x_3(=0.5)` must be an integer value, so two new constraints are created, `x_3=0` and `x_3=1`

Sub-problem B : Solution is found by adding `x_3=0`. Solution steps by BigM method Max `Z` `=` `` `6500` `x_1` ` + ` `7000` `x_2` ` + ` `2250` `x_3` ` + ` `2500` `x_4` subject to `` `700` `x_1` ` + ` `850` `x_2` ` + ` `300` `x_3` ` + ` `350` `x_4` ≤ `1200` `` `550` `x_1` ` + ` `550` `x_2` ` + ` `150` `x_3` ` + ` `200` `x_4` ≤ `700` `` `400` `x_1` ` + ` `350` `x_2` ` + ` `100` `x_3` ≤ `400` `` `` `x_1` ` + ` `` `x_2` ≥ `1` ` - ` `` `x_3` ` + ` `` `x_4` ≤ `1` `` `` `x_1` ≤ `1` `` `` `x_2` ≤ `1` `` `` `x_3` ≤ `1` `` `` `x_4` ≤ `1` `` `` `x_3` = `0` and `x_1,x_2,x_3,x_4 >= 0; ` Solution is Max `Z_(B)=8875` `(x_1=0,x_2=1,x_3=0,x_4=0.75)` and `Z_L=7000` `(x_1=0,x_2=1,x_3=0,x_4=0)` obtainted by the rounded off solution values.

Sub-problem C : Solution is found by adding `x_3=1`. Solution steps by BigM method Max `Z` `=` `` `6500` `x_1` ` + ` `7000` `x_2` ` + ` `2250` `x_3` ` + ` `2500` `x_4` subject to `` `700` `x_1` ` + ` `850` `x_2` ` + ` `300` `x_3` ` + ` `350` `x_4` ≤ `1200` `` `550` `x_1` ` + ` `550` `x_2` ` + ` `150` `x_3` ` + ` `200` `x_4` ≤ `700` `` `400` `x_1` ` + ` `350` `x_2` ` + ` `100` `x_3` ≤ `400` `` `` `x_1` ` + ` `` `x_2` ≥ `1` ` - ` `` `x_3` ` + ` `` `x_4` ≤ `1` `` `` `x_1` ≤ `1` `` `` `x_2` ≤ `1` `` `` `x_3` ≤ `1` `` `` `x_4` ≤ `1` `` `` `x_3` = `1` and `x_1,x_2,x_3,x_4 >= 0; ` Solution is This Problem has an infeasible solution, so this branch is terminated.

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The branch and bound diagram

A `x_1=0,x_2=1,x_3=0.5,x_4=0.375` `Z_A=9062.5` `Z_L=7000` Solution steps by BigM method
`x_3=0`		`x_3=1`
B `x_1=0,x_2=1,x_3=0,x_4=0.75` `Z_B=8875` `Z_L=7000` Solution steps by BigM method		C Infeasible Solution Solution steps by BigM method

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In Sub-problem B, `x_4(=0.75)` must be an integer value, so two new constraints are created, `x_4=0` and `x_4=1`

Sub-problem D : Solution is found by adding `x_4=0`. Solution steps by BigM method Max `Z` `=` `` `6500` `x_1` ` + ` `7000` `x_2` ` + ` `2250` `x_3` ` + ` `2500` `x_4` subject to `` `700` `x_1` ` + ` `850` `x_2` ` + ` `300` `x_3` ` + ` `350` `x_4` ≤ `1200` `` `550` `x_1` ` + ` `550` `x_2` ` + ` `150` `x_3` ` + ` `200` `x_4` ≤ `700` `` `400` `x_1` ` + ` `350` `x_2` ` + ` `100` `x_3` ≤ `400` `` `` `x_1` ` + ` `` `x_2` ≥ `1` ` - ` `` `x_3` ` + ` `` `x_4` ≤ `1` `` `` `x_1` ≤ `1` `` `` `x_2` ≤ `1` `` `` `x_3` ≤ `1` `` `` `x_4` ≤ `1` `` `` `x_3` = `0` `` `` `x_4` = `0` and `x_1,x_2,x_3,x_4 >= 0; ` Solution is Max `Z_(D)=7812.5` `(x_1=0.125,x_2=1,x_3=0,x_4=0)` and `Z_L=7000` `(x_1=0,x_2=1,x_3=0,x_4=0)` obtainted by the rounded off solution values.

Sub-problem E : Solution is found by adding `x_4=1`. Solution steps by BigM method Max `Z` `=` `` `6500` `x_1` ` + ` `7000` `x_2` ` + ` `2250` `x_3` ` + ` `2500` `x_4` subject to `` `700` `x_1` ` + ` `850` `x_2` ` + ` `300` `x_3` ` + ` `350` `x_4` ≤ `1200` `` `550` `x_1` ` + ` `550` `x_2` ` + ` `150` `x_3` ` + ` `200` `x_4` ≤ `700` `` `400` `x_1` ` + ` `350` `x_2` ` + ` `100` `x_3` ≤ `400` `` `` `x_1` ` + ` `` `x_2` ≥ `1` ` - ` `` `x_3` ` + ` `` `x_4` ≤ `1` `` `` `x_1` ≤ `1` `` `` `x_2` ≤ `1` `` `` `x_3` ≤ `1` `` `` `x_4` ≤ `1` `` `` `x_3` = `0` `` `` `x_4` = `1` and `x_1,x_2,x_3,x_4 >= 0; ` Solution is This Problem has an infeasible solution, so this branch is terminated.

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The branch and bound diagram

A `x_1=0,x_2=1,x_3=0.5,x_4=0.375` `Z_A=9062.5` `Z_L=7000` Solution steps by BigM method
`x_3=0`				`x_3=1`
B `x_1=0,x_2=1,x_3=0,x_4=0.75` `Z_B=8875` `Z_L=7000` Solution steps by BigM method				C Infeasible Solution Solution steps by BigM method
`x_4=0`		`x_4=1`
D `x_1=0.125,x_2=1,x_3=0,x_4=0` `Z_D=7812.5` `Z_L=7000` Solution steps by BigM method		E Infeasible Solution Solution steps by BigM method

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In Sub-problem D, `x_1(=0.125)` must be an integer value, so two new constraints are created, `x_1=0` and `x_1=1`

Sub-problem F : Solution is found by adding `x_1=0`. Solution steps by BigM method Max `Z` `=` `` `6500` `x_1` ` + ` `7000` `x_2` ` + ` `2250` `x_3` ` + ` `2500` `x_4` subject to `` `700` `x_1` ` + ` `850` `x_2` ` + ` `300` `x_3` ` + ` `350` `x_4` ≤ `1200` `` `550` `x_1` ` + ` `550` `x_2` ` + ` `150` `x_3` ` + ` `200` `x_4` ≤ `700` `` `400` `x_1` ` + ` `350` `x_2` ` + ` `100` `x_3` ≤ `400` `` `` `x_1` ` + ` `` `x_2` ≥ `1` ` - ` `` `x_3` ` + ` `` `x_4` ≤ `1` `` `` `x_1` ≤ `1` `` `` `x_2` ≤ `1` `` `` `x_3` ≤ `1` `` `` `x_4` ≤ `1` `` `` `x_3` = `0` `` `` `x_4` = `0` `` `` `x_1` = `0` and `x_1,x_2,x_3,x_4 >= 0; ` Solution is Max `Z_(F)=7000` `(x_1=0,x_2=1,x_3=0,x_4=0)` and `Z_L=7000` `(x_1=0,x_2=1,x_3=0,x_4=0)` obtainted by the rounded off solution values. This Problem has integer solution, so no further branching is required.

Sub-problem G : Solution is found by adding `x_1=1`. Solution steps by BigM method Max `Z` `=` `` `6500` `x_1` ` + ` `7000` `x_2` ` + ` `2250` `x_3` ` + ` `2500` `x_4` subject to `` `700` `x_1` ` + ` `850` `x_2` ` + ` `300` `x_3` ` + ` `350` `x_4` ≤ `1200` `` `550` `x_1` ` + ` `550` `x_2` ` + ` `150` `x_3` ` + ` `200` `x_4` ≤ `700` `` `400` `x_1` ` + ` `350` `x_2` ` + ` `100` `x_3` ≤ `400` `` `` `x_1` ` + ` `` `x_2` ≥ `1` ` - ` `` `x_3` ` + ` `` `x_4` ≤ `1` `` `` `x_1` ≤ `1` `` `` `x_2` ≤ `1` `` `` `x_3` ≤ `1` `` `` `x_4` ≤ `1` `` `` `x_3` = `0` `` `` `x_4` = `0` `` `` `x_1` = `1` and `x_1,x_2,x_3,x_4 >= 0; ` Solution is Max `Z_(G)=6500` `(x_1=1,x_2=0,x_3=0,x_4=0)` and `Z_L=6500` `(x_1=1,x_2=0,x_3=0,x_4=0)` obtainted by the rounded off solution values. This Problem has integer solution, so no further branching is required.

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The branch and bound diagram

A `x_1=0,x_2=1,x_3=0.5,x_4=0.375` `Z_A=9062.5` `Z_L=7000` Solution steps by BigM method
`x_3=0`						`x_3=1`
B `x_1=0,x_2=1,x_3=0,x_4=0.75` `Z_B=8875` `Z_L=7000` Solution steps by BigM method						C Infeasible Solution Solution steps by BigM method
`x_4=0`				`x_4=1`
D `x_1=0.125,x_2=1,x_3=0,x_4=0` `Z_D=7812.5` `Z_L=7000` Solution steps by BigM method				E Infeasible Solution Solution steps by BigM method
`x_1=0`		`x_1=1`
F `x_1=0,x_2=1,x_3=0,x_4=0` `Z_F=7000` `Z_L=7000` Solution steps by BigM method		G `x_1=1,x_2=0,x_3=0,x_4=0` `Z_G=6500` `Z_L=6500` Solution steps by BigM method

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The 0-1 Integer programming problem algorithm thus terminated and the optimal integer solution is :
`Z_(F)=7000` and `x_1=0,x_2=1,x_3=0,x_4=0`

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