Question

Show that the change-making problem (Exercise) can be formulated as an integer linear program. Can we solve this program as an LP, in the certainty that the solution will turn out to be integral (as in the case of bipartite matching)? Either prove it or give a counterexample.

Short answer

As little more than a result, it is a collection of non-negative integers represented as$n_1+n_2+\mathrm{...}+n_n$ for$n$ coins, as well as the number of coins must always be kept to a minimum.

Step-by-step solution

Consider the information

• During linear regression, every problem is actually parameters are stated as linear relationships, as well as the greatest profit or minimal cost is calculated utilizing linear relationships.

• It's a type of mathematical programming in which connections are optimised depending on limitations.

Numerical (Int) Linear Problem

• Obtaining the value of can be used to solve the integer issue.$n_i$ as one if the $n^{th}$ If a coin is chosen, the goal is to go to zero.

o Minimize$n_1+n_2+\mathrm{...}+n_n$

o Subject to$x_1{n}_1+x_2{n}_2+\mathrm{...}+x_n{n}_n=v$

• As either a result, the linear connection may be expressed as follows:

o Minimize${\sum }_{i=1}^n\sum n_i$

o Subject to${\sum }_{i=1}^n{x}_i{n}_i=v$

• Exactly integer values are contained inside the combination of something like a number of coins with their value.

As a result, the issue may be expressed as such an integers linear problem.

Linear programming:

“Yes”. This linear programme could be used to fix the issues..

Conclusion

• Given are the coins with denominations $x_1+x_2+\mathrm{...}+x_n$and the objective is to find the change for the value $\upsilon$.

• Linear programming can be used to solve the presented problem.

• It necessitates the use of the fewest possible coins.

• Suppose $n^i$denotes the number of times the $i^{th}$used $\left\{n_1+n_2+\dots +n_n\right\}$coins.

• Therefore, it is a set of non-negative integers for$n$coins written asand the number of coins must be minimized.

• That minimization should indeed be done under the constraint that the total value of the coins chosen is the same as the supplied value.$v$.

• Another criteria for such an optimization seems to be that the total of the coins chosen equals the provided value.