In Hard Times, how does Louisa Gradgrind's marriage to Bounderby set an example of patriachal and utilitarian "arrangaments" in Victorian society?

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This is a great question! It is clear that Dickens presents Bounderby and Louisa's marriage as an unhappy one. Note how it is juxtaposed in the text with another unhappy marriage - that of Stephen Blackpool to his drunken wife. Note how typically Mr. Gradgrind attempts to treat marriage like a logic problem that he would use to teach his philosophy of Facts in his school. When Louisa wants to know his opinion, Bounderby responds by asking her to consider it in the way that she has been taught to consider every other question, as being a matter of "Fact" alone, without allowing emotions to enter in to the equation:

"I would advise you (since you ask me) to consider this question, as you have been accustomed to consider every other question, simply as one of tangible Fact. The ignorant and the giddy may embarrass such subject with irrelevant fancies, and other absurdities that have no existence, properly viewed - really no existence - but it is no compliment to you to say, that you know better."

Thus it is that Gradgrind turns to empirical evidence to find an answer - he talks about some statistics concerning the relative age of husbands and wives to prove that the age difference would not prevent them having a happy marriage. Thus because of this and the fact that Louisa has not received any other offers of marriage, Gradgrind calculates that Louisa should accept. Louisa's marriage seems to be yet another vehicle for Dickens to illustrate the way that mechanisation has penetrated even the recesses of Victorian family life. Gradgrind makes his "calculation" without once thinking of or referring to love, thus highlighting the way that utilitarianism taken to its extreme converts human beings into machines. As Louisa responds with her decision, "What does it matter?" In a world where there is no room for emotion, what does it matter whom we marry?