Since the OP mentioned the T.U.L.I.P. outline, or at least two points thereof, it should be stated up front that not even Calvinists would deny that man was created with a "free will."
But the issue that seems to be overlooked more often than most is the "illusion" of "free will."
The OP asked:
If they believe in the doctrine of election and irresistable grace... how can they also believe in free will?
Has anybody taken the time to read what is written in the Theological Dictionary of the New Testament about the issue of "predestination' as it pertains to election, grace, and free will?
When looking at the most controversial verse in the Bible, Rom. 8:29, one single word stand out in prominance than even the word for predestination; "
ina."
According to "The New Analytical Greek Lexicon," Wesley J. Perschbacher, Hendrickson Publishing, Peabody, Mass., 01962, Copyright 1990, it means:
"
ina –
that, in order that, Mt. 19:13; Mk. 1:38; Jn. 1:22; 3:15; 17:1;
ina mhe,
that not, lest, Mt. 7:1; in the N.T. equivalent to
wste,
so that, so as that, Jn. 9:2. et al.; also, marking a simple circumstance,
the circumstance that, Mt. 10:25; Jn. 4:34; 6:29; 1 Jn. 4:17; 5:3; et al.;"
Now you may be asking, what in the world does this have to do with the subject at hand? Well, it has a great deal to do with it. Ethelbert Stauffer commenting in the TDNT says:
"But the impulse which makes all human action possible is the divine action. In Jesus this is the theological presupposition of all ethical imperatives, and it is worked out in detail by Paul and his followers. Here the guiding theological concepts of Apocalyptic ethics are worked out consistently. The divine creation is a calling forth in the original sense of the term. God’s historical work has the character of word. It is a word or summons directed to the will of man. In this sense, statements concerning God’s will and act lead on to final causes which speak of the possibilities and tasks that are posited of man, Rom. 8:29; 9:11 f.; 17, 23. But this determination of the goal is no mere fatalism. Predestination is orientated to the will of man. Nor is this determination of a will a deterministic enslavement. It is a voluntaristic awakening of the will. God wills the will of man. It is the will of man which He directs to His goals according to His own purposes. He is not directing a mere vessel with no will of its own. God’s will calls forth the will of man, and liberates it by taking procession of it. This predestinating will of God is fulfilled in the Christ event. This is again both God’s work and God’s word. It is directed to the will of man and it calls forth his action. th cavritiv ejste seswsmevnoi, oujk ex ergwn, ina mhv tiV kauchshtai aujtou gavr ejsmen poivhma, ktisqevnteV, ejn Cristw ihsou ejpi ergoiV ajgaqoiV, oijV prohtoivmasen oJ qeoV iwna ejn aujtoiV peripathvswmen, Eph. 2:8 f."
Theological Dictionary of the New Testament, By Gerhard Kittel, Translated by Geoffery W. Bromiley, D. Litt., D.D. Eerdmans Publishing Company, Grand Rapids, MI, , Vol. III, Q - K, Ina, Section B, Ethical Final Causes, Sub-sect. 2, In the New Testament, part b, p. 331-32, Ethelbert Stauffer commenting.
And certainly, a right understanding of mans supposedly "free will' needs explaining also.
God, Jesus, and Jeremiah all three say that it is the heart that is the seat of mans supposedly free will.
God said in Gen. 8:21 that mans heart is evil from his youth up. Jeremiah says that the heart is desperately wicked above all things, who can know it? (Jer. 17:9) And Jesus Himself said that everything that defileth a man comes from the heart. (Mt. 15:18-20)
The actions of the heart is what guides a mans actions. And if the heart is wicked, evil from his youth up, and the actions which defileth a man proceeds from the heart, then how can mans will be truly free?
It isn't. A man will only do what is in his heart. And a man cannot do what is right in God's eyes until he has been given a new heart. (Psa. 51:10)
Unless God draws a person, and the Holy Spirit is working from within to convict a person of their sin, how can they all of a sudden start doing whats right? They can't plain and simple.
"Can the Ethiopian change his skin, or the leopard his spots?
then may ye also do good, that are accustomed to do evil." -Jer. 13:23 (KJV)
Yes man has a free will, but until conviction, regeneration, and faith and repentance takes place first, his will is to do evil.
God Bless
Till all are one.