A Series of Lectures on Approximate Dynamic Programming Lecture 1 Dimitri P. Bertsekas Laboratory for Information and Decision Systems Massachusetts Institute of Technology University of Cyprus September 2017 Bertsekas (M.I.T.) Approximate Dynamic Programming 1 / 19
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A Series of Lectures on Approximate Dynamic Programming ... · kRu) where Q and R are positive definite symmetric matrices Bertsekas (M.I.T.) Approximate Dynamic Programming 11
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A Series of Lectures onApproximate Dynamic Programming
Lecture 1
Dimitri P. Bertsekas
Laboratory for Information and Decision SystemsMassachusetts Institute of Technology
Feedback policies: Rules that specify the control to apply at each possible state xk
that can occur
Major distinction: We minimize over sequences of functions of stateπ = {µ0, µ1, . . . , µN−1}, with uk = µk (xk ) ∈ Uk (xk ) - not sequences of controls{u0, u1, . . . , uN−1}
Cost of a policy π = {µ0, µ1, . . . , µN−1} starting at initial state x0
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
Imperfect-State Info Ch. 4
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
wk xk uk Demand at Period k Stock at Period k Stock at Periodk + 1
Cost of Period k Stock Ordered at Period k Inventory Systemr(uk) + cuk xk+1 = xk + u + k � wk
Stock at Period k +1 Initial State A C AB AC CA CD ABC
ACB ACD CAB CAD CDA
SA SB CAB CAC CCA CCD CBC CCB CCD
CAB CAD CDA CCD CBD CDB CAB
Do not Repair Repair 1 2 n�1 n p11 p12 p1n p1(n�1) p2(n�1)
...
p22 p2n p2(n�1) p2(n�1) p(n�1)(n�1) p(n�1)n pnn
2nd Game / Timid Play 2nd Game / Bold Play
1st Game / Timid Play 1st Game / Bold Play pd 1� pd pw 1� pw
0 � 0 1 � 0 0 � 1 1.5 � 0.5 1 � 1 0.5 � 1.5 0 � 2
System xk+1 = fk(xk, uk, wk) uk = µk(xk) µk wk xk
3 5 2 4 6 2
10 5 7 8 3 9 6 1 2
Finite Horizon Problems Ch. 1
Deterministic Problems Ch. 2
Stochastic Problems
Perfect-State Info Ch. 3
1
A Stage 1Subproblem
Solve the stage 1 subproblems (using the solution of stage 2 subproblems)At each state of stage 1, we record the optimal cost-to-go and the optimal decision
Approximate DP is primarily motivated by the often ENORMOUS computationaldemands of exact DP
Some perspectivesThe connection of theory and algorithms (convergence, rate of convergence,complexity, etc) is solid for exact DP and most of optimization
By contrast, for approximate DP, the connection of theory and algorithms is fragile
There is a great variety of approximate DP approaches
Some approximate DP algorithms have been able to solve impressively difficultproblems. We often do not fully understand why - a lot of research is ongoing
There are success stories without theory
There is theory without success stories
The theory available is interesting but may involve some assumptions not alwayssatisfied in practice
Implementation is often an art
The challenge is how to bring to bear the right mix from a broad array of methodsand theoretical ideas