1. Chapter 3 Linear Programming Problems  Optimization problems whose objective functions and constraints are.  Optimization problems whose objective functions and constraints are linear.   3.1.1 Definition A real-valued function f=f(x), x  R n is a if and only if there exits a c  R n such that A real-valued function f=f(x), x  R n is a linear function if and only if there exits a c  R n such that f(x) = c 1 x 1 +c 2 x 2 +... + c n x n (3.1) f(x) = c 1 x 1 +c 2 x 2 +... + c n x n (3.1)

2.   3.1.2 Definition A is an expression of the form: A linear equality is an expression of the form: c 1 x 1 +c 2 x 2 +... + c n x n = b (3.2) c 1 x 1 +c 2 x 2 +... + c n x n = b (3.2) A linear is defined similarly with  or ≥ in place of =. A linear inequality is defined similarly with  or ≥ in place of =.

3. Example 3.1.3

4. Conceptual Model... x1x1 x2x2 x3x3 xnxn c1x1c1x1 c2x2c2x2 c3x3c3x3 cnxncnxn

5. cjxjcjxj xjxj a 1j x j a 2j x j a mj x j... Return/cost Decision Constraints Activity j:

6. 3.2 A Standard Form  Notation and terminology: m = number of m = number of functional constraints n = number of n = number of decision variables b i = level of i, i=1,2,3,...,m b i = level of resource i, i=1,2,3,...,m x j = level of j, j=1,2,3,...,n x j = level of activity j, j=1,2,3,...,n z = value of z = value of objective function c j = per unit of activity j c j = benefit/cost per unit of activity j a ij = amount of by each unit of activity j. a ij = amount of resource consumed /produced by each unit of activity j.

7. And the problem is...............

8. Standard Form   opt=max   ~    b i ≥ 0, for all i

9. 3.3 Assumptions of LP Problems   Constant return to scale xjxj cjxjcjxj 1 cjcj xjxj Return (c j x j )

10.  :  Independence of activities: f(x 1,...,x n )  c 1 x 1 ...  c j x j ... + c n x n

11.   Continuous activity levels – For each activity, say j, the activity level x j is a continuous variable. In particular, x j is assumed to take only values. – For each activity, say j, the activity level x j is a continuous variable. In particular, x j is not assumed to take only integer values.  Quite often we a problem where the activity levels must be integers by a formulation where the activities are continuous variables.  Quite often we approximate a problem where the activity levels must be integers by a formulation where the activities are continuous variables.

12. RemarkRemark  Strictly speaking, in such cases it is necessary to justify the use of the approximation.

13. 3.4.1 Example  Welcome to Appex Corporation: We produce canned beans.  We have 2 canneries in Sydney: – one can produce up to 400 cans per day – one can produce up to 600 cans per day

14.  We have 3 warehouses in Brisbane – one needs at least 300 cans per day – one needs at least 200 cans per day – one needs at least 400 cans per day  The distances between canneries and warehouse are given (some unit of distance).

15. Distance Matrix Cannery 1 Cannery 2 Warehouse 1 Warehouse 2 Warehouse 3 754085 205575

16. TaskTask  Formulate a linear programming model to determine a minimum cost transportation schedule satisfying the union agreement and production capacities of the canneries.

17. AnalysisAnalysis  :  On the side: Sydney Brisbane Cannery 1 Cannery 2 warehouse 1 warehouse 2 warehouse 3

18.  :  Decision Variables: x ij := number of cans shipped from cannery i to warehouse j, i=1,2; j=1,2,3 x ij := number of cans shipped from cannery i to warehouse j, i=1,2; j=1,2,3  :  Objective function: f(x) := 75x 11 + 40x 12 + 85x 13 + 20x 21 + 55x 22 + 75x 23 f(x) := 75x 11 + 40x 12 + 85x 13 + 20x 21 + 55x 22 + 75x 23

19.  :  Constraints: : Production capacities: x 11 + x 12 + x 13  cannery 1  x 21 + x 22 + x 23  cannery 2 

20.  :  Union agreement: x 11 + x 21 ≥ 300 (warehouse 1) x 11 + x 21 ≥ 300 (warehouse 1) x 12 + x 22 ≥ 200 (warehouse 2) x 13 + x 23 ≥ 400 (warehouse 3)  :  Non-negativity: x 11,x 12,x 13,x 21,x 22,x 23 ≥0.  :  Perhaps even: x 11,x 12,x 13,x 21,x 22,x 23 

21. Complete Formulation