You can also make the denominator called D and regarding it then reveal what D is. Solving linear epigrams Feasible is when the solution is possible Slacks and surpluses EX. + EX. 10 XSL +ex.= + yell Yell represents slack Nonzero variables are called basic variables Zero variables are called non-basic variables Simplex algorithm Sensitivity and parametric analysis What happens when the object and consent change The constants next to the decision variables are called object coefficient Constants are called right hand side Constant*F Final Value: there are no As What is reduced cost??

Objective coefficient: is the constant next to F Allowable increase: this is the amount the objective coefficient can increase thou having to resolve the program, Allowable decrease: this is the amount the objective coefficient can decrease without having to resolve the program Remember if objective coefficient is still with in the range it the solution is binomial, then we don’t need to resolve. But because the objective coefficient has changed then the Constant*F has changed value and needs to be recalculated.

This can affect the objective function If two objective coefficient change then we have to resolve but if they change by the same amount then its still optimal as they are the same contours for the object function ?? In exam The constraint coefficient can’t change without resolve the system, xx + lay 230 2 and 1 and constraint coefficient , the 230 is the right hand side Shadow prices and right and side Difference between final value and costing R.

H side?? Need to add the shadow price * (new right had Side value- old right hand side value) In exam Change formulation If you buy 2 big Macs then a discount -. Bums , this is what is put into the objective function The constraint used so you only have this discount on every two big Macs BUMS GOB BUMS >= O, integer Summation notation need to see assignment 2 Aggregate planning page 5. , need electronic book Flows can be past on from previous weeks Orders also need to be accounted for Integer linear programming Branch and bound to solve integer programs The system is solved using non integer Then constant are apply to the variables to make it integer Egg= 1. 5 , and These espalier excessive as they do not exclude any possible values for XSL. This creates two new linear programs, the are nodes on a search tree All the nodes are then solved and the binomial solution is found Boding is used to identify optimality Staffing problem Remember to consider when people can start

Remember people are integer Cutting stock problem The solution is feasible if the number ordered is total produced A solution is better if is using the least number of rolls Example Xx -number of times use cut using pattern] , where J = {1 Objective function: min 5 SMS rolls = 2*xx + ex. pattern 2 results in 2 SCM rolls every cut and pattern 3 results in 5 SMS rolls for each cut XSL, xx>O , integer If we can make money from surplus rolls then 5 SMS rolls = 2*xx + – 150 150 is the order We then add into the objective function ex. -. 25*Y Integer is used, upper and lower bound

Ass if z=O 20 If all the supplies and Deana are integer , then the solutions flows are naturally integer Tables The cost are in the table and the total demand is on the sides Unbalanced problems If supply> demand We make dummy demand If demand > supply We make dummy supply , is this only infeasible the other is feasible?? In exam whether dummy cost is greater than zero, egg if there is a cost for not supplying Need tables to show this Sometimes we have constraints on supply to a destination, restricts this through a constraint Network problem Let Xii be the number Of carets Of beer transported form factory I to bar J , where I and J

Assignment problem Is to pair members of one IGRP wit hemmers of another group, where each member must be parried with a member of the other group Summation for this 1,2,3,4 Each m assigned to one women, do reverse for woman If we want to balance the problem we can add dummy man or women and put Co’s for every value Transshipment problem Transmission is when there are intermediate points at which the supplies an demand can be remounted Min the Sum the cost*arc There must be no flow lost at intermediate nodes, input = output from the interned nodes, no storage The start of each transshipment problem should eave one node and the end as well Undirected arcs Through put constant Need picture from book If We have inter values for Our upper bound and lower Bounds the result’s will be integer, as unnaturally integer Minimize cost with shortage Need picture form book 10. Incurring production costs with excess supply Maximum flow problem Like a transshipment problem but there are no costs, we want to maximize the flow through the network form the source to the sink Maximum flow as a special case of transshipment Make maximum profit transshipment problem, where ewe get a profit of $1 for each unit of flow leaving the soccer Minimum cut The value of a cut is the sumo the upper bound on those cars travel across the cut in the direction form the source to the sink, usually right Minimum cut theorem The maximum flow equal the minimum value of all cuts in the same network, in exam!! Things to ask Can get an electronic version of the notes, maybe filled in?

We to write why different types of problems are different Non-linear optimization Non-linear terms Log(x) Sing(x) In non-linear optimization where you start matters, we can only usually get locally optimal solutions not global, in exam of graph how to do this Solver an return “infeasible” even when a feasible solution exists but is hard to find Binary or integer variables can cause silver problems Constraints should be used to ensure function argument stay witlessness’s ranges Log(x) requires . 0001 not Do not formulate problems where there is a division by zero Page 14. 3, what is this??? Need help for -4/3 + 80/3 Don’t use absolute value function as solver needs derivatives. Abs,if(),min(),Max(),l/x, do these not have derivate??

Section 15 : Network and network optimization Arbitrage example General Nodes are the circles Arcs are the lines connecting the circles Directed arts are lines that point from one node to another undirected arc is when two nodes are have lines pointing to each other, need picture!! Length of arc is the cost of Wight associated with it Networks Adjacent nodes: the nodes are connected by an arc, egg O and 1 Path: a sequence of distinct arcs, it doesn’t need to be directed , the direction of the arcs doesn’t matter, (1 ,O) , (0,4) , is a path form 1 to 4 Directed path is a path of direction arcs just like a path Cycles/circuit: is a path form on node to the same node, we allays want the cycle to contain at least one arc.

O, 1) , (1 ,(3,0) Tour: a cycle involving all the nodes of the network, can’t do a tour because two isn’t connected to anything Connectedness: if there is a undirected path between every pair of nodes Subnetјarks: network which contains some of the nodes and arcs of the roaring network Trees: a connected cabinetwork with no undirected cycle, a node cannot have a path back to itself Spanning tree : is cabinetwork which maintains all the nodes of the roaring network and which is also a tree. N nodes, n-l arcs Complete network: where an car exist between ever possible pair f nodes Bipartite network: where nodes are divided in two groups with no cars between nodes of the same group, to find there look for what is the most connected nodes, and connected the least Shortest path problem Need to identify what the nodes arcs and acre weights are, as well as what we are trying to solve, in exam!!