Limiting factor analysis F5-Chapter 4 Limiting factor analysis Lecturer: Ji Weili 2018/7/17

Limiting factor analysis Make or buy decisions and scare resources Chapter Preview Limiting factor analysis Limiting factors Linear programming Slack and surplus Make or buy decisions and scare resources Shadow price 2018/7/17

Overview A revision of linear programming technique. 1 Some new concepts: slack, surplus, shadow price or dual price 2 2018/7/17

Limited resources A business needs to identify the optimal production plan for utilising its resources to maximise profit 2 techniques Single limiting factor Linear programming (>1 limiting factor) 2018/7/17

Single limiting factor analysis 1 Identify limiting factor(except sales demand) 2 Calculate contribution per unit 3 Calculate contribution per limiting factor 4 Rank products 5 Prepare a production plan 2018/7/17

Example: two potentially limiting factors step3 Plan 4 step2 Rank step1 Identify which of the limiting factors is a binding constraint. 2018/7/17

Make or buy decisions and scarce resources Combining internal and external production Decision rule---- minimise costs Example: 2.1.1 P107 2018/7/17

Linear programming steps 1 Define variables P110 let x = number of the Super produced let y = number of the Deluxe produced let p = contribution per month 2018/7/17

Linear programming steps 2 Establish constraints subject to: Machine hour 5x + 1.5y  400 Government restrictions x + y  150 Non-negativity x,y > 0 Formulate objective function Max p =100x + 200y 3 2018/7/17

Linear programming steps 4 Plot constraints on graph Find co-ordinates Machine hour 5x + 1.5y= 400 when x=0, y=267 y=0, x=80 Government restrictions x + y=150 When x=0, y=150 y=0, x= 150 Non-negativity x,y > 0 Set the inequalities to equations 2018/7/17 10

Linear programming steps 100 x y 200 300 50 150 5 Identify feasible region A The feasible area is OABC. B This is the area in which the solution will lie – likely to be on one of the points or on a line as want to maximise contribution. Can use letters or shade in the area C 11 2018/7/17

Linear programming steps 100 x y 200 300 50 150 5x+1.5y=400 Point out that the non-negativity is denoted by use of the positive quadrant of the graph x+y =150 12 2018/7/17

Linear programming steps 6 Plot objective function and identify optimal point p =100x + 200y Pick any value for p Say p= 10000 When x=0, y=50 When y=0, x=100 2018/7/17 13

Linear programming steps 100 x y 200 300 50 150 A B C Optimal point = A p =100x + 200y Plot the objective function and tell them to push it as far as possible to the edge of the feasible region to give the optimal point I.e. the maximum g and s and therefore the maximum contribution – use a ruler to keep it parallel Alternative approach is to work out the contribution at each feasible point to see which one gives maximum – so need to do simultaneous equations at each point 2018/7/17 14

Linear programming steps 7 Determine optimal solution The optimal point is A, where the profit line is passing through the intersection of x+y=150 with the y axis at (0,150). Optimal production plan: 0 Super models 150 Deluxe models Contribution: p=200*150=30000 2018/7/17 15

Linear programming Please note the format of your answers. Two methods to find the best solution—graphing or using simultaneous equations. Attention: You must draw the graph whatever. Let me see! 16

Slack and surplus When maximum availability of a resource is not used, slack occurs. When more than a minimum requirement is used, surplus occurs. Slack is associated with ≤ constraints and surplus with ≥ constraints. 2018/7/17 17

Limiting factors and shadow prices The shadow price or dual price is the extra contribution or profit that may be created by having one additional unit of the limiting factor at the original cost. Key points: 1)The shadow price represents the maximum premium above the basic rate that an organization should be willing to pay for one extra unit of a resource. 2018/7/17 18

Limiting factors and shadow prices 2)If a constraint is not binding at the optimal solution, the shadow price is zero. 3)Shadow prices are only valid for a small range. 4)Shadow prices provide a measure of the sensitivity of the effect of a unit change in a constraint. Example: calculating shadow prices P121-7.3 2018/7/17 19

End of Chapter 4 Thank You ! 2018/7/17