1. ~Gao zhengqing ~Zhou Ran ~Liu Jiamin Limiting factor analysis

2. . What are limiting factors? A limiting factor is any factor that is in scarce supply and that stops the organisation from expanding its activities further Example ： ~Demand ~Materials ~Labour ~Machine hours ~Money

3. One limiting factor usual objective is to maximise profit fixed costs are unaffected in short term Maximise contribution

4. One limiting factor-key factor analysis 1. Identify the scarce resource. 2. Calculate the contribution per unit for each product. 3. Calculate the contribution per unit of the scarce resource for each product. 4. Rank the products in order of the result from 3. 5. Allocate the resources using this ranking.

6. WX Co manufactures two products, Both products pass through two production departments, and. The organisation’s objective is to maximise Contribution to fixed costs. Product A is whereas product B is. There is for but is limited to per annum. The available in each department are restricted to per annum. Other relevant data are as follows. Machine hours required Product A Product B Variable cost per unit Product A Product B Mixing Hrs 0.06 0.08 Shaping Hrs 0.04 0.12 $ 1.30 1.70 A and B mixing shaping sold for $1.50 priced at $2.00 Unlimited demand 13,000 units product Ademand for B machine hours 2,400

8. Graphing the problem Step 1 Drawing the constraints Step 2 Establishing the feasible region Step 3 Finding the optimum allocation of resources 10000 20000 30000 40000 50000 60000 X Y 30000 20000 10000 0.06x+0.08y = 2,400 0.04x+0.12y = 2,400 y = 13,000 Feasible region 0.2x+0.3y = 3000 O

9. Using simultaneouse quations (a)Define variables x = number of units of Product A produced y = number of units of Product B produced (b)Establish the constraints 0.04x+0.12y ≤ 2,400 0.06x+0.08y ≤ 2,400 y ≤ 13,000 x ≥ 0,y ≥ 0 (c)Construct the objective function Maximise contribution(c):0.2x+0.3y Step 1

10. Step 2 Graph the problemStep 3 Establishing feasible region 10000 20000 30000 40000 50000 60000 X Y 30000 20000 10000 0.06x+0.08y = 2,400 0.04x+0.12y = 2,400 y = 13,000 Feasible region A B C D P O Y = 13,000 0.04x+0.12y = 2,400 0.04x+0.12(13,000) = 2,400 0.04x+1560 = 2,400 0.04x = 840 x = 21,000 When x = 21,000 and y = 13,000, total contribution = $8,100 When x = 24,000 and y = 12,000, total contribution = $8,400 Total contribution = $8,000 Step 4 Determine optimal solution

11. Slack and surplus Slack occurs when maximum availability of a resource or other constraining factor is not used. 线性规划中，小于等于 约束条件中未被使用的资源或能力的值成为 松弛量。 Surplus occurs when more then a minimum requirement is used. 线性规划 中，一个大于等于约束条件中超过资源或能 力最底限的部分称之为剩余量。

12. Example: Machine time(3*10)+(10*30)=330=availability Constraint is binding. Raw material(16*10)+(4*30)=280<400 There is a slack of 120kg of RM. Labour (6*10)+(6*30)=240=availability Constraint is binding. T=30>=12,so a surplus of 18 tins over the minimum production requirement. Slack is associated with <= constraints; Surplus is associated with >= constraints

13. Shadow prices The shadow price(dual price) of a limiting factor is the increase in value which would be created by having one additional unit of the limiting factor at its original cost. 影子价格就是，在最适化问题当中，当限制条件 放宽一个单位之后，最适解决方案的真实价值的 变化。在商业活动当中，影子价格是管理层愿意 为获取额外一个单位的既定资源，而多付出的最 大价格。

14. Thank you ~