1. The Economics of Production Make or Buy Decisions Capacity Expansion Learning-Curves Break-even Analysis Production Functions

2. Make or Buy? Let c 1 = unit purchase price c 2 = unit production cost (c 2 < c 1 ) K = fixed cost of production x = number of units required Produce if K + c 2 x  c 1 x orx  K / (c 1 - c 2 )

3. Make or Buy Example It costs the Maker Bi Company $20 a unit to purchase a critical part used in the manufacture of their primary product line – a thing-um-a-jig It is estimated that the part could be produced internally at a unit cost of $16 after incurring a fixed cost of $20,000 for the necessary equipment. What to do? a thing-um-a-jig

4. What to do? Let c 1 = $20 c 2 = $16 K = $200,000 x = number of units required Produce if x  K / (c 1 - c 2 ) = 20,000 /(20 – 16) = 5,000

5. Nonlinear Cost Function Let c 1 = unit purchase price c 2 = K + ax b where K, a, b > 0 x = number of units required If c 1 = 20 and c 2 = 20,000 + 100x.7, then

6. More on that Nonlinear Cost Function xMakeBuy 1900$39,730$38,000 1925$39,911$38,500 1950$40,092$39,000 1975$40,272$39,500 2000$40,451$40,000 2025$40,630$40,500 2035$40,701$40,700 2050$40,808$41,000 2075$40,985$41,500 2100$41,162$42,000 2125$41,338$42,500 2150$41,513$43,000 2175$41,688$43,500 2200$41,862$44,000 2225$42,036$44,500 2250$42,209$45,000 2275$42,381$45,500

7. Strategic Decisions Capacity Expansion Capacity Growth Planning –when to construct new facilities –where to locate facilities –how large to size a facility Economies of scale –advantage of expanding existing facilities –share plant, equipment, support personnel –avoid duplication at separate locations

8. Capacity Expansion competing objectives: maximize market share maximize capacity utilization time number units demand capacity leads demand time number units demand capacity lags demand

9. We need a model let D = annual increase in demand x = time interval (in yrs) between capacity increases r = annual discount rate, compounded continuously f(y) = cost of expansion of capacity y assume y = xD, then cost = C(x) = f(xD) [1 + e -rx + (e -2rx ) + (e -3rx ) + …] = f(xD) [1 + e -rx + (e -rx ) 2 + (e -rx ) 3 + …] = f(xD) / [1 – e -rx ] assume f(y) = ky a, then find the x that minimizes C(x)

10. A Diversion - the Geometric Series You see? It does converge.

11. Discounting – another diversion Consider the time value of money $1.00 today is worth more than a $1.00 next year How much more is it worth? If r = annual interest rate, then it is worth (1+r) $1.00 After two years, it is worth (1+r) 2 $1.00 (compounded) Compounded quarterly for 1 yr = Compounded continuously for one year = After t years =

12. More diversionary discounting A stream of costs: C 1, C 2, …, C n incurred at times t 1, t 2,…, t n has a present value of: Why can’t you show us an example? For an infinite planning horizon where x is the time between expansions:

13. The Example Chemical firm expanding at a cost ($M) of –where y is in tons per year. Demand is growing at the rate of D = 5,000 tons per year and future costs are discounted at a rate of r = 16 percent Find x that minimizes

14. Capacity Expansion Solution alternately set C’(x) = 0 solve for x.

15. Learning Curves Based upon the observation that unit labor hours or costs decrease for each additional unit produced Units produced Direct labor hrs per unit

16. Why does this happen? Employee learning reduced set-up times better routing and scheduling of material (WIP) improved tool design more efficient material handling equip. (MHE) reduced lead-times improved (simplified) product design production smoothing quality assurance revised plant layout increased machine utilization

17. Learning Curve (experience curves) Y(u) = labor hours to produce the u th unit assumeY(u) = au -b a = hours to produce the first unit b = rate at which production hours decline labor coming to work

18. Learning Curves Assume hours to produce unit 2n is a fixed percentage of the hours to produce unit n Then for an 80 percent learning curve: Observe the simple formula

19. Learning Curves least-squares analysis UnitDirect Labor Number - xHours -Y(x) 2035.8 4030.1 6027.3 8025.7 10024.1 Fit Y(x) = ax -b using Excel Y(x) = 74x -.243 2 -.243 =.845 or a 84.5% learning curve

20. Learning Curves Cumulative Cost hours to produce i th unit cumulative direct labor hrs to produce x units average unit hours to produce x units

21. Learning Curves Approximate Cumulative Cost

22. Example Y(x) = 74x -.243 2 -.243 =.845 or a 84.5% learning curve

23. Break-Even Analysis Let x = number of units produced and sold x = S -1 (unit selling price) S(x) = unit selling price F = fixed cost g(x) = variable cost to produce x units then break-even point occurs when revenue = cost; or S(x) x = F + g(x) and profit = revenue – cost or P(x) = S(x)x – [F + g(x)] Sam Even on a break

24. Break-Even Analysis x $ F Break even pt loss profit Max profit Revenue curve Cost curve loss Diminishing returns

25. Break-Even Analysis Demand Curve x S(x) S(x) = d + e x + f x 2 (quadratic) d, e, and f are constants to be determined

26. Break-Even Analysis Demand Curve x S(x) S(x) = d + e x + f x 2 d, e, and f are constants to be determined Approximate as linear S(x) = d + e x

27. Break-Even Analysis Unit Cost LetM = direct material unit cost ($/unit) L = direct labor rate ($/hour) B = factory burden rate Y(x) = direct labor hours to produce unit x C(x) = cost to produce unit x C(x) = M + L Y(x) + L B Y(x) = M + (1+B) L Y(x) = M + (1+B) L a x -b Learning curve effect

28. The Factory Burden Diversion Manufacturing Costs Factory burdenDirect costs Direct labor Direct material Indirect material Indirect labor Indirect expense -Supervision -Engineering -Maintenance -Heating -Lighting -Depreciation -Rent & Taxes -Office & janitorial supplies -Paint

29. Factory Burden - example Categoryannual cost Indirect material$ 6,120 Indirect labor 42,800 Indirect expenses 22,900 total$71,820 Product annual productionlabor hours ratewages A100,0001000$9/hr$9,000 B140,0001400$7/hr 9,800 C 80,0001600$7/hr 11,200 total4000 $30,000 burden rate = 71,820 /30,000 = 2.394 per direct labor $

30. Manufacturing Costs General Overhead Costs Profit Selling Price S(x) Administrative Costs Marketing Costs Development Costs Demands

31. Cumulative Cost g(x) = M x + L (1+B) T(x) = M x + L (1+B) [a x 1-b / (1-b)] total cost = F + M x + L (1+B) a x 1-b / (1-b) where F is a fixed cost to produce product x Unit cost: C(x) = M + (1+B) L a x -b Learning curve

32. Break-Even Analysis -Profit Profit = P(x) = S(x) x - [F + g(x)] letting S(x) = d + ex, e < 0 P(x) = (d + e x) x - F - M x - L (1+B) a x 1-b /(1-b) = d x + e x 2 - F - M x - g x 1-b where g = L (1+B) a /(1-b)

33. More Break-Even Analysis P(x) = (d - M) x + e x 2 - g x 1-b - F break-even: set P(x) = 0 and solve for x maximize profit: set and solve for x for e < 0, a max point can exist

34. Break-Even Analysis - example P(x) = d x + e x 2 - F - M x - g x -b+1 where g = (1+B) L a /(1-b) Data: d = 100 e = -.01 F = $100,000 M = $4 B =.5 L = $20 / hr a = 10 b =.60 P(x) = 100 x -.01 x 2 – 100,000 - 4 x – (1+.5) (20) (10) x.4 /.4 = 96x -.01x 2 –750 x.4 –100,000 2 -.6 = 66%

35. The Math

36. The Graph x = 1382 x = 4706

37. Production Functions A production function expresses the relationship between an organization's inputs and its outputs. It indicates, in mathematical or graphical form, what outputs can be obtained from various amounts and combinations of factor inputs. In its most general mathematical form, a production function is expressed as: Q = f(X 1,X 2,X 3,...,X n ) where: Q = quantity of output and X 1,X 2,X 3,...,X n = factor inputs (such as capital, labor, raw materials, land, technology, or management)

38. Production Functions There are several ways of specifying this function. One is as an additive production function: Q = a + bX 1 + cX 2 + dX 3,... where a,b,c, and d are parameters that are determined empirically. Another is as a Cobb-Douglas production functionCobb-Douglas Q = f(L,K,M) = A * (L alpha ) * (K beta ) * (M gamma ) where L = labor, K = capital, M = materials and supplies, and Q = units of product.

39. Cobb-Douglas Production Function Q = f(L,K,M) = A * (L alpha ) * (K beta ) * (M gamma ) Properties of the Cobb-Douglas production function: Decreasing returns to scale: alpha + beta + gamma < 1 Increasing returns to scale: alpha + beta + gamma > 1 Let C L, C K, and C M = the unit cost of labor, capital, and material, then C(L,K,M) = C L L + C K K + C M M is the total cost function

40. A Little Production Problem An interesting problem: Given a monthly budget of $B, how should the money be spent to obtain a specified output Q? Find L, K, and M where L = dollars spent on labor, K = dollars spent on facilities and equipment, and M = dollars spent on material I know I can work this one.

41. The Inevitable Example laborcapitalmaterial Aalphabetagamma 1000.30.20.4 LKMQ $8,333$66,667$25,000794,700RHS budget 111100,000 -3 100 -21 00

42. Stop the madness. Optimize your production system! profits homework: turn-in breakeven problem text: Chapter 1- 29, 30, 31, 32, 34, 35 36, 37,38, 43, 44