Chapter 10: Managing Economies of Scale in the Supply Chain: Cycle Inventory
Exercise Solutions
1. The economic order quantity is given by
2 DS . In this problem: hC
D = 109!00 "i.e. #00 units$day multiplied by #%! days$year& ' = (1000$order ) = h* = "0.2&"!00& = (100$unit$year 'o the +,- value is 1/0 units and the total yearly cost is (19/% The cycle inventory value is +,-$2 = 1/0$2 =0 orsheet 10.1 provides the solution to this problem. 2. "a& I3 the order quantity is 100 then the number o3 orders placed in a year are: D$- = 109!00$100 = 109!. 109!. 'o 109! orders are are placed each year at a cost o3 (1000$order (1000$order.. Thus the total order cost is (109!000. *ycle inventory = -$2 = 100$2 = !0 and the annual inventory cost is "!0&"0.2&"!00& = (!000 "b& I3 a load o3 100 units has to be optimal then corresponding order cost can be computed by using the 3ollo4ing e5pression: 2 DS
Q
hC
100 S
" 2&"109!00& S "0.2&"!00&
"100& 2 "0.2&"!00& "2&"109!00&
( .! per order
This analysis is sho4n in 4orsheet 1062. #. "a& e e 3irst consider the case o3 ordering separately: 7or supplier 8: ,rder quantity "-& =
2"20000&"00 100& "0.2&"!&
= 2 units$order
Total cost = order cost holding cost = "20000$2&"!00& "2$2&"0.2&"!& = (2 'imilarly 3or suppliers and * the order quantities are 1%/ and 99 and the associated total costs are (11 and (99 respectively. 'o the total cost is (%/#!
1
"b& In using complete aggregation 4e evaluate the order 3requency "n;& as 3ollo4s:
'o n; o3 the case is =
D A hC A
D B hC B DC hC C
2 S ; '; = 00 #"100& = (00 'o n; =
20000"0.2&"!& 2!00"0.2&"& 9000"0.2&"!& 2"00&
= orders$year
7or supplier 8: - = D$n = 20000$ = !000 units$order Total cost = order cost holding cost = "!00& "!000$2&"0.2&"!& = (!00 'imilarly 3or suppliers and * the order quantities are %2! and 22! and the associated total costs are (%!0 and (!1# respectively. 'o the total cost is (!%%#. orsheet 106# provides the solution to this problem
. "a& This is a quantity discount model and the decision is to identi3y the optimal order quantity in the presence o3 discounts. e evaluate the order quantities at di33erent unit prices using the economic order quantity equation as sho4n belo4:
7or price = (1.00 per unit 2" 20000&"00&"12& #09/ "0.2&"1& 'ince - < 19999
-= +,- =
e select - = 20000 "brea point& and evaluate the corresponding total cost 4hich includes purchase cost holding cost order cost "20000&"12& 20000 00 0.2 "0.9/& 20000 2
Total *ost = "20000&"12&"0.9/&
= (
219%0 'imilarly 4e evaluate the +,-s at prices o3 p = 0.9/ "- = #129/& and p = 0.9% "- = #1%2# 4hich is not in the range so use - = 0001&. The corresponding total costs are (21## and (2#%%0. 'o the optimal value o3 - = 0001 and the total cost is (2#%%0
2
The cycle inventory is -$2 = 0001$2 = 2000.! "b& I3 the manu3acturer did not o33er a quantity discount but sold all ply4ood at (0.9% per square 3oot then - = #1%2# and the total cost is ( 2###% This analysis is sho4n in 4orsheet 10 6
!. e solve this problem using a similar approach as in the previous case e5cept the equation used 3or computing the order quantity at a particular price level in the presence o3 marginal unit quantity discounts is as sho4n belo4:
2 D " S V i
- at 3or a price level * i =
qi C i &
hC i
7or price = (1.00 per unit 2"20000&"12&"00 0 "0&"1&& = #09/ "0.2&"1& 'ince - < 19999 4e adust - = 20000 and the corresponding total cost is ( 2%/00
-=
The same procedure is 3ollo4ed 3or the other unit prices and the optimal quantity is %#2% at a total cost o3 (22%%#. orsheet 10 6! sho4s the analysis and problem solution
%. In the case o3 no promotion 4e can use the +,- e5pression to compute the order quantity. 'o - =
2"1000&"!2&"200& 0.2!" 2&
%!0 units$order
In the presence o3 discount dD - = C d h d
-d =
CQ ; C d
0.2"1000&"!2& "2 0.2&0.2!
2"%!0& "2 0.2&
= #02 units$order
Dominic>s order given the short6term price reduction must be #02. orsheet 106% sho4s the solution to this problem
. In this problem the goal is to obtain an annual demand 3or 4hich T? costs are equal to ?T? costs. 8s the annual demand increases the optimal batch si@e gro4s maing T? more economical. 8bove the threshold obtained 7langer should use T?. elo4 t he threshold they should use ?T?. Thus 4e equate the t4o cost 3unctions as sho4n belo4:
#
T? *osts: 2" D &"!00&
,ptimal order quantity - T? =
8nnual order cost =
D
D
" 00&
QTL
QTL
8nnual holding cost =
Total *ost 3or T? =
"100&
QTL
8nnual trucing cost =
2
D QTL
"0.2&"!0&
"10&
"100&
D QTL
" 00&
QTL
2
"10&
?T? *osts: 2" D &"100& "0.2&"!0&
,ptimal order quantity - ?T? =
8nnual order cost =
D
8nnual trucing cost = 8nnual holding cost =
Total *ost 3or T? =
"100&
Q LTL
D "1&
Q LTL
2
D Q LTL
"10&
"100& D "1&
Q LTL
2
"10&
+quating the T? and ?T? costs results in a demand value o3 #0!%. I3 the demand goes beyond this value then the T? option 4ill prove economical and i3 the demand is belo4 this value then ?T? is the optimal choice. orsheet 106 solves this problem in +A*+? by using the solver option. "b& I3 the unit cost is increased to (100 then the ne4 threshold is %112. Thus as unit cost increases the ?T? option becomes pre3erable. "c& I3 the ?T? cost decreases to (0./ per unit then the ne4 threshold value becomes !.
/. "a& ?T? costs 4ith one supplier per truc: ,ptimal order quantity - T? =
2"#000&"100& = 2! units "0.2&"!0&
2! 12 = 0.9/ months #000
Time bet4een orders =
#000
"100& = (122! 2! 8nnual trucing cost = #000"1& = (#000 2! "10& = (122! 8nnual holding cost = 2 8nnual order cost =
Total *ost 3or T? = (!9
"b& T? costs 4ith one supplier per truc: ,ptimal order quantity - T? =
2"#000&"1000& "0.2&"!0&
= ! units
! 12 = #.1 months #000
Time bet4een orders =
#000
"100& = (#/ ! #000 "900& = (#/% 8nnual trucing cost = ! ! "10& = (#/# 8nnual holding cost = 2 8nnual order cost =
Total *ost 3or T? = (% "c& T? costs 4ith t4o suppliers per truc: In the presence o3 aggregation 4e solve 3or optimal order 3requency n;
'o n; o3 the case o3 2 suppliers is =
D1 hC 1
D
2
hC 2
2 S ; '; = /00 2"100& 2"100& = ( 1200 Thus n; =
"#000&"10& "#000&"10& 2"1200&
= ! orders$year
,ptimal order quantity "-& per supplier = D$n = %00 units ,rder cost per product =
#000 %00
"100& = (!00
!
8nnual trucing cost per product = 8nnual holding cost per product =
#000 %00
"/00 100"2&& $ 2 = (2!00
%00 "10& = (#000 2
Total *ost 3or T? = (%000 "d& The optimal number o3 suppliers that need to be grouped is 4ith an order quantity o3 90 units and total cost o3 (/99. The truc capacity o3 2000 units 4ould not be su33icient i3 more than suppliers are aggregated. "e& hen demand is #000 the aggregated T? option 4ith 3our suppliers is optimal and 4hen the demand decreases to 1!00 the ?T? option is optimal. 8s demand increases to 1/00 the aggregated T? option 4ith 3our suppliers is optimal. orsheet 106/ sho4s the results and analysis 3or this problem 9. e compute the total cost 3or the 3ast moving product and a similar approach can be utili@ed to evaluate the total costs 3or medium and slo4 moving products.
%
"a& 7ast moving products: +,- = - =
2"#0000&" 200&
Days o3 demand =
!"0.!&
#% #0000
8nnual setup cost =
"#%!& = 2
#0000 #%
8nnual holding cost =
= #% units$batch
" 200& = (1#2
#% 2
"0.!&"!& = (1#2
Total cost per product = (#% Total cost 3or all 3ast moving products "! products& = (1#20 'imilar analysis 3or the medium and slo4 products results in batch si@es o3 2191 and 9/0 respectively. "b& The total costs 3or three product groups are: 7ast moving = (1#20 Bedium moving = (2190/ 'lo4 moving = (#292 'o the total cost across all products is (#!22. "c& 7or the 3ast moving products the total time required is: #0000 100
#0000 #%
"0.!& =#0.# hours
'imilarly 3or the medium and slo4 moving products the number o3 hours needed is 122. and 2!.2 respectively. orsheet 1069 demonstrates these computations.
10. "a& In situations 4here 3ull trucloads are used the number o3 deliveries 3or large medium and small customers in a given year is ! 2 and 0. respectively 4hich is obtained by dividing annual demand by truc capacity in each case. 7or the ?arge customer: ,rder quantity = - = 12 units$order "truc capacity& The transportation cost 3or large customer is given by: n?"'s?& = !"/002!0& = (!2!0 The holding cost is given by: "12$2&"10000&"0.2!& = (1!000 'o the total cost is (202!0 The days o3 inventory carried at the large customer are: "12$2&"#%!&$%0 = # days o3 inventory 7or the medium and small customers the total costs are (1100 and (1!00 respectively and the inventory carried by these customers is 91 and 2 units respectively. Thus the overall cost o3 this plan 3or the three customers is (!#0!0 orsheet 10610 sho4s these evaluations. "b& In this case 4e evaluate separate +,-s 3or each o3 three cases. 7or the ?arge customer:
,rder quantity = - =
2 D" S s L & hC L
=
2"%0&"/00 2!0& 0.2!"10000&
= .1 units$order
Cumber o3 orders "n?& = D$- = %0$.1 = /.! orders$year The transportation cost 3or large customer is given by: n?"''?& = /.!"/002!0& = (// The holding cost is given by: ".1$2&"10000&"0.2!& = (// 'o the total cost is (1/ The days o3 inventory carried at the large customer are:
/
".1$2&"#%!&$%0 = 22 days o3 inventory 7or the medium and small customers the total costs are (1122! and (%/1 respectively and the inventory carried by these customers is # and !9 units respectively. Thus the overall cost o3 this plan 3or the three customers is (#!! "c& In this case 4e utili@e complete aggregation i.e. each truc has products that are shipped to all customers. In the presence o3 aggregation 4e solve 3or optimal order 3requency n;
'o n; o3 the case is =
D L hC L
D M hC M DS hC S
2S ; '; = /00 #"2!0& = (1!!0 'o n; =
%0"0.2!&"10000& 2"0.2!&"10000& /"0.2!&"10000& = /.% orders$year 2"1!!0&
7or the ?arge customer: ,rder quantity = - = D$n; = %0$/.% = %.9 units$order Transportation cost: n?"''?& = /.%"/002!0& = (90 The holding cost is given by: "%.9$2&"10000&"0.2!& = (/0 'o the total cost is (1!1 The days o3 inventory carried at the large customer are: "%.9$2&"#%!&$%0 = 21.2 days o3 inventory 7or the medium and small customers the total costs are (!%#% and (##1 respectively and the inventory carried by these customers is 21.2 and 21.2 units respectively. Thus the overall cost o3 this plan 3or the three customers is (2%02 "d& In the case o3 partial aggregation 4e evaluate relative delivery 3requency. In this case not every customer is supplied 4ith the product in every order. 'tep 1: 4e identi3y most 3requently ordered product assuming each product is ordered independently. 7or the large customer:
9
n L
=
hC L D L 2" S s L &
0.2!"10000&"%0& = /.! orders$year 2"/00 2!0&
=
7or the medium and small customers the order 3requency is !.# and #.1 respectively. Thus the most 3requent ordering o3 the product comes 3rom the large customer. 'tep 2: e identi3y the 3requency 4ith 4hich other customer orders are included into the most 3requently ordered. e evaluate
n M
and
n L
'ince 4e are already accounting 3or the 3i5ed cost 3or the large customer 4e only consider the product speci3ic costs 3or medium and small customers. Thus:
n M
=
hC M D M 2 s M
and similarly
0.2!"10000&"2& = 11 2" 2!0&
= = %.#
n L
e no4 evaluate the 3requency 4ith 4hich medium and small customers order relative to the large customer. m M
=
n L
'imilarly
n M
m S
= /.!$11 = 0. =< 4e round up to the closest integer i.e. 1
=2
'tep #: )aving decided the order 3requency 3or each customer 4e recalculate the order 3requency 3or the most 3requently ordering customer i.e. the l arge customer:
n=
D L hC L
D M hC M
2" S s M m M
DS hC S
s L m L &
=
%0"0.2!&"10000& 2"0.2!&"10000& /"0.2!&"10000& 2"/00 " 2!0 $ 1& " 2!0 $ 2&&
= 9.# orders$year 'tep : 7or medium and small customers 4e evaluate the order 3requency: nB = n$m B = 9.#$1 = 9.# n' = n$m ' = 9.#$2 = .%/ The total costs are evaluated as in the previous problem e5cept 3or the 3act that the order costs 3or medium and small customers only includes the product speci3ic costs. The total cost 3or tailored aggregation is ( 2%%9# These evaluations are sho4n in di33erent 4orsheets in 10610 11. "a& 7rom the retailer>s standpoint the optimal order quantity is:
10
-=
2"20000&"200& 0.2"!&
= 99/ units$order
etailer costs: ,rder costs = "20000$99/&"200& = (/99 )olding costs = "99/$2&"0.2&"!& = (/99 etailer total cost = (99/ *runchy>s costs: ,rder costs = "20000$99/&"1000& = (29! )olding costs = "99/$2&"0.2&"#& = (29#9 *runchy total cost = (2# Total cost = (#2#2 "b& In ointly optimi@ing the order quantity is: -=
2"20000&"200 1000& 0.2"!& 0.2"#&
= 1/9 units$order.
etailer costs: ,rder costs = (2!#0 )olding costs = (9/ etailer total cost = (1201 *runchy>s costs: ,rder costs = (12%9 )olding costs = (!%92 etailer total cost = (1/#1 Total cost = (#0#!/
11
"c& In this case 4e equate the total costs associated 4ith ordering at the +,- and the breapoint levels 3or the retailer in determining the discount level. The goal see option is utili@ed to obtain the discount per unit at brea point 4hich is equal to (0.0091. orsheet 10611 provides details o3 the analysis. 12. "a& Eiven that Demand is estimated to be equal to 2000000 F 2000p and the production costs 3or ,range is (100 per unit 4e get the optimal price by setting G equal to "2000000 2000"100&&$000 giving ,range a 4holesale price equal to (!!0. 8t this 4holesale price Eood uy 4ould set a retail price equal to "2000000 2000"!!0&&$000 or (!. Gro3its 3or ,range at this price 4ould be (202!00000 and Eood uy 4ould have a pro3it o3 (1012!0000. "b& I3 ,range o33ers a (0 discount to Eood uy then the ne4 price 4ould be "2000000 2000"!10&&$000 or (!!. Eood uy 4ould pass along (20 or !0H o3 the discount o33ered by ,range. orsheet 10612 provides details o3 the analysis. 1#. "a& Eood uy should purchase is lots equal to '-T"2D'&$h*J = '-TK"25!0000510000&$".25!!0&J = 90! "b& Eiven the (0 discount by ,range 3or the ne5t t4o 4ees Eood buy should adust its lot si@e to "0&"!0000&$"!!060&".2& "!!0590!&$"!!060& = 1%/1. +quation 10.1! The lot si@e increase about /%H. orsheet 1061# provides details o3 the analysis.
12
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