timing and inventory
Post on 29-Oct-2014
27 Views
Preview:
DESCRIPTION
TRANSCRIPT
Topic 11:
Supply Chain and Coordination
1. Stochastic EOQ models 2. The reorder point & the order up to model3. Simulation of inventory models4. Supply chain coordination with LP 6. Aggregate planning7. The aggregate planning formulation with LP
There are four commonly employed ways in whichmanagement may express its willingness or tolerance for
running out of inventory:
1. Cycle Service Level: of the probability of not runningout of stock in one cycle.
2. Fill Rate: probability that demand may be filled fromstock
3. Number of stockouts per year
4. Proportion of days in stock
Criteria often used as cost surrogates
Continuous Review of Inventory (ROP or Q System)
R
Inve
ntor
y
Time
L
LL
Place an order for Q units whenever inventory withdrawal brings inventory
position to R
BDR L
The reorder point (R) is normally calculated by adding some level of safety stock (B) to the expected demand over leadtime
Numerical Example of a Q system
Demand over leadtime = N(36,15) and CSL = 90%
36
15
1-CSL
B B=1.28(15)=19.2
R R=36+19.2=55.2
Periodic Review of Inventory (PRS or P System)In
vent
ory
Time
LL
Review an item’s inventory position every P time periods. At that time,
place and order to replenish to T units
BDT LP
The up-to order level(T) is normally calculated by adding some level of safety stock (B) to the expected demand over the protection interval (L + P)
T
P
Numerical Example of a P System
D = N(40,15) per weekL = 3 weeksCSL = 80%
What P is required to approximate the cost tradeoffs of a 400 unit EOQ?
What is the desired level of T?
weeks1040
400
56545520)54(84.)13(40
54)310(*152
TLP
Numerical Example
• D = N(15,6) units per weekA = $50 per order
• h = $12 per unit per year L = 2 weeks CSL = 80%
• Assume Continuous review System
• A. What is the EOQ?
• B. What is the desired B?
• C. What is the desired R ?
6.8012
50))52(15(2
18.7)5.8(845.
5.82*6* 22
B
LtL
18.3718.7)2(15
Numerical Example (cont.)
• D = N(15,6) units per weekA = $50 per order
• h = $12 per unit per year L = 2 weeks CSL = 80%
• Assume Periodic Review
• A. What value of P provides approximate EOQ tradeoffs?
• B. What is the desired
• B and T?
weeksweeks 54.552780
6.80
4.1184.13)15(7
4.13)9.15(845.
9.157*6
7
2
T
B
weeksLP
LP
Advantages of P and Q Systems
• Continuous (Q) systems
– Carry less safety stock
– Order size is constant
– Individualize replenishment intervals
– Suited to quantity discounts and capacity limitations
• Periodic (P) systems
– Less need to take additional physical inventory
– Fixed replenishment intervals
– Can coordinate replenishment of multiple items
A Reorder Point (ROP) Inventory Model
• GEN;• LIMITS,10,,,10;• EQUIVALENCE,{ { INVPOS, XX[1] } };• EQUIVALENCE,{ { ONHAND, XX[2] } };• EQUIVALENCE,{ { BACKORD, XX[3] } };• EQUIVALENCE,{ { LOST, XX[4] } };• EQUIVALENCE,{ { ORDERS, XX[6] } };• EQUIVALENCE,{ { ROPOINT, XX[8] } };• EQUIVALENCE,{{QUANT,XX[9]}};• INITIALIZE,0.0,10000,YES;• INTLC,{{INVPOS,6},{ONHAND,6},{ROPOINT,4},{QUANT,12}};• TIMST,,INVPOS,"INVENTORY POSITION",0,0.0,1.0;• TIMST,,ONHAND,"INVENTORY ON HAND",0,0.0,1.0;• TIMST,,BACKORD,"BACKORDERS",0,0.0,1.0;• TIMST,,LOST,"DEMAND LOST",0,0.0,1.0;• TIMST,,ORDERS,"ORDERS PLACED",0,0.0,1.0;• NET;• FIN;
An Order-Up-To Inventory Model
• GEN,"UPTO MODEL",,,,YES,YES;• LIMITS,10,,,10;• EQUIVALENCE,{ { INVPOS, XX[1] } };• EQUIVALENCE,{ { ONHAND, XX[2] } };• EQUIVALENCE,{ { BACKORD, XX[3] } };• EQUIVALENCE,{ { LOST, XX[4] } };• EQUIVALENCE,{ { UPTO, XX[5] } };• EQUIVALENCE,{ { PERIOD, XX[7] } };• EQUIVALENCE,{ { QUANT, ATRIB[1] } };• INITIALIZE,0.0,10000,YES;• INTLC,{{INVPOS,6},{ONHAND,6},{UPTO,12},{PERIOD,6}};• TIMST,,INVPOS,"INVENTORY POSITION",0,0.0,1.0;• TIMST,,ONHAND,"INVENTORY ON HAND",0,0.0,1.0;• TIMST,,BACKORD,"BACKORDERS",0,0.0,1.0;• TIMST,,LOST,"DEMAND LOST",0,0.0,1.0;• NET;• FIN;
What is the bullwhip effect?
• Demand variability increases as you move up the supply chain from customers towards supply
Customer
RetailerDistributorFactoryTier 1 SupplierEquipment
Bullwhip effect in autos to machine tools
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
% c
hang
e in
dem
and
GDP = solid line
Source:Anderson, Fine and Parker (1996)
Autos Machine tools
Bullwhip effect in the US PC supply chain
Semiconductor
1995 1996 1997 1998 1999 2000 2001
-40%
-20%
0%
20%
40%
60%
80%
PC
SemiconductorEquipment
Changes indemand
Semiconductor
1995 1996 1997 1998 1999 2000 2001
-40%
-20%
0%
20%
40%
60%
80%
PC
SemiconductorEquipment
Changes indemand
Annual percentage changes in demand (in $s) at three levels of the semiconductor supply chain: personal computers, semiconductors and semiconductor manufacturing equipment.
Consequences of the bullwhip effect
• Inefficient production or excessive inventory.
• Low utilization of the distribution channel.
• Necessity to have capacity far exceeding average demand.
• High transportation costs.
• Poor customer service due to stockouts.
Causes of the bullwhip effect
• Order synchronization– Multiple retailers who tend to order around the same time period– Manufacturers responding to an MRP system that place raw material orders at the
beginning of the month
• Order batching– In order to save on shipping or ordering costs, firms order a full pallet or full truck load
• Trade promotions and forward buying– Supplier offers a discount on product ordered in a specific time period– Supplier offers a quantity discount– A retailer orders a large quantity intending to take advantage of a discount and sells
excess product to a second retailer (this strategy is called diversion)
• Reactive and over-reactive ordering– A retailer who is not sure that demand is stable over time may act aggressively when
faced with periods of lower or higher than expected demand
• Shortage gaming– A retailer who wants to insure product from an under-capacitated supplier may over
order expecting to only receive a portion of the ordered quantity
Strategies to combat the bullwhip effect
• Information sharing:
– Collaborative Planning, Forecasting and Replenishment (CPFR)
• Smooth the flow of products
– Coordinate with retailers to spread deliveries evenly.
– Reduce minimum batch sizes.
– Smaller and more frequent replenishments (EDI).
• Eliminate pathological incentives
– Every day low price
– Restrict returns and order cancellations
– Order allocation based on past sales in case of shortages
• Vendor Managed Inventory (VMI): delegation of stocking decisions
Supply Chain Synchronization and Linear Programming
The Transportation Problem: a general formulation of a class of problemsrelated to the supply and distribution of goods and services across a network. Generally, the transportation problem is concerned with the most cost effective(or cost minimizing) way to supply several demand locations (nodes) from morethan one supply location (nodes)
Example
Special transportation concerns:
Route (or arcs) that have a maximum capacity
Routes that cannot be traversed
The Transshipment Problem: a more generalized version of thetransportation problem in which intermediate, transshipment, nodes are addedto the network. Transshipment nodes are often used to model warehouses,material transfer locations, or junctions for mixed mode delivery of goods andservices.
Example
Special transshipment concerns:
Backwards or sidewards movement in the network
Capacity limitations of the transshipment nodes
Special transshipment concerns:
Quaker Oats has begun manufacturing, in two of its plants, a new granola productmade of three parts oats, two parts raisins and one part almonds. Two oat vendorsand two almond vendors have been identified, but only one reliable vendor of raisinscould be found. The supply of raw materials and the shipped costs are provided
Vendor Supply intons
Cost to Plant 1 Cost to Plant 2
Oat 1 25,000 $100 $110
Oat 2 30,000 $105 $95
Raisin 50,000 $550 $525
Almond 1 9,000 $1,050 $1,150
Almond 2 10,000 $1,200 $1,100
Quaker ships to three distribution facilities. The shipping cost of completed (6-ton)pallets of product and the demand at each distribution facility are provided
Hannaford Quaker WalMart Plant Capacity
Plant 1 $100 $65 $90 9,500
Plant 2 $95 $70 $90 8,500
Demand 2,500 5,000 10,000
Quaker Oats has begun manufacturing, in two of its plants, a new granola productmade of three parts oats, two parts raisins and one part almonds. Two oat vendors
EXAMPLE
O1
O2
R
A1
A2
Q1
Q2
H
D
W
25,000
30,000
50,000
9,000
10,000
2,500
5,000
10,000
9,500
8,500
Formulation
Minimize Z= 100O1Q1+110O1Q2+105O2Q1+95O2Q2+550RQ1+525RQ2+1050A1Q1+1150A1Q2+1200A2Q1+1100A2Q2+100Q1H+65Q1D+90Q1W+95Q2H+70Q2D+90Q2W
Subject To: C1 1O1Q1+1O1Q2<=25000C2 1O2Q1+1O2Q2<=30000C3 1RQ1+1RQ2<=50000C4 1A1Q1+1A1Q2<=9000C5 1A2Q1+1A2Q2<=10000C6 1Q1H+1Q2H>=2500C7 1Q1D+1Q2D>=5000C8 1Q1W+1Q2W>=10000C9 1O1Q1+1O2Q1-3Q1H-3Q1D-3Q1W=0C10 1O1Q2+1O2Q2-3Q2H-3Q2D-3Q2W=0C11 1RQ1-2Q1H-2Q1D-2Q1W=0C12 1RQ2-2Q2H-2Q2D-2Q2W=0C13 1A1Q1+1A2Q1-1Q1H-1Q1D-1Q1W=0C14 1A1Q2+1A2Q2-1Q2H-1Q2D-1Q2W=0C15 Q1H+Q1D+Q1W<=9500C16 Q2H+Q2D+Q2W<=8500
Combined Report for GranolaDecision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 O1Q1 25,000.00 100.0000 2,500,000.00 0 basic -M 105.00002 O1Q2 0 110.0000 0 20.0000 at bound 90.00 M3 O2Q1 2,000.00 105.0000 210,000.00 0 basic 100.00 M4 O2Q2 25,500.00 95.0000 2,422,500.00 0 basic -M 105.00005 RQ1 18,000.00 550.0000 9,900,000.00 0 basic 535.00 M6 RQ2 17,000.00 525.0000 8,925,000.00 0 basic -M 540.00007 A1Q1 9,000.00 1,050.00 9,450,000.00 0 basic 1,020.00 1,200.00008 A1Q2 0 1,150.00 0 50.0000 at bound 1,100.00 M9 A2Q1 0 1,200.00 0 150.0000 at bound 1,050.00 M10 A2Q2 8,500.00 1,100.00 9,350,000.00 0 basic -M 1,130.000011 Q1H 0 100.0000 0 5.0000 at bound 95.00 M12 Q1D 5,000.00 65.0000 325,000.00 0 basic -2,465.00 70.000013 Q1W 4,000.00 90.0000 360,000.00 0 basic 85.0000 95.000014 Q2H 2,500.00 95.0000 237,500.00 0 basic -2,465.00 100.000015 Q2D 0 70.0000 0 5.0000 at bound 65.00 M16 Q2W 6,000.00 90.0000 540,000.00 0 basic 85.00 95.0000
Objective Function (Min.) = 44,220,000.0000
Left Hand Right Hand Slack Shadow Allowable AllowableConst Side Direction Side Surplus Price Min. RHS Max. RHS
1 C1 25,000.00 <= 25,000.00 0 -5.00 22,500.0000 27,000.00002 C2 27,500.00 <= 30,000.00 2,500.00 0 27,500.0000 M3 C3 35,000.00 <= 50,000.00 15,000.00 0 35,000.0000 M4 C4 9,000.00 <= 9,000.00 0 0 9,000.0000 M5 C5 8,500.00 <= 10,000.00 1,500.00 0 8,500.0000 M6 C6 2,500.00 >= 2,500.00 0 2,560.00 1,833.3330 2,500.00007 C7 5,000.00 >= 5,000.00 0 2,530.00 4,333.3340 5,000.00008 C8 10,000.00 >= 10,000.00 0 2,555.00 9,333.3330 10,000.00009 C9 0 = 0 0 105.00 -2,000.0000 2,500.000010 C10 0 = 0 0 95.00 -25,500.0000 2,500.000011 C11 0 = 0 0 550.00 -18,000.0000 15,000.000012 C12 0 = 0 0 525.00 -17,000.0000 15,000.000013 C13 0 = 0 0 1,050.00 -9,000.0000 014 C14 0 = 0 0 1,100.00 -8,500.0000 1,500.000015 C15 9,000.00 <= 9,500.00 500.00 0 9,000.0000 M16 C16 8,500.00 <= 8,500.00 0 -30.00 8,500.0000 9,166.6670
Combined Report for GranolaDecision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 O1Q1 25,000.00 100.0000 2,500,000.00 0 basic -M 105.00002 O1Q2 0 110.0000 0 20.0000 at bound 90.00 M3 O2Q1 2,000.00 105.0000 210,000.00 0 basic 100.00 M4 O2Q2 25,500.00 95.0000 2,422,500.00 0 basic -M 105.00005 RQ1 18,000.00 550.0000 9,900,000.00 0 basic 535.00 M6 RQ2 17,000.00 525.0000 8,925,000.00 0 basic -M 540.00007 A1Q1 9,000.00 1,050.00 9,450,000.00 0 basic 1,020.00 1,200.00008 A1Q2 0 1,150.00 0 50.0000 at bound 1,100.00 M9 A2Q1 0 1,200.00 0 150.0000 at bound 1,050.00 M10 A2Q2 8,500.00 1,100.00 9,350,000.00 0 basic -M 1,130.000011 Q1H 0 100.0000 0 5.0000 at bound 95.00 M12 Q1D 5,000.00 65.0000 325,000.00 0 basic -2,465.00 70.000013 Q1W 4,000.00 90.0000 360,000.00 0 basic 85.0000 95.000014 Q2H 2,500.00 95.0000 237,500.00 0 basic -2,465.00 100.000015 Q2D 0 70.0000 0 5.0000 at bound 65.00 M16 Q2W 6,000.00 90.0000 540,000.00 0 basic 85.00 95.0000
Objective Function (Min.) = 44,220,000.0000
Left Hand Right Hand Slack Shadow Allowable AllowableConst Side Direction Side Surplus Price Min. RHS Max. RHS
1 C1 25,000.00 <= 25,000.00 0 -5.00 22,500.0000 27,000.00002 C2 27,500.00 <= 30,000.00 2,500.00 0 27,500.0000 M3 C3 35,000.00 <= 50,000.00 15,000.00 0 35,000.0000 M4 C4 9,000.00 <= 9,000.00 0 0 9,000.0000 M5 C5 8,500.00 <= 10,000.00 1,500.00 0 8,500.0000 M6 C6 2,500.00 >= 2,500.00 0 2,560.00 1,833.3330 2,500.00007 C7 5,000.00 >= 5,000.00 0 2,530.00 4,333.3340 5,000.00008 C8 10,000.00 >= 10,000.00 0 2,555.00 9,333.3330 10,000.00009 C9 0 = 0 0 105.00 -2,000.0000 2,500.000010 C10 0 = 0 0 95.00 -25,500.0000 2,500.000011 C11 0 = 0 0 550.00 -18,000.0000 15,000.000012 C12 0 = 0 0 525.00 -17,000.0000 15,000.000013 C13 0 = 0 0 1,050.00 -9,000.0000 014 C14 0 = 0 0 1,100.00 -8,500.0000 1,500.000015 C15 9,000.00 <= 9,500.00 500.00 0 9,000.0000 M16 C16 8,500.00 <= 8,500.00 0 -30.00 8,500.0000 9,166.6670
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
Mon
day
Tu
esd
ayA Multi-Period Transshipment
Problem
1. Inventories and multi-period planning2. Limitations on shipping quantities3. Changes in demand 4. Multimode shipping5. Returns6. Reverse logistics
Aggregate Production Planning (APP)
A macro approach to operational planning that focuses on overall capacity. APP links higher-level facility planning to lower level scheduling decisions within a medium-term planning horizon (2 to 18 months).
Aggregate Production Planning (APP)
Material RequirementsPlanning (MRP)
Master ProductionScheduling (MPS)
Shop FloorControl
Purchasing
DemandPlanning
and Forecasting
CapacityPlanning
(CRP) Material and Capacity Plans
APP (cont)
• Role of Aggregate Planning– Long-term planning function
– Strategic preparation for tactical actions
• Aggregate Planning Issues
– Staffing -hiring, firing, training
– Procurement - supplier contracts for materials, components
– Sub-Contracting - capacity vendoring
– Marketing - promotional activities
Basic Aggregate Planning (cont.)
CottonShirts
Men’sWomen’s Boy’s Girl’s
StyleA
StyleB
StyleC
Week1 Week2 Week3 Week4 Week5
Level of Aggregation
- product - time
Typical managerial objectives Minimize production costsMaximize profitsMaximize customer serviceMinimize inventoryMinimize changes in production rates Minimize changes in work-force levelsMaximize utilization of plant and equipment
Reactive alternativesAdjusting the workforce (hire, fire and layoff)Overtime and undertimeVacation schedules Anticipation inventorySubcontractorsBackorder, backlogs and stockouts
Aggressive alternatives (adjusting the demand patterns)Complementary productsCreative pricing
Anticipation Inventory
Pure APP Planning Strategies
Time
Un
its
Demand
Production
Pure APP Planning Strategies
Time
Un
its
Demand & Production
Tools and Methods Used to Develop Aggregate Plans
1. Trial and error (most commonly used technique)
2. Mathematical Techniques
3. Simulation (more complex trial and error?)
A Basic Linear Programming Formulation
• The heart of the APPLP formulation is the production constraint
• (Beginning Inventory) + Production – (Ending Inventory) = Demand
• Or I1 + P - I2 = D
• Chained over multiple periods by inventory flow
• I1 + P1 - I2 = D1
• I2 + P2 - I3 = D2
• I3 + P3 - I4 = D3
• etc…
I B P O S F I B
F Dt t t t t t t t
t t
1 1
Where:t= Number of Discrete Planning PeriodI = Units of InventoryP= Units of Production in Regular TimeO= Units of Production in OvertimeS= Units of Production SubcontractedD= Units of DemandF= Units of Filled DemandB= Units Backordered
A More Complete Production Constraint
LP Example 1:
Maximize -1I1-1I2-1I3-1I4-1I5-3B1-3B2-3B3-3B4-3B5-12P1-14P2-14P3-16P4-18O1- 20O2-20O3-22O4- 20S1-20S2-20S3-20S4+80F1+80F2+80F3+80F4
Subject to: 1I1-1I2-1B1+1B2+1P1+1O1+1S1-1F1=01I2-1I3-1B2+1B3+1P2+1O2+1S2-1F2=01I3-1I4-1B3+1B4+1P3+1O3+1S3-1F3=01I4-1I5-1B4+1B5+1P4+1O4+1S4-1F4=01F1<=2001F2<=2501F3<=3501F4<=250
Combined Report
Decision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 I1 800.0000 -1.0000 -800.0000 0 basic -4.0000 02 I2 600.0000 -1.0000 -600.0000 0 basic -4.0000 03 I3 350.0000 -1.0000 -350.0000 0 basic -4.0000 04 I4 0 -1.0000 0 -1.0000 at bound -M 05 I5 0 -1.0000 0 -4.0000 at bound -M 3.00006 B1 0 -3.0000 0 -4.0000 at bound -M 1.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -4.0000 at bound -M 1.00009 B4 0 -3.0000 0 -3.0000 at bound -M 010 B5 250.0000 -3.0000 -750.0000 0 basic -4.0000 011 P1 0 -12.0000 0 -11.0000 at bound -M -1.000012 P2 0 -14.0000 0 -12.0000 at bound -M -2.000013 P3 0 -14.0000 0 -11.0000 at bound -M -3.000014 P4 0 -16.0000 0 -13.0000 at bound -M -3.000015 O1 0 -18.0000 0 -17.0000 at bound -M -1.000016 O2 0 -20.0000 0 -18.0000 at bound -M -2.000017 O3 0 -20.0000 0 -17.0000 at bound -M -3.000018 O4 0 -22.0000 0 -19.0000 at bound -M -3.000019 S1 0 -20.0000 0 -19.0000 at bound -M -1.000020 S2 0 -20.0000 0 -18.0000 at bound -M -2.000021 S3 0 -20.0000 0 -17.0000 at bound -M -3.000022 S4 0 -20.0000 0 -17.0000 at bound -M -3.000023 F1 200.0000 80.0000 16,000.0000 0 basic 1.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 2.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 3.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 3.0000 M
Objective Function (Max.) = 81,500.0000
Decision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 I1 800.0000 -1.0000 -800.0000 0 basic -4.0000 02 I2 600.0000 -1.0000 -600.0000 0 basic -4.0000 03 I3 350.0000 -1.0000 -350.0000 0 basic -4.0000 04 I4 0 -1.0000 0 -1.0000 at bound -M 05 I5 0 -1.0000 0 -4.0000 at bound -M 3.00006 B1 0 -3.0000 0 -4.0000 at bound -M 1.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -4.0000 at bound -M 1.00009 B4 0 -3.0000 0 -3.0000 at bound -M 010 B5 250.0000 -3.0000 -750.0000 0 basic -4.0000 011 P1 0 -12.0000 0 -11.0000 at bound -M -1.000012 P2 0 -14.0000 0 -12.0000 at bound -M -2.000013 P3 0 -14.0000 0 -11.0000 at bound -M -3.000014 P4 0 -16.0000 0 -13.0000 at bound -M -3.000015 O1 0 -18.0000 0 -17.0000 at bound -M -1.000016 O2 0 -20.0000 0 -18.0000 at bound -M -2.000017 O3 0 -20.0000 0 -17.0000 at bound -M -3.000018 O4 0 -22.0000 0 -19.0000 at bound -M -3.000019 S1 0 -20.0000 0 -19.0000 at bound -M -1.000020 S2 0 -20.0000 0 -18.0000 at bound -M -2.000021 S3 0 -20.0000 0 -17.0000 at bound -M -3.000022 S4 0 -20.0000 0 -17.0000 at bound -M -3.000023 F1 200.0000 80.0000 16,000.0000 0 basic 1.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 2.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 3.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 3.0000 M
Objective Function (Max.) = 81,500.0000
Combined Report (cont.)
Left Hand Right Hand Slack Shadow Allowable AllowableConstraint Side Direction Side or Surplus Price Min. RHS Max. RHS
1 C1 0 = 0 0 -1.0000 -800.0000 M2 C2 0 = 0 0 -2.0000 -600.0000 M3 C3 0 = 0 0 -3.0000 -350.0000 M4 C4 0 = 0 0 -3.0000 -250.0000 M5 C5 200.0000 <= 200.0000 0 79.0000 0 M6 C6 250.0000 <= 250.0000 0 78.0000 0 M7 C7 350.0000 <= 350.0000 0 77.0000 0 M8 C8 250.0000 <= 250.0000 0 77.0000 0 M
Left Hand Right Hand Slack Shadow Allowable AllowableConstraint Side Direction Side or Surplus Price Min. RHS Max. RHS
1 C1 0 = 0 0 -1.0000 -800.0000 M2 C2 0 = 0 0 -2.0000 -600.0000 M3 C3 0 = 0 0 -3.0000 -350.0000 M4 C4 0 = 0 0 -3.0000 -250.0000 M5 C5 200.0000 <= 200.0000 0 79.0000 0 M6 C6 250.0000 <= 250.0000 0 78.0000 0 M7 C7 350.0000 <= 350.0000 0 77.0000 0 M8 C8 250.0000 <= 250.0000 0 77.0000 0 M
LP Example 2:
Maximize -1I1-1I2-1I3-1I4-1I5-3B1-3B2-3B3-3B4-3B5-12P1-14P2- 14P3-16P4-18O1-20O2-20O3-22O4-20S1-20S2-20S3-20S4+80F1+80F2+80F3+80F4
Subject to: 1I1-1I2-1B1+1B2+1P1+1O1+1S1-1F1=01I2-1I3-1B2+1B3+1P2+1O2+1S2-1F2=01I3-1I4-1B3+1B4+1P3+1O3+1S3-1F3=01I4-1I5-1B4+1B5+1P4+1O4+1S4-1F4=01F1<=2001F2<=2501F3<=3501F4<=250
I1=0
Combined Report
Decision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 I1 0 -1.0000 0 0 basic -M M2 I2 0 -1.0000 0 -4.0000 at bound -M 3.00003 I3 0 -1.0000 0 -4.0000 at bound -M 3.00004 I4 0 -1.0000 0 -4.0000 at bound -M 3.00005 I5 0 -1.0000 0 -4.0000 at bound -M 3.00006 B1 0 -3.0000 0 -15.0000 at bound -M 12.00007 B2 200.0000 -3.0000 -600.0000 0 basic -3.0000 1.00008 B3 450.0000 -3.0000 -1,350.0000 0 basic -3.0000 1.00009 B4 800.0000 -3.0000 -2,400.0000 0 basic -3.0000 1.000010 B5 1,050.0000 -3.0000 -3,150.0000 0 basic -3.0000 1.000011 P1 0 -12.0000 0 0 at bound -M -12.000012 P2 0 -14.0000 0 -5.0000 at bound -M -9.000013 P3 0 -14.0000 0 -8.0000 at bound -M -6.000014 P4 0 -16.0000 0 -13.0000 at bound -M -3.000015 O1 0 -18.0000 0 -6.0000 at bound -M -12.000016 O2 0 -20.0000 0 -11.0000 at bound -M -9.000017 O3 0 -20.0000 0 -14.0000 at bound -M -6.000018 O4 0 -22.0000 0 -19.0000 at bound -M -3.000019 S1 0 -20.0000 0 -8.0000 at bound -M -12.000020 S2 0 -20.0000 0 -11.0000 at bound -M -9.000021 S3 0 -20.0000 0 -14.0000 at bound -M -6.000022 S4 0 -20.0000 0 -17.0000 at bound -M -3.000023 F1 200.0000 80.0000 16,000.0000 0 basic 12.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 9.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 6.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 3.0000 M
Objective Function (Max.) = 76,500.0000 (Note: Alternate Solution Exists!!)
Decision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 I1 0 -1.0000 0 0 basic -M M2 I2 0 -1.0000 0 -4.0000 at bound -M 3.00003 I3 0 -1.0000 0 -4.0000 at bound -M 3.00004 I4 0 -1.0000 0 -4.0000 at bound -M 3.00005 I5 0 -1.0000 0 -4.0000 at bound -M 3.00006 B1 0 -3.0000 0 -15.0000 at bound -M 12.00007 B2 200.0000 -3.0000 -600.0000 0 basic -3.0000 1.00008 B3 450.0000 -3.0000 -1,350.0000 0 basic -3.0000 1.00009 B4 800.0000 -3.0000 -2,400.0000 0 basic -3.0000 1.000010 B5 1,050.0000 -3.0000 -3,150.0000 0 basic -3.0000 1.000011 P1 0 -12.0000 0 0 at bound -M -12.000012 P2 0 -14.0000 0 -5.0000 at bound -M -9.000013 P3 0 -14.0000 0 -8.0000 at bound -M -6.000014 P4 0 -16.0000 0 -13.0000 at bound -M -3.000015 O1 0 -18.0000 0 -6.0000 at bound -M -12.000016 O2 0 -20.0000 0 -11.0000 at bound -M -9.000017 O3 0 -20.0000 0 -14.0000 at bound -M -6.000018 O4 0 -22.0000 0 -19.0000 at bound -M -3.000019 S1 0 -20.0000 0 -8.0000 at bound -M -12.000020 S2 0 -20.0000 0 -11.0000 at bound -M -9.000021 S3 0 -20.0000 0 -14.0000 at bound -M -6.000022 S4 0 -20.0000 0 -17.0000 at bound -M -3.000023 F1 200.0000 80.0000 16,000.0000 0 basic 12.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 9.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 6.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 3.0000 M
Objective Function (Max.) = 76,500.0000 (Note: Alternate Solution Exists!!)
LP Example 3:
Maximize -1I1-1I2-1I3-1I4-1I5-3B1-3B2-3B3-3B4-3B5-12P1-14P2-14P3-16P4-18O1-20O2-20O3-22O4-20S1-20S2-20S3-20S4+80F1+80F2+80F3+80F4
Subject to: 1I1-1I2-1B1+1B2+1P1+1O1+1S1-1F1=01I2-1I3-1B2+1B3+1P2+1O2+1S2-1F2=01I3-1I4-1B3+1B4+1P3+1O3+1S3-1F3=01I4-1I5-1B4+1B5+1P4+1O4+1S4-1F4=01F1<=2001F2<=2501F3<=3501F4<=250I1=0
B5<=10
Combined Report
Decision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 I1 0 -1.0000 0 0 basic -M M2 I2 840.0000 -1.0000 -840.0000 0 basic -1.0000 3.00003 I3 590.0000 -1.0000 -590.0000 0 basic -1.0000 3.00004 I4 240.0000 -1.0000 -240.0000 0 basic -2.0000 3.00005 I5 0 -1.0000 0 -16.0000 at bound -M 15.00006 B1 0 -3.0000 0 -15.0000 at bound -M 12.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -4.0000 at bound -M 1.00009 B4 0 -3.0000 0 -4.0000 at bound -M 1.000010 B5 10.0000 -3.0000 -30.0000 0 basic -15.0000 M11 P1 1,040.0000 -12.0000 -12,480.0000 0 basic -12.0000 012 P2 0 -14.0000 0 -1.0000 at bound -M -13.000013 P3 0 -14.0000 0 0 at bound -M -14.000014 P4 0 -16.0000 0 -1.0000 at bound -M -15.000015 O1 0 -18.0000 0 -6.0000 at bound -M -12.000016 O2 0 -20.0000 0 -7.0000 at bound -M -13.000017 O3 0 -20.0000 0 -6.0000 at bound -M -14.000018 O4 0 -22.0000 0 -7.0000 at bound -M -15.000019 S1 0 -20.0000 0 -8.0000 at bound -M -12.000020 S2 0 -20.0000 0 -7.0000 at bound -M -13.000021 S3 0 -20.0000 0 -6.0000 at bound -M -14.000022 S4 0 -20.0000 0 -5.0000 at bound -M -15.000023 F1 200.0000 80.0000 16,000.0000 0 basic 12.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 13.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 14.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 15.0000 M
Objective Function (Max.) = 69,820.0000 (Note: Alternate Solution Exists!!)
Decision Solution Unit Cost Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 I1 0 -1.0000 0 0 basic -M M2 I2 840.0000 -1.0000 -840.0000 0 basic -1.0000 3.00003 I3 590.0000 -1.0000 -590.0000 0 basic -1.0000 3.00004 I4 240.0000 -1.0000 -240.0000 0 basic -2.0000 3.00005 I5 0 -1.0000 0 -16.0000 at bound -M 15.00006 B1 0 -3.0000 0 -15.0000 at bound -M 12.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -4.0000 at bound -M 1.00009 B4 0 -3.0000 0 -4.0000 at bound -M 1.000010 B5 10.0000 -3.0000 -30.0000 0 basic -15.0000 M11 P1 1,040.0000 -12.0000 -12,480.0000 0 basic -12.0000 012 P2 0 -14.0000 0 -1.0000 at bound -M -13.000013 P3 0 -14.0000 0 0 at bound -M -14.000014 P4 0 -16.0000 0 -1.0000 at bound -M -15.000015 O1 0 -18.0000 0 -6.0000 at bound -M -12.000016 O2 0 -20.0000 0 -7.0000 at bound -M -13.000017 O3 0 -20.0000 0 -6.0000 at bound -M -14.000018 O4 0 -22.0000 0 -7.0000 at bound -M -15.000019 S1 0 -20.0000 0 -8.0000 at bound -M -12.000020 S2 0 -20.0000 0 -7.0000 at bound -M -13.000021 S3 0 -20.0000 0 -6.0000 at bound -M -14.000022 S4 0 -20.0000 0 -5.0000 at bound -M -15.000023 F1 200.0000 80.0000 16,000.0000 0 basic 12.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 13.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 14.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 15.0000 M
Objective Function (Max.) = 69,820.0000 (Note: Alternate Solution Exists!!)
Work Force Capacity Constraints
• The APPLP can incorporate short-term capacity adjustments
• For example, a work force can be increased or diminished
• (Workers in p1) = (Workers in p0) + (p0 hires) – (p0 layoffs)
• or Wi – Hi + Fi = W0
• and chained over multiple periods by work force level
• W1 - H1 + L1 = W0
• W2 - W1 - H2 + L2 = 0
• W3 - W2 - H3 + L3 = 0
LP Example 4:
Maximize -1I1-1I2-1I3-1I4-1I5-3B1-3B2-3B3-3B4-5B5- 12P1-14P2-14P3-16P4-18O1-20O2-20O3-22O4- 20S1-20S2-20S3-20S4+80F1+80F2+80F3+80F4-250W1- 250W2-250W3-250W4-500H1-500H2-500H3-500H4-1000L1-1000L2- 1000L3-1000L4
Subject to: 1I1-1I2-1B1+1B2+1P1+1O1+1S1-1F1=01I2-1I3-1B2+1B3+1P2+1O2+1S2-1F2=01I3-1I4-1B3+1B4+1P3+1O3+1S3-1F3=01I4-1I5-1B4+1B5+1P4+1O4+1S4-1F4=0
1F1<=2001F2<=2501F3<=3501F4<=250
1I1=01B5<=10
1W1-1H1+1L1=20-1W1+1W2-1H2+1L2=0-1W2+1W3-1H3+1L3=0-1W3+1W4-1H4+1L4=0
1P1-10W1<=01P2-10W2<=01P3-10W3<=01P4-10W4<=01O1-2W1<=01O2-2W2<=01O3-2W3<=01O4-2W4<=0
Combined Report
Decision Solution Unit Cost or Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 I1 0 -1.0000 0 0 basic -M M2 I2 40.0000 -1.0000 -40.0000 0 basic -2.0000 03 I3 0 -1.0000 0 -1.0000 at bound -M 04 I4 0 -1.0000 0 -1.0000 at bound -M 0.00005 I5 0 -1.0000 0 -21.0000 at bound -M 20.00006 B1 0 -3.0000 0 -22.0000 at bound -M 19.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -3.0000 at bound -M 09 B4 0 -3.0000 0 -3.0000 at bound -M 0.000010 B5 10.0000 -5.0000 -50.0000 0 basic -20.0000 M11 P1 200.0000 -12.0000 -2,400.0000 0 basic -19.0000 114.800012 P2 200.0000 -14.0000 -2,800.0000 0 basic -20.0000 95.000013 P3 200.0000 -14.0000 -2,800.0000 0 basic -20.0000 76.000014 P4 200.0000 -16.0000 -3,200.0000 0 basic -20.0000 55.000015 O1 40.0000 -18.0000 -720.0000 0 basic -19.0000 616.000016 O2 10.0000 -20.0000 -200.0000 0 basic -20.0000 -19.000017 O3 40.0000 -20.0000 -800.0000 0 basic -20.0000 430.000018 O4 0 -22.0000 0 -2.0000 at bound -M -20.000019 S1 0 -20.0000 0 -1.0000 at bound -M -19.000020 S2 0 -20.0000 0 0 at bound -M -20.000021 S3 110.0000 -20.0000 -2,200.0000 0 basic -21.0000 -20.000022 S4 40.0000 -20.0000 -800.0000 0 basic -21.0000 -17.000023 F1 200.0000 80.0000 16,000.0000 0 basic 19.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 20.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 20.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 20.0000 M27 W1 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 1,018.000028 W2 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 840.000029 W3 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 650.000030 W4 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 460.000031 H1 0 -500.0000 0 -1,268.0000 at bound -M 768.000032 H2 0 -500.0000 0 -1,090.0000 at bound -M 590.000033 H3 0 -500.0000 0 -900.0000 at bound -M 400.000034 H4 0 -500.0000 0 -710.0000 at bound -M 210.000035 L1 0 -1,000.0000 0 -232.0000 at bound -M -768.000036 L2 0 -1,000.0000 0 -410.0000 at bound -M -590.000037 L3 0 -1,000.0000 0 -600.0000 at bound -M -400.000038 L4 0 -1,000.0000 0 -790.0000 at bound -M -210.0000
Objective Function (Max.) = 47,990.0000 (Note: Alternate Solution Exists!!)
Decision Solution Unit Cost or Total Reduced Basis Allowable AllowableVariable Value Profit c(j) Contribution Cost Status Min. c(j) Max. c(j)
1 I1 0 -1.0000 0 0 basic -M M2 I2 40.0000 -1.0000 -40.0000 0 basic -2.0000 03 I3 0 -1.0000 0 -1.0000 at bound -M 04 I4 0 -1.0000 0 -1.0000 at bound -M 0.00005 I5 0 -1.0000 0 -21.0000 at bound -M 20.00006 B1 0 -3.0000 0 -22.0000 at bound -M 19.00007 B2 0 -3.0000 0 -4.0000 at bound -M 1.00008 B3 0 -3.0000 0 -3.0000 at bound -M 09 B4 0 -3.0000 0 -3.0000 at bound -M 0.000010 B5 10.0000 -5.0000 -50.0000 0 basic -20.0000 M11 P1 200.0000 -12.0000 -2,400.0000 0 basic -19.0000 114.800012 P2 200.0000 -14.0000 -2,800.0000 0 basic -20.0000 95.000013 P3 200.0000 -14.0000 -2,800.0000 0 basic -20.0000 76.000014 P4 200.0000 -16.0000 -3,200.0000 0 basic -20.0000 55.000015 O1 40.0000 -18.0000 -720.0000 0 basic -19.0000 616.000016 O2 10.0000 -20.0000 -200.0000 0 basic -20.0000 -19.000017 O3 40.0000 -20.0000 -800.0000 0 basic -20.0000 430.000018 O4 0 -22.0000 0 -2.0000 at bound -M -20.000019 S1 0 -20.0000 0 -1.0000 at bound -M -19.000020 S2 0 -20.0000 0 0 at bound -M -20.000021 S3 110.0000 -20.0000 -2,200.0000 0 basic -21.0000 -20.000022 S4 40.0000 -20.0000 -800.0000 0 basic -21.0000 -17.000023 F1 200.0000 80.0000 16,000.0000 0 basic 19.0000 M24 F2 250.0000 80.0000 20,000.0000 0 basic 20.0000 M25 F3 350.0000 80.0000 28,000.0000 0 basic 20.0000 M26 F4 250.0000 80.0000 20,000.0000 0 basic 20.0000 M27 W1 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 1,018.000028 W2 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 840.000029 W3 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 650.000030 W4 20.0000 -250.0000 -5,000.0000 0 basic -482.0000 460.000031 H1 0 -500.0000 0 -1,268.0000 at bound -M 768.000032 H2 0 -500.0000 0 -1,090.0000 at bound -M 590.000033 H3 0 -500.0000 0 -900.0000 at bound -M 400.000034 H4 0 -500.0000 0 -710.0000 at bound -M 210.000035 L1 0 -1,000.0000 0 -232.0000 at bound -M -768.000036 L2 0 -1,000.0000 0 -410.0000 at bound -M -590.000037 L3 0 -1,000.0000 0 -600.0000 at bound -M -400.000038 L4 0 -1,000.0000 0 -790.0000 at bound -M -210.0000
Objective Function (Max.) = 47,990.0000 (Note: Alternate Solution Exists!!)
Combined Report (cont.)
Left Hand Right Hand Slack Shadow Allowable AllowableConstraint Side Direction Side or Surplus Price Min. RHS Max. RHS
1 C1 0 = 0 0 -19.0000 -10.0000 30.00002 C2 0 = 0 0 -20.0000 -10.0000 30.00003 C3 0 = 0 0 -20.0000 -110.0000 M4 C4 0 = 0 0 -20.0000 -40.0000 M5 C5 200.0000 <= 200.0000 0 61.0000 190.0000 230.00006 C6 250.0000 <= 250.0000 0 60.0000 240.0000 280.00007 C7 350.0000 <= 350.0000 0 60.0000 240.0000 M8 C8 250.0000 <= 250.0000 0 60.0000 210.0000 M9 C9 0 = 0 0 18.0000 0 10.000010 C10 10.0000 <= 10.0000 0 15.0000 0 50.000011 C11 20.0000 = 20.0000 0 -768.0000 18.7500 20.454512 C12 0 = 0 0 -590.0000 -2.5000 1.000013 C13 0 = 0 0 -400.0000 -20.0000 4.000014 C14 0 = 0 0 -210.0000 -20.0000 4.000015 C15 0 <= 0 0 7.0000 -30.0000 10.000016 C16 0 <= 0 0 6.0000 -30.0000 10.000017 C17 0 <= 0 0 6.0000 -200.0000 110.000018 C18 0 <= 0 0 4.0000 -200.0000 40.000019 C19 0 <= 0 0 1.0000 -30.0000 10.000020 C20 -30.0000 <= 0 30.0000 0 -30.0000 M21 C21 0 <= 0 0 0 -40.0000 110.000022 C22 -40.0000 <= 0 40.0000 0 -40.0000 M
Left Hand Right Hand Slack Shadow Allowable AllowableConstraint Side Direction Side or Surplus Price Min. RHS Max. RHS
1 C1 0 = 0 0 -19.0000 -10.0000 30.00002 C2 0 = 0 0 -20.0000 -10.0000 30.00003 C3 0 = 0 0 -20.0000 -110.0000 M4 C4 0 = 0 0 -20.0000 -40.0000 M5 C5 200.0000 <= 200.0000 0 61.0000 190.0000 230.00006 C6 250.0000 <= 250.0000 0 60.0000 240.0000 280.00007 C7 350.0000 <= 350.0000 0 60.0000 240.0000 M8 C8 250.0000 <= 250.0000 0 60.0000 210.0000 M9 C9 0 = 0 0 18.0000 0 10.000010 C10 10.0000 <= 10.0000 0 15.0000 0 50.000011 C11 20.0000 = 20.0000 0 -768.0000 18.7500 20.454512 C12 0 = 0 0 -590.0000 -2.5000 1.000013 C13 0 = 0 0 -400.0000 -20.0000 4.000014 C14 0 = 0 0 -210.0000 -20.0000 4.000015 C15 0 <= 0 0 7.0000 -30.0000 10.000016 C16 0 <= 0 0 6.0000 -30.0000 10.000017 C17 0 <= 0 0 6.0000 -200.0000 110.000018 C18 0 <= 0 0 4.0000 -200.0000 40.000019 C19 0 <= 0 0 1.0000 -30.0000 10.000020 C20 -30.0000 <= 0 30.0000 0 -30.0000 M21 C21 0 <= 0 0 0 -40.0000 110.000022 C22 -40.0000 <= 0 40.0000 0 -40.0000 M
APP Example
Frank-N-Stuff Manufacturing makes light fixtures. The production manager needs help preparing an aggregateproduction plan for the year 2002. Expected demand and the number of weekends in each period is provided in thetable below.
Projected demand in 2002 Number of weekendsPeriod 1 2000 13Period 2 1800 12Period 3 4500 13Period 4 1200 11
It takes 4 labor hours to produce one fixture. The contract with the Labor Union specifies that all employees arepaid 8 hours a day each workday and management assumes that there are 55 regular workdays available eachperiod. No workday overtime is allowed. However, each employee can be asked to work up to 8 hours of overtimeeach weekend if needed. All demand must be satisfied by year’s end (end of year backlog is not allowed). Ifnecessary, subcontracting can be used. Currently there are 10 employees working in the plant.
Inventory Holding Cost (product/period) $8Backlog Cost (product/period) $16Subcontracting Cost (product) $150Hiring & Training Cost (worker) $2500Layoff Cost (worker) $1000Regular Time Cost (worker/hour) $15Overtime Cost (worker/hour) $20
a. Prepare an aggregate plan that will minimize total costs and meet the demand.
b. Prepare an aggregate plan that will minimize total costs, meet the demand and keep regular productionconstant throughout the year (i.e., the number of chairs produced in regular time must be the same for allfour periods)
top related