Linear Programming
• Linear programming is a
mathematical technique that
allows the decision maker to
allocate scarce resources in such
a way as to optimize an
objective of interest. It is linear
because the relationships
involved are linear.
• Problem Formulation
• Graphical Analysis
• Spreadsheet Solution
• Sensitivity Analysis
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HMP654/EXECMAS
Linear Programming
Case Problem - (A) p. 99
The administrator of a surgical specialty division of Northpark University Hospital and
Medical Center is studying the potential impact of a planned marketing program on two highvolume, short-stay inpatient procedures offered by the division to its Medicare patient
population. These procedures are labeled DRG-1 and DRG-2 in the Hospital’s case-mix coding
system.
The division’s hospital service unit has excess capacity for additional inpatients, with
capacity expressed in terms of inpatient days, nursing service hours, and diagnostic procedures
available per month on the unit. The administrator knows the amount of reimbursement from the
Medicare agency for each DRG.
The Hospital’s accounting system provides the Medicare reimbursement amounts as
follows:
DRG-1
DRG-2
$ 2,125
$ 2,500
Northpark University Hospital and Medical Center has developed a cost-accounting
information system that provides the following information on unit costs:
Inpatient days
Nursing service
Diagnostic procedures
$ 600 per day
$ 40 per hour
$ 75 per procedure
The Hospital’s management engineering staff has also carried out a cost-finding study on
high-volume DRGs for use in the case-mix analysis management information system. The study
provides information on resources consumed for each of the two DRGs under review:
Average length of stay
Nursing hours
Diagnostic procedures
DRG-1
2 days
10
3
DRG-2
1 day
30
4
Excess service capacity on the surgical specialty unit is as follows:
Inpatient days
Nursing service hours
Diagnostic procedures
120 per month
900 per month
360 per month
Working with a management engineer from the Hospital, the administrator wants to
1. compute the optimum mix of the two DRGs that would maximize net revenue within
the excess capacity available on the division’s nursing unit
2. determine where to place the emphasis in the planned marketing campaign
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HMP654/EXECMAS
Linear Programming - Formulation
• Identify problem’s objective
• Identify decision variables
• Express objective as a linear
combination of decision
variables
• Express constraints as linear
combinations of decision
variables
• Identify upper or lower bounds
on the decision variables
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HMP654/EXECMAS
Linear Programming - Formulation
• Identify problem’s objective
– Maximize Total Net Revenue
• Identify Decision Variables
– X1 = Number of DRG-1
procedures performed in a
month
– X2 = Number of DRG-2
procedures performed in a
month
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HMP654/EXECMAS
Linear Programming - Formulation
• Express objective as a linear
combination of decision variables
Benefit - Cost Analysis
Per patient
DRG-1
DRG-2
$2,125
$2,500
Inpatent days
Nursing srevice
Diagnostic procedures
$1,200
$400
$225
$600
$1,200
$300
Total Cost
$1,825
$2,100
$300
$400
Revenue
Cost:
Net Revenue
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HMP654/EXECMAS
Linear Programming - Formulation
The objective function is:
Max Z = 300X1 + 400X2
• Express constraints as linear
combinations of decision
variables
2X1 +
X2 < 120 (Inpatient days)
10X1 + 30X2 < 900 (Nursing hours)
3X1 + 4X2 < 360 (Diagnostic procs.)
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HMP654/EXECMAS
Linear Programming - Formulation
• Identify upper or lower bounds
on the decision variables
X1 > 0
X2 > 0
(non-negativity
constraints)
• Complete l.p. formulation:
Max Z = 300X1 + 400X2
subject to
2X1 +
X2 < 120
10X1 + 30X2 < 900
3X1 + 4X2 < 360
X1, X2 > 0
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HMP654/EXECMAS
Linear Programming
Graphical Analysis
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HMP654/EXECMAS
Linear Programming
Graphical Analysis
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HMP654/EXECMAS
Linear Programming
Graphical Analysis
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HMP654/EXECMAS
Linear Programming
Graphical Analysis
Unique optimal solution
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HMP654/EXECMAS
Linear Programming
Graphical Analysis
Multiple Optimal Solutions
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HMP654/EXECMAS
Linear Programming
Graphical Analysis
Infeasible Problem
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HMP654/EXECMAS
Linear Programming
Graphical Analysis
Unbounded Problem
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HMP654/EXECMAS
Linear Programming
Spreadsheet Modeling
• Organize the data for the model on the
spreadsheet.
• Reserve separate cells in the
spreadsheet to represent each decision
variable in the algebraic model.
• Create a formula in a cell in the
spreadsheet that corresponds to the
objective function in the algebraic
model.
• For each constraint in the algebraic
model, create a formula in a cell in the
spreadsheet that corresponds to the
left-hand-side (LHS) of the constraint.
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HMP654/EXECMAS
Linear Programming
Spreadsheet Modeling
Objective
Coefficients
(C)
Objective
Function
(F)
Decision
Variables
(V)
Constraints
LHS
(F)
Constraints
Coefficients
(C)
Constraints
RHS
(C)
(V) - Variables
(C) - Constants
(F) - Formulas
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HMP654/EXECMAS
Linear Programming
Spreadsheet Modeling
Feasible Solution
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HMP654/EXECMAS
Linear Programming
Spreadsheet Modeling
Infeasible Solution
Case Problem - Spreadsheet Solution
Inpatient Days Constraint
No. of Procedures
DRG-1
20
Length of Stay per DRG (Days)
DRG-2
45
Profit Contribution
DRG-1
$300
DRG-1
2
DRG-2
1
Nursing Hours Constraint
DRG-2
$400
Used
85
Available
120
Used
1550
Available
900
Used
240
Available
360
Infeasible!
Nursing Hours per DRG
DRG-1
10
DRG-2
30
Net Revenue
Diagnostic Procedures Constraint
Profit
$24,000
Diagnostic Procedures per DRG
DRG-1
3
DRG-2
4
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HMP654/EXECMAS
Linear Programming
Spreadsheet Modeling
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HMP654/EXECMAS
Linear Programming
Spreadsheet Modeling
Optimal Solution
Case Problem - Spreadsheet Solution
Inpatient Days Constraint
No. of Procedures
DRG-1
54
Length of Stay per DRG (Days)
DRG-2
12
Profit Contribution
DRG-1
$300
DRG-1
2
DRG-2
1
Used
120
Available
120
Used
900
Available
900
Used
210
Available
360
Nursing Hours Constraint
DRG-2
$400
Nursing Hours per DRG
DRG-1
10
DRG-2
30
Net Revenue
Diagnostic Procedures Constraint
Profit
$21,000
Diagnostic Procedures per DRG
DRG-1
3
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DRG-2
4
HMP654/EXECMAS
Linear Programming
(From Warner and Holloway, 1978)
The dietary department of a hospital feels that patients on a “regular” diet should obtain the
following amount of nutrition daily:




between 2,600 and 4,000 calories,
at least 72 grams of protein
no more than 100 grams of fat, and
at least 12 mg of iron
The dietary department (let’s suppose) has eight food items it can serve to meet these requirements.
The cost per pound of each food item and its contribution to each of the four nutritional requirements are
given in the following table (the figures are not actual). What combination and amounts of the food items
will provide the nutritional requirements at lowest food cost?
i
1
2
3
4
5
6
7
8
Food Item
Flour
Milk
Beef
Chicken
Kidney Beans
Spinach
Fish
Potatoes
Cost/lb
.40
.35
2.50
.80
.80
.90
3.50
.40
Calories/lb
860
440
500
310
280
140
402
613
Protein/lb
15
31
60
63
51
41
78
30
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Fat/lb
16
39
110
84
12
4
34
20
Iron/lb
3
6
10
14
8
12
18
4
HMP654/EXECMAS
Linear Programming
• Identify problem’s objective
– Minimize total cost of diet.
• Identify the decision variables
Xi = No. of lbs. of food item i to be
included in the optimal diet.
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HMP654/EXECMAS
Linear Programming
• State the objective function as
a linear combination of the
decision variables.
Min Z = 0.40X1 + 0.35X2 + 2.50X3 +
0.80X4 + 0.80X5 + 0.90X6 +
3.50X7 + 0.40X8
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HMP654/EXECMAS
Linear Programming
• State the constraints as linear
combinations of the decision
variables.
calories
860X1 + 440X2 + 500X3 + 310X4 + 280X5 +
140X6 + 402X7 + 613X8 > 2,600
860X1 + 440X2 + 500X3 + 310X4 + 280X5 +
140X6 + 402X7 + 613X8 < 4,000
protein
15X1 + 31X2 + 60X3 + 63X4 + 51X5 + 41X6 +
78X7 + 30X8 > 72
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HMP654/EXECMAS
Linear Programming
fat
16X1 + 39X2 + 110X3 + 84X4 + 12X5 + 4X6 +
34X7 + 20X8 < 100
iron
3X1 + 6X2 + 10X3 + 14X4 + 8X5 + 12X6 +
18X7 + 4X8 > 12
• Identify any upper or lower
bounds on the decision
variables.
Xi > 0
i = 1,....,8
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HMP654/EXECMAS
Linear Programming
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HMP654/EXECMAS
Linear Programming
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HMP654/EXECMAS
Linear Programming
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HMP654/EXECMAS
Linear Programming Transportation Problem
Case Problem (A) p. 123
The director of Pharmacy Services for Tri-State Health Services Corporation is
responsible for management oversight of a contract with a major supplier of intravenous
fluids for the Corporation. There are four hospitals in the system supplied by the contract.
The supplier has three warehouses in the region from which the IV fluids can be obtained.
Transportation is by truck, and shipment costs are based upon the number of miles
traveled for delivery.
The director is preparing an order to be placed with the supplier for next month’s
shipment of fluids to each hospital in the system. Requirements are as follows:
Hospital A
Hospital B
Hospital C
Hospital D
1,200 cases needed
1,500 cases needed
2,300 cases needed
2,400 cases needed
The quantities available for shipment from each of the three warehouse locations
operated by the supplier, obtained by a phone call to the supplier’s shipping manager, are
as follows:
Warehouse #1
Warehouse #2
Warehouse #3
1,800 cases of IV fluids available to ship
2,400 cases of IV fluids available to ship
3,200 cases of IV fluids available to ship
The contract between Tri-State and the supplier calls for a standard shipment cost
(by truck) of $0.02 per case of IV fluid for each mile traveled in making the delivery. The
distances between each supply point and each destination are as follows:
Hospital
Warehouse #1
Warehouse #2
Warehouse #3
A
60
30
50
B
50
80
70
C
110
100
140
D
40
50
90
In placing the order, the director wants to include a shipment request that will
minimize transportation costs for the system.
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HMP654/EXECMAS
Linear Programming Transportation Problem
• Identify problem’s objective
– Minimize Transportation Costs
for the System.
• Identify the decision variables
– No. of cases of IV fluids shipped
from Warehouse i to Hospital j.
i = 1,..,3; j = A,...,D
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HMP654/EXECMAS
Linear Programming Transportation Problem
W3
X3A
A
X1A
B
X2B
W2
W1
D
C
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HMP654/EXECMAS
Linear Programming Transportation Problem
• State the objective function as a
linear combination of the
decision variables.
Cost contribution = $0.02 x distance (i,j)
Min Z = 1.20X1A + 1.00X1B + 2.20X1C + 0.80X1D +
0.60X2A + 1.60X2B + 2.00X2C + 1.00X2D +
1.00X3A + 1.40X3B + 2.80X3C + 1.80X3D
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HMP654/EXECMAS
Linear Programming Transportation Problem
• State the constraints as linear
combinations of the decision
variables.
Demand:
Hospital A
X1A + X2A + X3A = 1,200
Hospital B
X1B + X2B + X3B = 1,500
Hospital C
X1C + X2C + X3C = 2,300
Hospital D
X1D + X2D + X3D = 2,400
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HMP654/EXECMAS
Linear Programming Transportation Problem
Supply:
Warehouse #1
X1A + X1B + X1C + X1D < 1,800
Warehouse #2
X2A + X2B + X2C + X2D < 2,400
Warehouse #3
X3A + X3B + X3C + X3D < 3,200
• Identify any upper or lower
bounds on the decision
variables.
Xi,j > 0
i = 1,..,3; j = A,...,D
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HMP654/EXECMAS
Linear Programming Transportation Problem
Complete Formulation:
Min Z = 1.20X1A + 1.00X1B + 2.20X1C + 0.80X1D +
0.60X2A + 1.60X2B + 2.00X2C + 1.00X2D +
1.00X3A + 1.40X3B + 2.80X3C + 1.80X3D
s.t.
X1A + X2A + X3A = 1,200
X1B + X2B + X3B = 1,500
X1C + X2C + X3C = 2,300
X1D + X2D + X3D = 2,400
X1A + X1B + X1C + X1D < 1,800
X2A + X2B + X2C + X2D < 2,400
X3A + X3B + X3C + X3D < 3,200
Xi,j > 0
i = 1,..,3; j = A,...,D
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HMP654/EXECMAS
Linear Programming Transportation Problem
Spreadsheet Model
Transportation Assignment (no. of cases)
Hospital
A
B
Warehouse #1
Warehouse #2
Warehouse #3
Shipped to
Cases needed
0
1,200
0
1,500
C
D
0
2,300
0
2,400
Shipped
from
0
0
0
Avail.
supply
1,800
2,400
3,200
Distances
Warehouse #1
Warehouse #2
Warehouse #3
Cost per case per mile
A
60
30
50
Hospital
B
50
80
70
C
110
100
140
$0.02
D
40
50
90
Total Cost
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$0.00
HMP654/EXECMAS
Linear Programming Transportation Problem
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HMP654/EXECMAS
Linear Programming Transportation Problem
Optimal Solution
Transportation Assignment (no. of cases)
Hospital
A
B
Warehouse #1
0
0
Warehouse #2
0
0
Warehouse #3
1,200
1,500
Shipped to
Cases needed
1,200
1,200
1,500
1,500
C
0
2,300
0
2,300
2,300
D
1,800
100
500
Shipped
from
1,800
2,400
3,200
Avail.
supply
1,800
2,400
3,200
2,400
2,400
Distances
Warehouse #1
Warehouse #2
Warehouse #3
Cost per case per mile
A
60
30
50
Hospital
B
50
80
70
$0.02
C
110
100
140
D
40
50
90
Total Cost
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$10,340.00
HMP654/EXECMAS
Linear Programming Assignment Problem
Case Problem (A) p. 131
A clinical laboratory has four different types of tests to be carried out using
automated testing equipment. Four technicians are to be assigned to complete these tests.
The salary rates (per hour) for the technicians are as follows:
Technician A
Technician B
Technician C
Technician D
$7.00
$6.75
$8.25
$7.50
Because of different levels of training and experience, the technicians require
different amounts of time to complete each test as follows:
Technician A
Technician B
Technician C
Technician D
Test 1
30 minutes
25
18
30
Test 2
75
60
90
80
Test 3
15
18
12
20
Test 4
45
60
45
50
Your task is to assign the technicians to the automated equipment in the most costeffective manner.
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HMP654/EXECMAS
Linear Programming Assignment Problem
• Identify problem’s objective
– Minimize Total Costs
• Identify the decision variables
Xi,j = 1 if technician i is assigned to test j.
Xi,j = 0 if technician i is not assigned to
test j.
i = A,...,D ; j = 1,...,4
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HMP654/EXECMAS
Linear Programming Assignment Problem
• State the objective function as a
linear combination of the
decision variables.
Cost contribution = salary(i) x time (i,j)/60
Min Z = 3.50XA1 + 8.75XA2 + 1.75XA3 + 5.25XA4 +
2.81XB1 + 6.75XB2 + 2.03XB3 + 6.75XB4 +
2.48XC1 + 12.38XC2 + 1.65XC3 + 6.19XC4 +
3.75XD1 + 10XD2 +2.50XD3 + 6.25XD4
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HMP654/EXECMAS
Linear Programming Assignment Problem
• State the constraints as linear
combinations of the decision
variables.
– Each technician must be assigned
to one, and only one test.
XA1 + XA2 +XA3 + XA4 = 1
XB1 + XB2 + XB3 + XB4 = 1
XC1 + XC2 + XC3 + XC4 = 1
XD1 + XD2 + XD3 + XD4 =1
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HMP654/EXECMAS
Linear Programming Assignment Problem
– To each test, one, and only one
technician must be assigned
XA1 + XB1 + XC1 + XD1 = 1
XA2 + XB2 + XC2 + XD2 = 1
XA3 + XB3 + XC3 + XD3 = 1
XA4 + XB4 + XC4 + XD4 = 1
• Identify any upper or lower
bounds on the decision
variables.
Xi,j > 0
i = A,..,D; j = 1,...,4
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HMP654/EXECMAS
Linear Programming Assignment Problem
Complete Formulation:
Min Z = 3.50XA1 + 8.75XA2 + 1.75XA3 + 5.25XA4 +
2.81XB1 + 6.75XB2 + 2.03XB3 + 6.75XB4 +
2.48XC1 + 12.38XC2 + 1.65XC3 + 6.19XC4 +
3.75XD1 + 10XD2 + 2.50XD3 + 6.25XD4
s.t.
XA1 + XA2 +XA3 + XA4 = 1
XB1 + XB2 + XB3 + XB4 = 1
XC1 + XC2 + XC3 + XC4 = 1
XD1 + XD2 + XD3 + XD4 =1
XA1 + XB1 + XC1 + XD1 = 1
XA2 + XB2 + XC2 + XD2 = 1
XA3 + XB3 + XC3 + XD3 = 1
XA4 + XB4 + XC4 + XD4 = 1
Xi,j > 0
i = A,..,D; j = 1,...,4
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HMP654/EXECMAS
Linear Programming Assignment Problem
Spreadsheet Model
Technician Assignment
Test 1
Test 2
Test 3
Test 4
Technician A
Technician B
Technician C
Technician D
Sum
0
0
0
Total Cost
$0
0
Time to Complete Tests (minutes)
Technician A
Technician B
Technician C
Technician D
Sum
0
0
0
0
Salary Rates (per hour)
Test 1
30
25
18
30
Test 2
75
60
90
80
Test 3
15
18
12
20
Test 4
45
60
45
50
Test 1
$3.50
$2.81
$2.48
$3.75
Test 2
$8.75
$6.75
$12.38
$10.00
Test 3
$1.75
$2.03
$1.65
$2.50
Test 4
$5.25
$6.75
$6.19
$6.25
Technician A
Technician B
Technician C
Technician D
$7.00
$6.75
$8.25
$7.50
Cost per Assignment
Technician A
Technician B
Technician C
Technician D
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HMP654/EXECMAS
Linear Programming Assignment Problem
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HMP654/EXECMAS
Linear Programming Assignment Problem
Optimal Solution
Technician Assignment
Technician A
Technician B
Technician C
Technician D
Test 1
0
0
1
0
Sum
1
Test 2
0
1
0
0
1
Test 3
0
0
0
1
Test 4
1
0
0
0
1
Total Cost
$16.98
1
Time to Complete Tests (minutes)
Technician A
Technician B
Technician C
Technician D
Sum
1
1
1
1
Salary Rates (per hour)
Test 1
30
25
18
30
Test 2
75
60
90
80
Test 3
15
18
12
20
Test 4
45
60
45
50
Test 1
$3.50
$2.81
$2.48
$3.75
Test 2
$8.75
$6.75
$12.38
$10.00
Test 3
$1.75
$2.03
$1.65
$2.50
Test 4
$5.25
$6.75
$6.19
$6.25
Technician A
Technician B
Technician C
Technician D
$7.00
$6.75
$8.25
$7.50
Cost per Assignment
Technician A
Technician B
Technician C
Technician D
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HMP654/EXECMAS