Investigation of the influence of product variety on inventories in hospitals

Aravazhi, Agaraoli; Helgheim, Berit Irene; Jæger, Bjørn

doi:10.2478/emj-2020-0003

Figures & Tables

Product usage for a product group containing three products

PRODUCT USAGE NUMBER	U₁	U₂	U₃
0	0	0	0
1	1	0	0
2	0	1	0
3	1	1	0
4	0	0	1
5	1	0	1
6	0	1	1
7	1	1	1

Initial details of products within the non-sterile product group

	DEMAND	AVG. UNIT COST
NSP₁	260	NOK* 0.172
NSP₂	461	NOK 0.127
NSP₃	162	NOK 0.133

Cost when products are reduced from product group

DESIGN POINT		(1,1)			(2,1)			(3,1)			(3,2)			(4,1)			(5,1)			(5,2)			(6,1)			(6,2)
Description		NSP₁ is used. NSP₂ & NSP₃ are substituted by NSP₁			NSP₂ is used. NSP₁ & NSP₃ are substituted by NSP₂			NSP₁ &NSP₂ are used. NSP₃ is substituted by NSP₁			NSP1& NSP2 are used. NSP3 is substituted by NSP2			NSP₃ is used. NSP₁ & NSP₂ are substituted by NSP₃			NSP₁ & NSP₃ are used. NSP₂ is substituted by NSP₁			NSP₁ & NSP₃ are used. NSP₂ is substituted by NSP₃			NSP₂ & NSP₃ are used. NSP₁ is substituted by NSP₂			NSP₂ & NSP₃ are used. NSP₁ is substituted by NSP₃
Product		NSP₁	NSP₂	NSP₃	NSP₁	NSP2	NSP3	NSP₁	NSP2	NSP3	NSP₁	NSP2	NSP3	NSP₁	NSP2	NSP3	NSP₁	NSP2	NSP3	NSP₁	NSP2	NSP3	NSP₁	NSP2	NSP3	NSP₁	NSP2	NSP3
Initial Demand	(Units)	260	0	0	0	461	0	260	461	0	260	461	0	0	0	162	260	0	162	260	0	162	0	461	162	0	461	162
Added Demand	(Units)	523	0	0	0	422	0	162	0	0	0	162	0	0	0	721	461	0	0	0	0	461	0	260	0	0	0	260
Modified Demand	(Units)	883	0	0	0	883	0	422	461	0	260	623	0	0	0	883	721	0	162	260	0	623	0	721	162	0	461	422
Logistics Cost of Products	(NOK)	133.39	0.00	0.00	0.00	114.73	0.00	92.25	82.94	0.00	72.43	96.4	0.00	0.00	0.00	117.31	120.54	0.00	50.28	72.43	0.00	98.55	0.00	103.69	50.28	0.00	82.94	81.12
Total Cost of Products	(NOK)	285.19	0.00	0.00	0.00	226.94	0.00	164.8	141.52	0.00	117.13	175.56	0.00	0.00	0.00	234.82	244.49	0.00	71.84	117.13	0.00	181.46	0.00	195.31	71.84	0.00	141.52	137.28
Logistics Cost for Product Group	(NOK)	133.39			114.73			175.19			168.83			117.31			170.82			170.98			153.97			164.06
Total Cost for Product Group	(NOK)	285.19			226.94			306.32			292.69			234.82			316.33			298.59			267.15			278.8
% saving Logistics Cost		35.14			44.21			14.81			17.91			42.96			16.94			16.86			25.13			20.23
% saving Total Cost		13.71			31.33			7.31			11.44			28.95			4.29			9.65			19.17			15.64

Substitution factor of the product group

	SP₁	SP₂
SP₁	1	2
SP₂	1	1

Overall results

		STERILE	NON-STERILE	TOTAL
Current Process	Number of Products	1645	686	2331
	Logistics Cost (NOK)	2 027 437	1 240 102	3 267 540
	Total Cost (NOK)	9 890 439	6 334 964	16 225 404
All products used with optimised Inventory	Number of Products	1645	686	2331
	% reduction in number of Products	0.00	0.00	0.00
	Logistics Cost (NOK)	663 507	245 603	909 110
	% reduction in logistics Cost	67.27	80.19	72.18
	Total Cost (NOK)	7 651 830	5 572 990	13 224 820
	% reduction in total Cost	22.63	12.03	18.49
Substitution Model	Number of Products	1524	545	2069
	% reduction in number of Products	7.36	20.55	11.24
	Logistics Cost (NOK)	636 228	215 468	851 696
	% reduction in logistics Cost	68.62	82.62	73.93
	Total Cost (NOK)	7 123 596	5 483 927	12 607 523
	% reduction in total Cost	27.97	13.43	22.30

Number of possible product usage and design points based on size of product groups

SIZE OF PRODUCT GROUP	NO. OF POSSIBLE PRODUCT USAGE	No. OF DESIGN POINTS
1	1	1
2	3	3
3	7	10
5	31	196
8	255	41393
10	1023	2237921
15	32767	1.39 * 10¹¹

Design points for a product group of three products

DESIGN POINTS	U₁	U₂	U₃	W_1J	W_2J	W_3J
1,1	1	0	0	0 1 1	0 0 0	0 0 0
2,1	0	1	0	0 0 0	1 0 1	0 0 0
3,1	1	1	0	0 0 1	0 0 0	0 0 0
3,2	1	1	0	0 0 0	0 0 1	0 0 0
4,1	0	0	1	0 0 0	0 0 0	1 1 0
5,1	1	0	1	0 1 0	0 0 0	0 0 0
5,2	1	0	1	0 0 0	0 0 0	0 1 0
6,1	0	1	1	0 0 0	1 0 0	0 0 0
6,2	0	1	1	0 0 0	0 0 0	1 0 0
7,1	1	1	1	0 0 0	0 0 0	0 0 0

Cost when all the products in the product group are used

DESIGN POINT: (7, 1)	NSPj	NSP₂	NSP₃	TOTAL
Demand (Units)	260	461	162	883
Logistics Cost (NOK)	72.43	82.94	50.28	205.66
Total Cost (NOK)	117.13	141.52	71.84	330.49

Initial data of the product group

	DEMAND	AVG. UNIT COST
SP₁	50	NOK 54.95
SP₂	37	NOK 73.53

Cost when both products are used

DESIGN POINT: (3,1)	SP₁	Sp₂	TOTAL
Demand (Units)	50	37	87
Logistics Cost (NOK)	577.20	576.11	1 153.31
Total Cost (NOK)	3 324.46	3 296.70	6 621.15

Cost when only one product is used

DESIGN POINT		(1,1)		(2,1)
Description		SP₁ is used. SP₂ is substituted by SP₁		SP₂ is used. SP₁ is substituted by SP₂
Product		SP ₁	SP₂	SP₁	SP₂
Initial Demand	(Units)	50	0	0	37
Added Demand	(Units)	74	0	0	50
Modified Demand	(Units)	124	0	0	87
Logistics Cost of Products	(NOK)	903.11	0.00	0.00	876.99
Total Cost of Products	(NOK)	7 716.30	0.00	0.00	7 274.05
Logistics Cost for Product Group	(NOK)	903.11		876.99
Total Cost for Product Group	(NOK)	7 716.30		7 274.05
% saving Logistics Cost		21.69		23.96
% saving Total Cost		- 16.54		-9.86

Notations used in the substitution model

NOTATION	DESCRIPTION
N	Total number of products within the product group
i, j	Product number within the product group; i, j ∈ (1,...,N)
P	Notation used for product usage number
M	Notation used for the design point
Uip$U_{i}^{p}$	{1,0,$\left\{ \begin{matrix}1, \\0, \\\end{matrix} \right.$if product i within the product group is used in the product usage number Otherwise
UiM$U_{i}^{M}$	{1,0,$\left\{ \begin{matrix}1, \\0, \\\end{matrix} \right.$if product / within the product group is used in the design point M Otherwise
UjM$U_{j}^{M}$	{1,0,$\left\{ \begin{matrix}1, \\0, \\\end{matrix} \right.$if product j within the product group is used in the design point M Otherwise
WjiM$W_{ji}^{M}$	if product j within the product group is used in the design point M where UjM$U_{j}^{M}${1,0,$\left\{ \begin{matrix}1, \\0, \\\end{matrix} \right.$Otherwise
SijM$S_{ij}^{M}$	{X,0,$\left\{ \begin{matrix}\text{X,} \\0, \\\end{matrix} \right.$if x units of product i can substitute 1 unit of product j within the product group at the design point M Otherwise
	Where x is a real number
RijM$R_{ij}^{M}$	{1,if SijM≠00,if SijM=0$\left\{ \begin{array}{*{35}{l}}1, & \text{if}\,S_{ij}^{M}\ne 0 \\0, & \text{if}\,S_{ij}^{M}=0 \\\end{array} \right.$
IDi$I{{D}_{i}}$	Initial demand of product i within the product group
CDiM$CD_{i}^{M}$	Modified demand of product i within the product group at the design point M with the replacement factor
DiM$D_{i}^{M}$	Modified demand of product i within the product group at the design point M with the conversion factor
LCiM$LC_{i}^{M}$	Logistics cost of product i within the product group at the design point M
TCiM$TC_{i}^{M}$	Total cost of product i within the product group at the design point M
LCM$L{{C}^{M}}$	Summation of logistics cost of products in the product group at the design point M
TCM$T{{C}^{M}}$	Summation of the total cost of products in the product group at the design point M

Investigation of the influence of product variety on inventories in hospitals

Figures & Tables

Product usage for a product group containing three products

Initial details of products within the non-sterile product group

Cost when products are reduced from product group

Substitution factor of the product group

Overall results

Number of possible product usage and design points based on size of product groups

Design points for a product group of three products

Cost when all the products in the product group are used

Initial data of the product group

Cost when both products are used

Cost when only one product is used

Notations used in the substitution model

Paradigm

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