Operational Complexity of Supplier-Customer Systems Measured by Entropy—Case Studies
<p>Plot of functions <math display="inline"> <semantics> <mrow> <msubsup> <mi>S</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>d</mi> </mrow> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>S</mi> <mrow> <mtext>BGS</mtext> </mrow> </msub> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> over (<span class="html-italic">c</span>,<span class="html-italic">d</span>) <math display="inline"> <semantics> <mi>ϵ</mi> </semantics> </math> Ω for given <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> .</p> "> Figure 2
<p>Curves representing the intersection <math display="inline"> <semantics> <mrow> <msubsup> <mi>S</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>d</mi> </mrow> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mstyle mathsize="140%" displaystyle="true"> <mo>∩</mo> </mstyle> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>S</mi> <mrow> <mtext>BGS</mtext> </mrow> </msub> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> over (<span class="html-italic">c</span>,<span class="html-italic">d</span>) <math display="inline"> <semantics> <mi>ϵ</mi> </semantics> </math> Ω.</p> "> Figure 3
<p>Information scheme of a supplier-customer system.</p> "> Figure 4
<p>S1fA: concrete TrC 16–20, time gaps <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>o</mi> </mrow> </msub> </mrow> </semantics> </math>; (<b>a</b>) EDF; (<b>b</b>) values with outlier, continuous piecewise linear function.</p> "> Figure 5
<p>S2fA: concrete TrC 16–20, time gaps <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>o</mi> </mrow> </msub> </mrow> </semantics> </math>; (<b>a</b>) EDF; (<b>b</b>) discrete values without outlier.</p> "> Figure 6
<p>S1fA: solid brick CP 290 × 140 × 65, time gaps <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>o</mi> </mrow> </msub> </mrow> </semantics> </math>; (<b>a</b>) EDF; (<b>b</b>) empirical frequencies.</p> "> Figure 7
<p>S2fA: Solid brick CP 290 × 140 × 65, time gaps <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>o</mi> </mrow> </msub> </mrow> </semantics> </math>; (<b>a</b>) EDF; (<b>b</b>) values with outlier, continuous piecewise linear function.</p> "> Figure 8
<p>3-D bar plot of entropy ratios <span class="html-italic">h</span> of products (1—concrete, 2—solid brick, 3—building block) summarized by suppliers S1fA and S2fA laterally.</p> "> Figure 9
<p>3-D bar plot of entropy ratios <span class="html-italic">h</span> of products (1—blouses, 2—dresses, 3—skirts) summarized by suppliers S1fB and S2fB laterally.</p> "> Figure 10
<p>3-D bar plot of entropy ratios <span class="html-italic">h</span> of products (1—Tank G100, 2—Mast LTA, 3—Exchanger P12) summarized by suppliers S1fC and S2fC laterally.</p> "> Figure 11
<p><math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>o</mi> </mrow> </msub> </mrow> </semantics> </math> values; (<b>a</b>) from unfiltered data, set [<span class="html-italic">b<sub>d</sub></span>, <span class="html-italic">b<sub>u</sub></span>] = [0, 0]; (<b>b</b>) EDF.</p> "> Figure 12
<p><math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>o</mi> </mrow> </msub> </mrow> </semantics> </math> values; (<b>a</b>) from filtered data, set [<span class="html-italic">b<sub>d</sub></span>, <span class="html-italic">b<sub>u</sub></span>] = [0, 7]; (<b>b</b>) EDF.</p> "> Figure 13
<p><math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>o</mi> </mrow> </msub> </mrow> </semantics> </math> values; (<b>a</b>) from filtered data, set [<span class="html-italic">b<sub>d</sub></span>, <span class="html-italic">b<sub>u</sub></span>] = [0, 14]; (<b>b</b>) EDF.</p> "> Figure 14
<p>Entropy ratio <span class="html-italic">h</span>(<span class="html-italic">b</span>) = <span class="html-italic">H</span>(<span class="html-italic">b</span>)/<span class="html-italic">H<sub>u</sub></span>(<span class="html-italic">b</span>) from data filtered by [0, <span class="html-italic">b</span>], <span class="html-italic">b</span> = 0, 1, …,14 days.</p> "> Figure 15
<p>Fractal character of <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>o</mi> </mrow> </msub> </mrow> </semantics> </math> values plot containing all deliveries from 1 January 2007 till 31 December 2010.</p> ">
Abstract
:1. Introduction
2. Theoretical Background
2.1. Entropy Used to Measure Operational Complexity
2.2. Information Scheme of Supplier-Customer Systems and Basic Structure of a Problem-Oriented Database
2.3. Definition of Variables
- Recast such data into a suitable probabilistic system with N system states, in general;
- Calculate all probabilities introduced, i.e., p1, …, pN.
3. Operational Complexity of Supplier-Customer Systems—Case Studies
- Case-collected data sheets are extracted from problem-oriented database properly either by structured query language (SQL) processing of reports generated by MIS, or manually, in the simplest case;
- Check all excerpted data for logical consistency;
- Statistical processing of the excerpted data and computation of entropy, e.g., issuing histograms (HIS), empirical distribution functions (EDF), and other additional numerical and/or graphical outputs, if necessary.
- (i)
- Sort and scale {yk} by affine map in order to get {xk} of a random variable X identically distributed as Y, but with dom(X) = [0, 1].
- (ii)
- Extract all repeating values form {xk} in order to get strictly increasing subset {xi}, i = 1, …, N, 0 ≤ x1 < x2 < … < xN ≤ 1, with frequencies {fi}, i = 1, ..., N of values which actually define system states.
- (iii)
- Calculate EDF F(ξ) = P(ξ < x), x {xi}, with dom(F(.)) = range(F(.)) = [0, 1], and HIS, called an empirical frequency function alternatively, which gives relative frequencies {pi} = {fi/K}, i = 1, …, N.
- (iv)
- Compute entropy and other related quantities, basically using Equation (2a,b) for calculation of H, and Hu, or equivalents S and Su based upon natural logarithms log(x) alternatively.
- To: order issue time, instead of i,oTi,
- Td: delivery time, instead of i,dTi,
3.1. Medium-Sized Building Engineering Company FA
3.2. Small-Sized Fashion Shop FB
3.3. Medium-Sized Mechanical Engineering Company FC
3.4. Small-Sized Lubricant Shop FD
3.5. Top-Medium-Sized Mechanical Engineering Company FE
3.6. Short Comparison of the Analyzed Study Cases
4. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|
103 | 113 | 86 | 118 | 94 | 101 | 95 | 88 | 101 | 101 |
p1 | p2 | p3 | p4 | p5 | p6 | p7 | p8 | p9 | p10 |
---|---|---|---|---|---|---|---|---|---|
0.103 | 0.113 | 0.086 | 0.118 | 0.094 | 0.101 | 0.095 | 0.088 | 0.101 | 0.101 |
– | – | Quantity | Time |
---|---|---|---|
(A) Supplier side | scheduled production | s,sQi, i = 1, ..., n | s,sTi, i = 1, ..., n |
actual production | s,pQi, i = 1, ..., n | s,pTi, i = 1, ..., n | |
(B) Interface | forecast | i,fQi, i = 1, ..., n | i,fTi, i = 1, ..., n |
order | i,oQi, i = 1, ..., n | i,oTi, i = 1, ..., n | |
delivery | i,dQi, i = 1, ..., n | i,dTi, i = 1, ..., n | |
(C) Customer side | scheduled production | c,sQi, i = 1, ..., n | c,sTi, i = 1, ..., n |
actual production | c,pQi, i = 1, ..., n | c,pTi, i = 1, ..., n |
Supplier : Product | H | Hu | h = H/Hu |
---|---|---|---|
S1fA : Concrete | 2.55792 | 5.04439 | 0.507082 |
S2fA : Concrete | 2.66536 | 5.95420 | 0.447643 |
S1fA : Solid brick | 2.76999 | 5.08746 | 0.544474 |
S2fA : Solid brick | 2.62193 | 5.08746 | 0.515370 |
S1fA : Building block | 2.80140 | 4.52356 | 0.619292 |
S2fA : Building block | 2.93795 | 4.95420 | 0.593023 |
Supplier : Product | H | Hu | h = H/Hu |
---|---|---|---|
S1fB : Blouses | 1.96692 | 4.00000 | 0.491729 |
S2fB : Blouses | 1.22791 | 6.04439 | 0.203148 |
S1fB : Dresses | 1.85475 | 4.45943 | 0.415917 |
S2fB : Dresses | 0.932112 | 4.52356 | 0.206057 |
S1fB : Skirts | 1.22791 | 6.04439 | 0.203148 |
S2fB : Skirts | 1.33920 | 6.37504 | 0.210069 |
Supplier : Product | H | Hu | h = H/Hu |
---|---|---|---|
S1fC : Tank G100 | 2.31212 | 7.29462 | 0.316962 |
S2fC : Tank G100 | 2.69223 | 7.29462 | 0.369071 |
S1fC : Mast LTA | 3.24267 | 6.45943 | 0.502005 |
S2fC : Mast LTA | 3.34846 | 6.45943 | 0.518383 |
S1fC : Exchanger P12 | 2.31212 | 7.29462 | 0.316962 |
S2fC : Exchanger P12 | 2.52078 | 7.29462 | 0.345568 |
b | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 9 |
h(b) | 0.327 | 0.327 | 0.324 | 0.324 | 0.320 | 0.319 | 0.302 | 0.235 | 0.224 |
b | 10 | 11 | 12 | 13 | 14 | ||||
h(b) | 0.212 | 0.207 | 0.198 | 0.178 | 0.139 |
Study Case No. | 1 | 2 | 3 | 4 | 5 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Firm | FA | FB | FC | FD | FE | ||||||
Commodity | C1fa | C2fa | C3fa | C1fb | C2fb | C3fb | C1fc | C2fc | C3fc | Cfd | many |
Optimal supplier | S2fa | S2fa | S2fa | S2fb | S2fb | S1fb | S1fc | S1fc | S1fc | Sfd | Sfe |
min h | 0.448 | 0.515 | 0.593 | 0.203 | 0.206 | 0.203 | 0.317 | 0.502 | 0.317 | 0.327 | - |
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Lukáš, L.; Hofman, J. Operational Complexity of Supplier-Customer Systems Measured by Entropy—Case Studies. Entropy 2016, 18, 137. https://doi.org/10.3390/e18040137
Lukáš L, Hofman J. Operational Complexity of Supplier-Customer Systems Measured by Entropy—Case Studies. Entropy. 2016; 18(4):137. https://doi.org/10.3390/e18040137
Chicago/Turabian StyleLukáš, Ladislav, and Jiří Hofman. 2016. "Operational Complexity of Supplier-Customer Systems Measured by Entropy—Case Studies" Entropy 18, no. 4: 137. https://doi.org/10.3390/e18040137