text:
# Inducing cancer to enemy politicians, spies, world-leaders

Would a state be able (with currently available technology) to cause cancer to its enemies, which it can't silence in a different way? (e.g. staged accidents, assassinations, imprisonment)

Hugo Chavez said regarding his cancer:

It would not be strange if they had developed the technology to induce cancer and nobody knew about it until now ... I don't know. I'm just reflecting

What he is missing is that the odds of a person getting cancer are about 50%, with 90% of cancers occurring after the age of 45. The cancers he refers too could be explained by sheer luck.

But then again mice with suppressed immune systems, can easily be infected with cancer.

The catch is:

• it must be done without physical contact with the victim
• the cancer has to develop fast enough (within a few years or less)
• the operation must go (relatively) unnoticed by the victim. So, no Acute Radiation Sickness (ARS) allowed.

For example, by exposing the victim to carcinogens (viruses, radiation, etc) and at the same time irradiating regularly the victim with not-too-high doses with x-ray lasers from a large distance in order to keep its immune system suppressed. The radiation could come in bursts every few weeks or months, in order to avoid ARS.

Perhaps the carcinogens can be skipped if the immune system is kept suppressed for too long. For example AIDS patients have higher chances of getting specific cancers:

People infected with HIV are several thousand times more likely than uninfected people to be diagnosed with Kaposi sarcoma

Bonus question:

• Any counter-measures for this type of stealthy-assassination? A detection of radiation would be considered a counter-measure, since it would no longer be stealthy.

Note: The answer to the main question should include as many details as possible regarding the methods that could be used, using only currently available technology, or technology that is very likely to be available within a couple of years from now.

Also, the method doesn't have to always work; 30% chance of causing cancer would still be enough.

• Cancer isn't a quick disease. How quickly were you wanting your victims to be incapacitated? – Snow Nov 30 '16 at 13:17
• @Pete Indeed, though there are some aggressive types of cancer. Also, it doesn't have to induce cancer out of the blue. It could simply increase the growth of existing cancerous cells (since the average human has 50% chance of getting cancer), meaning it wouldn't take that long. And it doesn't have to always be successful. If 1 in 3 attempts succeed at a cost of a few million.. it would still be worth using. "How quickly " - a couple of years. Probably less than 4 would be ideal, because of presidency duration in many countries. – Fermi paradox Nov 30 '16 at 13:53
• The US spent millions on a heart attack gun, seems better than having a way to make someone who is annoying you now get cancer and die years later. – Snowlockk Nov 30 '16 at 13:54
• Sadly, I have experience of glioblastomas being fatal within a year. However, I wouldn't be able to suggest any plausible method of intentionally causing one other than using a "faulty" x-ray machine. I'm also not sure how effective a "weapon" would be given the 30% chance of killing someone in a four year period. You'd assume that whatever threat the person has posed would be possibly negated after that time. – Snow Nov 30 '16 at 14:07
• @Fermi, I was thinking about pre-sensitizing the victim, and then provoking an anaphylactic hyper-reaction, weeks or months later. (With, for instance an uncommon but not unheard of mold spore into the HVAC system.) Happy to chat -- if I can figure out chat here. (I'm new.) – Catalyst Nov 30 '16 at 22:28

Alexander Litvinenko was killed, probably by the Russian government, by getting him to ingest a significant quantity of radioactive Polonium-210. He did not die of cancer, he died of acute radiation poisoning.

Cancer does not kill fast, so if you are trying to slowly poison/kill your enemies over time (a year or so), this same sort of radiation dosing would be effective. Polonium-210 was hard to detect despite the fact that Polonium is not common (or present) in the human body. It does not give off much gamma radiation (only on 0.001% of all decays); instead giving off alpha particles that caused significant damage to Litvinenko's internal organs while not penetrating to the outside where they could be detected. The medical and police examinations of Litvinenko did not detect the radiation poisoning, it was only when samples were send to the UK's Atomic Weapons Establishment that the Polonium was identified.

A low level dose from certain other radio-nuclides could be even harder to detect. Polonium-210's activity ($\lambda$) is 8.36$\times10^{-8}$ 1/s; that means that $1 - e^{-\lambda} \approx \lambda$ or 8.36$\times10^{-6}$ % of the material decays every second. We can find a radio-nuclide that has a lower activity and is thus harder to detect. It will have to decay by alpha or beta decays so that it is difficult to detect outside of the body when ingested. It will have to also decay to a relatively common isotope so that the daughter nuclide isn't easily detected. If the poisoning element was a common biological element, it would be difficult to chemically separate from materials already in your body.

# Possible poisons

Lets look at this list and a chart of nuclides to find some possibilities.

-Sodium 22 has activity of 1.22$\times10^{-8}$ (about 1/8 of polonium) undergoes beta decay, delivers 500 eV x-ray radiation about 0.1% of the time, is the 9th common element in the body, and decays to stable $^{22}$Ne which is not common but is found at 0.138 ppm dissolved in blood. Sodium-22 is hard to detect in sodium form, and doesn't release many gammas, but does release positrons that can cause annihilation events and follow-on gamma release that would be suspicious. It also might lead to suspiciously high dissolved neon levels in the blood. However, I don't know how many doctors would perform gas chromotography of the blood as part of diagnosis of any symptoms.

-Strontium 90 has activity of 1.22$\times10^{-8}$ (about 1/80) of plutonium, undergoes beta decay, releases zero gammas, is the 17th most common element in the body, and decays to $^{90}$Y. This isn't great because Yttrium-90 itself is not stable, and will decay within hours to $^{90}$Zr. This second decay does release gammas, 200-400 keV ones 97% of the time, but 2.3 MeV ones .002% of the time. The high energy gammas will occur at 1/40 the frequency as the already hard to detect Polonium gammas; the lower energy ones are lower energy and harder to detect, but will occur about a thousand times more frequently than polonium. I don't know if this would be net easier or net harder to detect. In any case, Zirconium is rare in the human body and might stand out. An advantage of Strontium is its chemical similarity to Calcium, so the body will naturally try to deposit in in bones, giving a very good chance of causing a particularly debilitating case of myeloma.

-Phosphorous 32 has activity of 8.10$\times10^{-7}$ (about 10 times polomium), undergoes beta decay, delivers zero energy as gamma radiation, is the 6th common element in the body, and decays to stable $^{32}$S. This is the most common isotope of sulfur, and the 8th most common element in the body. Phosporous-32 will act quickly, blurring the line between acute radiation poisoning and cancer. Its energy delivery is 1/10 that of polonium, but it decays 10 times faster so net energy delivery is about the same. This will kill in weeks. It does not release gammas at all. Its decay products are identical to the 8th most common constituent of the body and is therefore all-but impossible to detect.

# Conclusion

There are three decent options for poisioning. Keep in mind, this is with a 10 microgram dose, equivalent to the one that killed Litvenenko.

• Strontium-90 will almost certainly cause deadly bone marrow cancer in a few years, but might be detectable due to its significant low energy gamma activity and decay into the rare element Zirconium.
• Sodium-22 releases x-rays that are unlikely to be detected, and will cause chronic radiation poisoning in about a year. However, the body may be apt at flushing sodium, limiting the damage; the decay product Neon might be detectable in the blood; and the positron annihilations may or may not be detectable.
• Phosphorous-32 releases no gammas, is metabolized directly into ATP and DNA for a whole body radiation kill, and has an undetectable decay product. However, this would kill in a matter of days leaving little question that the victim has been poisoned somehow.

Pick your poison, and happy hunting.

# Use Stuxnet

Basically, you indirectly supply state-of-the-art medical facilities to your intended target. Let's say a CAT scanner or some other kind of diagnostic device.

When your target is observed going in for a check up, boot up your Stuxnet worm and overload the radiation dose (while logging the correct dose amount to cover up).

Repeat on a yearly/quarterly basis as required.

# Bonus Answer (detection of assassination attempt)

Although people tend to trust computer logs implicitly, a suspicious security team may wish to use a radiation detector during use to ensure that the x-ray machine (or whatever) is really delivering the dose it says it is to the victim in question.

• It just takes a bit of psychology. Print a few news stories of how unwell someone is ("Has El Presidente suffered a minor stroke?") and they'll react to disprove those rumours. It might take a few feet on the ground and some subtle tactics, but it could work out. – Snow Nov 30 '16 at 14:36

How to induce cancer to someone, with current technology? The answer is so simple, that you will not like it, so please bear with me (because I will not limit myself to answer no. 1):

The answer is: make the target start smoking!

Yep, that easy. And perfectly within reach.

Tobacco inoculates you with up to 70 carcinogens, being the no. 1 avoidable cause of cancer, inducing cancers from the lung, larynx, head and neck, bladder, kidney, esophagus, stomach and colon. However, as you know, even smokers haven't got as nearly as 100% rate of cancer. The percentage is something as 10-30%.

So this just goes to show you how hard it is to induce cancer the way you want. Because cancer is a disease that is caused by mutations on the DNA that will make the cells go haywire and proliferate uncontrollably. But even after the mutation, the cells have lots of mechanisms to heal that mutation, be it DNA repair systems, be it immune responses against the tumor cells.

And carcinogen induces DNA mutations, either directly or indirectly.

This means that a person that is not exposed to a carcinogen may develop cancer, and a person exposed to a carcinogen may not. It's a matter of probability, of likelihood.

To increase the cancer rates to non-negliglible numbers, you would have to opt between two strategies: a) Expose the target to a single, but super-high dose of the carcinogen (e.g. radiation from an atomic bomb) b) Expose the target to low, but continuous / repeated, doses of the carcinogen

Both of these would increase the rate of detection.

So let's see some ways we could do this, with technologies available today... even though none will be as simple and accessible as tobacco.

1) Exposure to chemical carcinogens

This is the easiest way and it doesn't need much sophisticated technologies to achieve.

For example, sneakly cover the air conditioning of the rooms where the target works with a layer of asbestos. This will increase the risk of lung and pleural cancer.

Or taint the meals of the target with known carcinogens. For instance, make them eat only cereals contaminated with the Aspergillus fungus, that produces aflatoxin, that increases the rate of liver cancer.

Or make your own cocktail of chemical carcinogens and put them on the food or air supplies.

Pros: The access to these carcinogens may be extremely simple, depending on the carcinogen.

Cons: The efficacy will be extremely low. You would need to constantly taint the food and air of the target, which would increase the risk of detection.

This is your prefered method, but I don't think it is feasible. Contrary to what Pete has said, I have no knowledge of solid evidence demonstrating that constant exposure to CT scans (or to radiation produced by CT machines) increase the risk of cancer significantly... if it means doing a semestral or trimestral CT scan.

Even radiotherapy, with significantly higher radiation doses, will not increase this risk with great orders of magnitude, even though radiotherapy is more limited in time and is fine-tuned to reduce the risk of radiation-induced cancer to the max.

But still, you would need very high doses of radiation AND you would need them to be focused on a very concentrated and constant spot in the body. Even today, radioncologists need to make "armors" and "masks" that completely imobilize the patient and need to constantly recallibrate the machine in order to be certain that the beam is always affecting the same area, or else the therapeutic benefit gets compromised.

There is just no way to achieve this on a moving target with current technology, without using such a dose of radiation that would affect all the neighborhood, increasing the chances of detection.

But there is a way that you can make radiation find your target. Kingledion has beat me to the punch with that. Just make the person ingest substances containing radiation releasing isotopes. They will decay at regular intervals and innoculate a dose of radiation at each interval.

You can concentrate the radiation on a specific organ by binding the radioisotopes to molecules that concentrate on particular regions of the body.

For example, the only place on our body that uses iodine is the thyroid. Doctors use 131 iodine as a means to concentrate radiation on the thyroid.

Another example, doctors use radionuclides bound to a prostate specific antigen on diagnostic PET. This antigen concentrates the radiation on the prostate, making it easier to detect prostatic cancer by measuring the radiation released.

Kingledion mentioned strontium 90, which is analogous to calcium, which would concentrate the radiation on the bone. This would be helpful, because the bone marrow (the place where you produce the blood) is a place with a high celular division rate and therefore, very sensitive to cancer induction (leukemias, etc...)

Cons: Radiation may be detected,as kingledion as mentioned.

4) Immunity

It is true that immunosupression may lead to cancer, because then there will be no immune cells to detect cancer cells and destroy them. However, that level of immunosupression would be hard to induce without extremely high doses of myelotoxics or radiation. Furthermore, it would be easily detected before inducing cancer... because the target would contract many infectious diseases beforehand, that would make him/her see a doctor, which would easily detect the immunosupression with a blood test.

So I don’t think that’s feasible.

5) Epigenetic tampering

We don’t solely depend on genetics. We also depend on epigenetics. Meaning, all the cells in our organism possess the totality of our genetic code. But which genes we express, on which organs and at which situations depend on epigenetics.

Epigenetics means that we may turn on or off a gene, by adding molecule radicals (namely methyl radicals) to the DNA sequence immediately preceding that gene.

So, if a cancer is promoted by a mutated gene, we could epigenetically turn it off. And vice-versa... we could epigenetically turn on genes that would increase the likelihood of cancer.

At this time, we don’t have a way to manipulate epigenetics at our will. Every anticancer epigenetic drug that I know of has failed on clinical trials. But it is not a stretch to believe that this technology will be available in a couple of years.

Cons: You would need to develop this drug on highly specialized laboratories. Either you make a drug that just tampers with all your epigenetic make-up (and so the consequences will be unpredictable)... or you develop a drug that epigenetically focuses just on one gene, but then you will have a problem. Because you will need to make that drug available to the target on a sufficient dose. Since you can’t make physical contact with him, it means you can’t innoculate the drug. So the drug should be able to traverse the digestive system without being digested and be absorbed on a sufficient dose to achieve the effect. I don’t think this is feasible for a stealthy assassination.

Remember tobacco? Everyone knows that it causes cancer! Yet everyone still does it.

Why? Because amongst the carcinogens you have nicotine, which is also a neurotransmitter that causes a sense of well-being and therefore, addiction.

Processed meat is a carcinogen, but who can live without that yummy hamburger? You just need to have it. Just one won’t make a difference, right? Cancer just happens to the others, right? You can stop whenever you want, right?

Just make sure that you couple your carcinogen with an addictive substance or behavior. Thereby you will ensure that the target will be exposed to the carcinogen on continuous and regular doses. The target will BEG you for the carcinogen.

In fact, the target will make the job for you. They will rationalize everything, everything, to be able to continue their addiction. This is truer if you make the addictive drug have a hangover effect, that will cause depression or fatigue, so the target will no longer be able to live without it.

This is easy. Every neurological substance that induces addiction may induce (sometimes fatal) abstinence syndromes.

Heck, at a certain time, you can even forget about stealth. The target will be crawling up to you. If he asks for anyone else's help, he/she will stop getting the drug. He/She will not have that, especially if he/she is a dictator that won't have anyone deny him/her something.

Just make your evil chemists develop a carcinogen and highly addictive neurotransmitter, like nicotine.

7) Infection

This is, according to my opinion, the best strategy. Viruses may cause cancer because they are genetic invaders, that will cause DNA damage. Bacteria may cause cancer, because they may produce carcinogens internally.

Any microbiological laboratory would have the technology to genetically alter bacteria or viruses in order to turn them cancerigenous, if they would set their mind on it.

Helicobacter pylori is a bacterium that infects nearly half of the world’s population and it increases the risk of gastric cancer. We know there are strains that are more cancerigenous than others. Taint the food with a genetically altered H. pylori and you’re good to go.

Or why not alter a non-cancerigenous bacteria that is ubiquitous on our intestinal flora, like an E. coli?

Or create a hybrid flu virus. It would be airborne and extremely virulent. But give it genes that would incorporate on the host’s DNA in order to induce mutations. Or make it hijack the cellular machinery to produce and release carcinogens on circulation.

Pros: You would have to inoculate the target just once, thereby decreasing chances of detection. It could be airborne or foodborne (or, if you don't count that as physical contact, sexually transmited - just hire and inoculate a prostitute). Once inside the host, the microbe would self-perpetuate, without the need for more actions from the killer. You could just sit back and enjoy.

In the case of viruses, doctors wouldn’t be able to detect them, because they would be new viruses. They would only be found if the doctors knew a priori what viral antigens they should be looking for.

Cons Theoretically, a doctor could eventually detect the infection.

Pro pro But even if they did diagnose accurately those infections, that doesn’t mean they could treat them. Viral infections are extremely difficult to treat. And you could genetically engineer your bacteria with antibiotic-resistance plasmids.

• Good answer. I would just like to caution that epigenetics as a true scientific discipline is in its infancy, and so most epigenetic theories you hear around are probably pseudoscientific. I don't think there is any evidence as of yet of how 'epigenetic tampering' could be accomplished, so it isn't really in the range of currently available technology, as the OP specifies. – kingledion Nov 30 '16 at 21:52
• @kingledion: Absolutely right. I did mention that on my comment. But I also said: "But it is not a stretch to believe that this technology will be available in a couple of years". Which, if current research proceeds on the current way, it could. – Pedro Gabriel Nov 30 '16 at 21:59

In the news there was real-world case of man with HIV being infected with cancer from his parasite. His parasite had a cancer, but then its cancer cells invaded his body too because his immune system was too weak to stop them.

http://www.iflscience.com/health-and-medicine/parasitic-worms-cancer-cells-caused-tumors-hiv-positive-man/

So the answer is simple. Induce cancer in a parasite by using radiation and then get your enemy infected by said parasite. But for this plan to work immunity of your enemy politician must be already weak (old age, bad genes, HIV-positive, etc).


text:
# American Institute of Mathematical Sciences

• Previous Article
Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method
• JIMO Home
• This Issue
• Next Article
Capital-constrained supply chain with multiple decision attributes: Decision optimization and coordination analysis
October  2019, 15(4): 1857-1879. doi: 10.3934/jimo.2018126

## An economic order quantity for deteriorating items with allowable rework of deteriorated products

 Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran

* Corresponding author

Received  December 2017 Revised  April 2018 Published  July 2019

This paper presents an inventory model for deteriorating items with variable demand when shortage is permitted and quantity discount in purchase cost, and rework on deteriorating products are also allowed. The main idea of this research is to study the effects of the discount and the rework on the inventory costs. In this paper, it is assumed that for a certain quantity of purchased items, the seller would offer a discount and the manager would have the choice to either accept the discount or dismiss. On the other hand, there is also a similar decision-making scenario, where the manager makes a decision to reduce the total costs by using the rework and reducing the shortage periods or reducing the total costs by ignoring the rework cost and increasing the shortage periods. The implementation of the mathematical model is illustrated with a numerical example and sensitivity analysis describes the effects of the parameters on the total costs. The results show that the rework will decrease the total costs of the inventory system, significantly.

Citation: Mahdi Karimi, Seyed Jafar Sadjadi, Alireza Ghasemi Bijaghini. An economic order quantity for deteriorating items with allowable rework of deteriorated products. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1857-1879. doi: 10.3934/jimo.2018126
##### References:
 [1] S. Aggarwal and C. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, Journal of the Operational Research Society, 46 (1995), 658-662. Google Scholar [2] A. K. Bhunia, A. A. Shaikh and R. Gupta, A study on two-warehouse partially backlogged deteriorating inventory models under inflation via particle swarm optimisation, International Journal of Systems Science, 46 (2015), 1036-1050. doi: 10.1080/00207721.2013.807385. Google Scholar [3] A. K. Bhunia, A. A. Shaikh and L. Sahoo, A two-warehouse inventory model for deteriorating item under permissible delay in payment via particle swarm optimisation, International Journal of Logistics Systems and Management, 24 (2016), 45-69. doi: 10.1504/IJLSM.2016.075662. Google Scholar [4] A. K. Bhunia, A. A. Shaikh, G. Sharma and S. Pareek, A two storage inventory model for deteriorating items with variable demand and partial backlogging, Journal of Industrial and Production Engineering, 32 (2015), 263-272. doi: 10.1080/21681015.2015.1046508. Google Scholar [5] M. Bounkhel, Nonlinear receding horizon control of production inventory systems with deteriorating items, Yugoslav Journal of Operations Research, 18 (2018), 37-45. doi: 10.2298/YJOR0801037B. Google Scholar [6] C. K. Chan, W. H. Wong, A. Langevin and Y. Lee, An integrated production-inventory model for deteriorating items with consideration of optimal production rate and deterioration during delivery, International Journal of Production Economics, 189 (2017), 1-13. doi: 10.1016/j.ijpe.2017.04.001. Google Scholar [7] C.-T. Chang, Inventory models with stock-and pricedependent demand for deteriorating items based on limited shelf space, Yugoslav Journal of Operations Research, 20 (2016).Google Scholar [8] C.-T. Chang, L.-Y. Ouyang and J.-T. Teng, An EOQ model for deteriorating items under supplier credits linked to ordering quantity, Applied Mathematical Modelling, 27 (2003), 983-996. doi: 10.1016/S0307-904X(03)00131-8. Google Scholar [9] H. J. Chang, A partial backlogging inventory model for non-instantaneous deteriorating items with stock-dependent consumption rate under inflation, Yugoslav Journal of Operations Research, 20 (2016).Google Scholar [10] S.-C. Chen and J.-T. Teng, Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis, European Journal of Operational Research, 243 (2015), 566-575. doi: 10.1016/j.ejor.2014.12.007. Google Scholar [11] Z. Chen and B. R. Sarker, Integrated production-inventory and pricing decisions for a single-manufacturer multi-retailer system of deteriorating items under JIT delivery policy, The International Journal of Advanced Manufacturing Technology, 89 (2017), 2099-2117. Google Scholar [12] K. D. Choudhury, B. Karmakar, M. Das and T. K. Datta, An inventory model for deteriorating items with stock-dependent demand time-varying holding cost and shortages, Opsearch, 52 (2015), 55-74. doi: 10.1007/s12597-013-0166-x. Google Scholar [13] V. Choudri, M. Venkatachalam and S. Panayappan, Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money, Journal of Industrial and Management Optimization, 12 (2016), 1153-1172. doi: 10.3934/jimo.2016.12.1153. Google Scholar [14] R. R. Chowdhury, S. Ghosh and K. Chaudhuri, An inventory model for deteriorating items with stock and price sensitive demand, International Journal of Applied and Computational Mathematics, 1 (2015), 187-201. doi: 10.1007/s40819-014-0011-9. Google Scholar [15] P. Ghare and G. Schrader, An inventory model for deteriorating item for exponentially deteriorating items, Journal of Industrial Engineering, 14 (1963), 238-243. Google Scholar [16] Y. Ghiami and T. Williams, A two-echelon production-inventory model for deteriorating items with multiple buyers, International Journal of Production Economics, 159 (2015), 233-240. doi: 10.1016/j.ijpe.2014.09.017. Google Scholar [17] M. Ghoreishi, G.-W. Weber and A. Mirzazadeh, An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation-and selling price-dependent demand and customer returns, Annals of Operations Research, 226 (2015), 221-238. doi: 10.1007/s10479-014-1739-7. Google Scholar [18] S. K. Ghosh, T. Sarkar and K. Chaudhuri, A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand, American Journal of Mathematical and Management Sciences, 34 (2015), 147-161. doi: 10.1080/01966324.2014.980870. Google Scholar [19] S. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. Google Scholar [20] K.-L. Hou, An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting, European Journal of Operational Research, 168 (2006), 463-474. doi: 10.1016/j.ejor.2004.05.011. Google Scholar [21] T.-P. Hsieh and C.-Y. Dye, Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand, European Journal of Operational Research, 262 (2017), 136-150. doi: 10.1016/j.ejor.2017.03.038. Google Scholar [22] Y.-F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the operational Research Society, 54 (2003), 1011-1015. doi: 10.1057/palgrave.jors.2601588. Google Scholar [23] C. K. Jaggi, K. Aggarwal and S. K. Goel, Optimal order policy for deteriorating items with inflation induced demand, International Journal of Production Economics, 10 (2006), 707-714. doi: 10.1016/j.ijpe.2006.01.004. Google Scholar [24] Jaggi, Pareek, Goel and Nidhi, An inventory model for deteriorating items with ramp type demand under fuzzy environment, International Journal of Logistics Systems and Management, 22 (2015), 436-463. doi: 10.1504/IJLSM.2015.072748. Google Scholar [25] A. Jamal, B. Sarker and S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of the operational Research Society, 48 (1997), 826-833. Google Scholar [26] D. K. Jana, B. Das and M. Maiti, Multi-item partial backlogging inventory models over random planning horizon in random fuzzy environment, Applied Soft Computing, 21 (2014), 12-27. Google Scholar [27] N. Kumar and S. Kumar, Effect of learning and salvage worth on an inventory model for deteriorating items with inventory-dependent demand rate and partial backlogging with capability constraints, Uncertain Supply Chain Management, 4 (2016), 123-136. doi: 10.5267/j.uscm.2015.11.002. Google Scholar [28] S. Kumar and U. Rajput, Fuzzy inventory model for deteriorating items with time dependent demand and partial backlogging, Applied Mathematics, 6 (2015), Article ID 54567, 13 pages. doi: 10.4236/am.2015.63047. Google Scholar [29] S. Kumar, A. K. Singh and M. K. Patel, Optimization of Weibull deteriorating items inventory model under the effect of price and time dependent demand with partial backlogging, Sadhana, 41 (2016), 977-984. Google Scholar [30] Y. Li, S. Zhang and J. Han, Dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items, Automatica, 76 (2017), 200-213. doi: 10.1016/j.automatica.2016.11.003. Google Scholar [31] J.-J. Liao, A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity, Applied Mathematical Modelling, 31 (2007), 1690-1699. doi: 10.1016/j.apm.2006.05.003. Google Scholar [32] G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550. doi: 10.1016/j.eswa.2011.09.044. Google Scholar [33] R. Maihami, B. Karimi and S. M. Ghomi, Pricing and Inventory Control in a Supply Chain of Deteriorating Items: A Non-cooperative Strategy with Probabilistic Parameters, International Journal of Applied and Computational Mathematics, 3 (2017), 2477-2499. doi: 10.1007/s40819-016-0250-z. Google Scholar [34] W. A. Mandal and S. Islam, Fuzzy inventory model for weibull deteriorating items, with time depended demand, shortages, and partially backlogging, Pak. J. Stat. Oper. Res., 12 (2016), 101-109. doi: 10.18187/pjsor.v12i1.1153. Google Scholar [35] V. K. Mishra, Inventory Model of Deteriorating Items with Revenue Sharing on Preservation Technology Investment under Price Sensitive Stock Dependent Demand, International Journal of Mathematical Modelling and Computations, 21 (2016), 37-48.Google Scholar [36] D. J. Mohanty, R. S. Kumar and A. Goswami, A two-warehouse inventory model for non-instantaneous deteriorating items over stochastic planning horizon, Journal of Industrial and Production Engineering, 33 (2016), 516-532. doi: 10.1080/21681015.2016.1176964. Google Scholar [37] B. Mukherjee and A. Bansal, An Approach for Developing an Optimum Quantity Discount Policy of Deteriorating Items Inventory Transportation System, International Journal of Innovative Technology and Research, 5 (2017), 5811-5816.Google Scholar [38] B. Naik and R. Patel, Deteriorating items inventory model with different deterioration rates for imperfect quality items and shortages, International Journal of Computational and Applied Mathematics, 12 (2017), 273-284. Google Scholar [39] S. Pal, G. Mahapatra and G. Samanta, A production inventory model for deteriorating item with ramp type demand allowing inflation and shortages under fuzziness, Economic Modelling, 46 (2015), 334-345. doi: 10.1016/j.econmod.2014.12.031. Google Scholar [40] X. Pan and S. Li, Optimal control of a stochastic production-inventory system under deteriorating items and environmental constraints, International Journal of Production Research, 53 (2015), 607-628. doi: 10.1080/00207543.2014.961201. Google Scholar [41] S. Papachristos and K. Skouri, An optimal replenishment policy for deteriorating items with time-varying demand and partial-exponential type-backlogging, Operations Research Letters, 27 (2000), 175-184. doi: 10.1016/S0167-6377(00)00044-4. Google Scholar [42] P. Pathma Raja, A study on factors affecting quality deterioration in housing construction industry in Klang Valley, in Open University Malaysia.Google Scholar [43] M. Pervin, G. C. Mahata and S. K. Roy, An inventory model with declining demand market for deteriorating items under a trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251. doi: 10.1080/17509653.2015.1081082. Google Scholar [44] M. Pervin, S. K. Roy and G.-W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460. doi: 10.1007/s10479-016-2355-5. Google Scholar [45] M. Pervin, S. K. Roy and G.-W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002. Google Scholar [46] R. S. Rajan and R. Uthayakumar, Optimal pricing and replenishment policies for instantaneous deteriorating items with backlogging and trade credit under inflation, Journal of Industrial Engineering International, 13 (2017), 428-443. Google Scholar [47] A. Roy, Fuzzy inventory model for deteriorating items with price dependent demand, International Journal of Management Science and Engineering Management, 10 (2015), 237-241. doi: 10.1080/17509653.2014.959086. Google Scholar [48] G. Samanta, A production inventory model with deteriorating items and shortages, Yugoslav Journal of Operations Research, 14 (2016).Google Scholar [49] P. Samanta, J. Das and S. Indrajitsingha, Fuzzy Inventory Model for Two Parameter Weibull Deteriorating Items, (2017).Google Scholar [50] B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2017), 3138-3151. doi: 10.1016/j.apm.2012.07.026. Google Scholar [51] N. H. Shah and L. E. Cárdenas-Barrón, Retailer's decision for ordering and credit policies for deteriorating items when a supplier offers order-linked credit period or cash discount, Applied Mathematics and Computation, 259 (2015), 569-578. doi: 10.1016/j.amc.2015.03.010. Google Scholar [52] H. Shavandi, H. Mahlooji and N. E. Nosratian, A constrained multi-product pricing and inventory control problem, Applied Soft Computing, 12 (2012), 2454-2461. Google Scholar [53] S. W. Shinn, H. Hwang and S. P. Sung, Joint price and lot size determination under conditions of permissible delay in payments and quantity discounts for freight cost, European Journal of Operational Research, 91 (1996), 528-542. Google Scholar [54] M. Srichandan, Inventory Model of Deteriorating Items for Linear Holding Cost with Time Dependent Demand, (2015).Google Scholar [55] A. A. Taleizadeh, An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments, Applied Mathematical Modelling, 38 (2014), 5357-5366. doi: 10.1016/j.apm.2014.02.014. Google Scholar [56] A. A. Taleizadeh and M. Nematollahi, An inventory control problem for deteriorating items with back-ordering and financial considerations, Applied Mathematical Modelling, 38 (2014), 93-109. doi: 10.1016/j.apm.2013.05.065. Google Scholar [57] A. A. Taleizadeh, M. Noori-daryan and L. E. Cárdenas-Barrón, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295. doi: 10.1016/j.ijpe.2014.09.009. Google Scholar [58] R. Tat, A. A. Taleizadeh and M. Esmaeili, Developing economic order quantity model for non-instantaneous deteriorating items in vendor-managed inventory (VMI) system, International Journal of Systems Science, 46 (2015), 1257-1268. doi: 10.1080/00207721.2013.815827. Google Scholar [59] S. Tayal, S. Singh and R. Sharma, An inventory model for deteriorating items with seasonal products and an option of an alternative market, Uncertain Supply Chain Management, 3 (2015), 69-86. doi: 10.5267/j.uscm.2014.8.003. Google Scholar [60] J.-T. Teng, L. E. Cárdenas-Barrón, H.-J. Chang, J. Wu and Y. Hu, Inventory lot-size policies for deteriorating items with expiration dates and advance payments, Applied Mathematical Modelling, 40 (2016), 8605-8616. doi: 10.1016/j.apm.2016.05.022. Google Scholar [61] J.-T. Teng and C.-T. Chang, Optimal manufacturer's replenishment policies in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 195 (2009), 358-363. doi: 10.1016/j.ejor.2008.02.001. Google Scholar [62] J.-T. Teng, L.-Y. Ouyang and L.-H. Chen, A comparison between two pricing and lot-sizing models with partial backlogging and deteriorated items, International Journal of Production Economics, 105 (2007), 190-203. doi: 10.1016/j.ijpe.2006.03.003. Google Scholar [63] J. T. Teng, H. Yang and L. Ouyang, On an EOQ model for deteriorating items with time-varying demand and partial backlogging, Journal of the Operational Research Society, 54 (2003), 432-436. Google Scholar [64] J. T. Teng, M. S. Chern and H. L. Yang, An optimal recursive method for various inventory replenishment models with increasing demand and shortages, Naval Research Logistics, 44 (1997), 791-806. doi: 10.1002/(SICI)1520-6750(199712)44:8<791::AID-NAV6>3.0.CO;2-5. Google Scholar [65] S. Tiwari, L. E. Cárdenas-Barrón, A. Khanna and C. K. Jaggi, Impact of trade credit and inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-warehouse environment, International Journal of Production Economics, 176 (2016), 154-169. doi: 10.1016/j.ijpe.2016.03.016. Google Scholar [66] S. Tiwari, C. K. Jaggi, A. K. Bhunia, A. A. Shaikh and M. Goh, Two-warehouse inventory model for non-instantaneous deteriorating items with stock-dependent demand and inflation using particle swarm optimization, Annals of Operations Research, 254 (2017), 401-423. doi: 10.1007/s10479-017-2492-5. Google Scholar [67] W.-C. Wang, J.-T. Teng and K.-R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, European Journal of Operational Research, 232 (2014), 315-321. doi: 10.1016/j.ejor.2013.06.027. Google Scholar [68] X. Xu, Q. Bai and M. Chen, A comparison of different dispatching policies in two-warehouse inventory systems for deteriorating items over a finite time horizon, Applied Mathematical Modelling, 41 (2017), 359-374. doi: 10.1016/j.apm.2016.08.024. Google Scholar [69] H.-L. Yang, Two-warehouse partial backlogging inventory models for deteriorating items under inflation, International Journal of Production Economics, 103 (2006), 362-370. doi: 10.1016/j.ijpe.2005.09.003. Google Scholar [70] P.-C. Yang, Pricing strategy for deteriorating items using quantity discount when demand is price sensitive, European Journal of Operational Research, 157 (2004), 389-397. doi: 10.1016/S0377-2217(03)00241-8. Google Scholar [71] Q. Zhang, H. Dong, J. Luo and A. Segerstedt, Supply chain coordination with trade credit and quantity discount incorporating default risk, International Journal of Production Economics, 153 (2014), 352-360. doi: 10.1016/j.ijpe.2014.03.019. Google Scholar

show all references

##### References:
 [1] S. Aggarwal and C. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, Journal of the Operational Research Society, 46 (1995), 658-662. Google Scholar [2] A. K. Bhunia, A. A. Shaikh and R. Gupta, A study on two-warehouse partially backlogged deteriorating inventory models under inflation via particle swarm optimisation, International Journal of Systems Science, 46 (2015), 1036-1050. doi: 10.1080/00207721.2013.807385. Google Scholar [3] A. K. Bhunia, A. A. Shaikh and L. Sahoo, A two-warehouse inventory model for deteriorating item under permissible delay in payment via particle swarm optimisation, International Journal of Logistics Systems and Management, 24 (2016), 45-69. doi: 10.1504/IJLSM.2016.075662. Google Scholar [4] A. K. Bhunia, A. A. Shaikh, G. Sharma and S. Pareek, A two storage inventory model for deteriorating items with variable demand and partial backlogging, Journal of Industrial and Production Engineering, 32 (2015), 263-272. doi: 10.1080/21681015.2015.1046508. Google Scholar [5] M. Bounkhel, Nonlinear receding horizon control of production inventory systems with deteriorating items, Yugoslav Journal of Operations Research, 18 (2018), 37-45. doi: 10.2298/YJOR0801037B. Google Scholar [6] C. K. Chan, W. H. Wong, A. Langevin and Y. Lee, An integrated production-inventory model for deteriorating items with consideration of optimal production rate and deterioration during delivery, International Journal of Production Economics, 189 (2017), 1-13. doi: 10.1016/j.ijpe.2017.04.001. Google Scholar [7] C.-T. Chang, Inventory models with stock-and pricedependent demand for deteriorating items based on limited shelf space, Yugoslav Journal of Operations Research, 20 (2016).Google Scholar [8] C.-T. Chang, L.-Y. Ouyang and J.-T. Teng, An EOQ model for deteriorating items under supplier credits linked to ordering quantity, Applied Mathematical Modelling, 27 (2003), 983-996. doi: 10.1016/S0307-904X(03)00131-8. Google Scholar [9] H. J. Chang, A partial backlogging inventory model for non-instantaneous deteriorating items with stock-dependent consumption rate under inflation, Yugoslav Journal of Operations Research, 20 (2016).Google Scholar [10] S.-C. Chen and J.-T. Teng, Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis, European Journal of Operational Research, 243 (2015), 566-575. doi: 10.1016/j.ejor.2014.12.007. Google Scholar [11] Z. Chen and B. R. Sarker, Integrated production-inventory and pricing decisions for a single-manufacturer multi-retailer system of deteriorating items under JIT delivery policy, The International Journal of Advanced Manufacturing Technology, 89 (2017), 2099-2117. Google Scholar [12] K. D. Choudhury, B. Karmakar, M. Das and T. K. Datta, An inventory model for deteriorating items with stock-dependent demand time-varying holding cost and shortages, Opsearch, 52 (2015), 55-74. doi: 10.1007/s12597-013-0166-x. Google Scholar [13] V. Choudri, M. Venkatachalam and S. Panayappan, Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money, Journal of Industrial and Management Optimization, 12 (2016), 1153-1172. doi: 10.3934/jimo.2016.12.1153. Google Scholar [14] R. R. Chowdhury, S. Ghosh and K. Chaudhuri, An inventory model for deteriorating items with stock and price sensitive demand, International Journal of Applied and Computational Mathematics, 1 (2015), 187-201. doi: 10.1007/s40819-014-0011-9. Google Scholar [15] P. Ghare and G. Schrader, An inventory model for deteriorating item for exponentially deteriorating items, Journal of Industrial Engineering, 14 (1963), 238-243. Google Scholar [16] Y. Ghiami and T. Williams, A two-echelon production-inventory model for deteriorating items with multiple buyers, International Journal of Production Economics, 159 (2015), 233-240. doi: 10.1016/j.ijpe.2014.09.017. Google Scholar [17] M. Ghoreishi, G.-W. Weber and A. Mirzazadeh, An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation-and selling price-dependent demand and customer returns, Annals of Operations Research, 226 (2015), 221-238. doi: 10.1007/s10479-014-1739-7. Google Scholar [18] S. K. Ghosh, T. Sarkar and K. Chaudhuri, A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand, American Journal of Mathematical and Management Sciences, 34 (2015), 147-161. doi: 10.1080/01966324.2014.980870. Google Scholar [19] S. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. Google Scholar [20] K.-L. Hou, An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting, European Journal of Operational Research, 168 (2006), 463-474. doi: 10.1016/j.ejor.2004.05.011. Google Scholar [21] T.-P. Hsieh and C.-Y. Dye, Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand, European Journal of Operational Research, 262 (2017), 136-150. doi: 10.1016/j.ejor.2017.03.038. Google Scholar [22] Y.-F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the operational Research Society, 54 (2003), 1011-1015. doi: 10.1057/palgrave.jors.2601588. Google Scholar [23] C. K. Jaggi, K. Aggarwal and S. K. Goel, Optimal order policy for deteriorating items with inflation induced demand, International Journal of Production Economics, 10 (2006), 707-714. doi: 10.1016/j.ijpe.2006.01.004. Google Scholar [24] Jaggi, Pareek, Goel and Nidhi, An inventory model for deteriorating items with ramp type demand under fuzzy environment, International Journal of Logistics Systems and Management, 22 (2015), 436-463. doi: 10.1504/IJLSM.2015.072748. Google Scholar [25] A. Jamal, B. Sarker and S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of the operational Research Society, 48 (1997), 826-833. Google Scholar [26] D. K. Jana, B. Das and M. Maiti, Multi-item partial backlogging inventory models over random planning horizon in random fuzzy environment, Applied Soft Computing, 21 (2014), 12-27. Google Scholar [27] N. Kumar and S. Kumar, Effect of learning and salvage worth on an inventory model for deteriorating items with inventory-dependent demand rate and partial backlogging with capability constraints, Uncertain Supply Chain Management, 4 (2016), 123-136. doi: 10.5267/j.uscm.2015.11.002. Google Scholar [28] S. Kumar and U. Rajput, Fuzzy inventory model for deteriorating items with time dependent demand and partial backlogging, Applied Mathematics, 6 (2015), Article ID 54567, 13 pages. doi: 10.4236/am.2015.63047. Google Scholar [29] S. Kumar, A. K. Singh and M. K. Patel, Optimization of Weibull deteriorating items inventory model under the effect of price and time dependent demand with partial backlogging, Sadhana, 41 (2016), 977-984. Google Scholar [30] Y. Li, S. Zhang and J. Han, Dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items, Automatica, 76 (2017), 200-213. doi: 10.1016/j.automatica.2016.11.003. Google Scholar [31] J.-J. Liao, A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity, Applied Mathematical Modelling, 31 (2007), 1690-1699. doi: 10.1016/j.apm.2006.05.003. Google Scholar [32] G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550. doi: 10.1016/j.eswa.2011.09.044. Google Scholar [33] R. Maihami, B. Karimi and S. M. Ghomi, Pricing and Inventory Control in a Supply Chain of Deteriorating Items: A Non-cooperative Strategy with Probabilistic Parameters, International Journal of Applied and Computational Mathematics, 3 (2017), 2477-2499. doi: 10.1007/s40819-016-0250-z. Google Scholar [34] W. A. Mandal and S. Islam, Fuzzy inventory model for weibull deteriorating items, with time depended demand, shortages, and partially backlogging, Pak. J. Stat. Oper. Res., 12 (2016), 101-109. doi: 10.18187/pjsor.v12i1.1153. Google Scholar [35] V. K. Mishra, Inventory Model of Deteriorating Items with Revenue Sharing on Preservation Technology Investment under Price Sensitive Stock Dependent Demand, International Journal of Mathematical Modelling and Computations, 21 (2016), 37-48.Google Scholar [36] D. J. Mohanty, R. S. Kumar and A. Goswami, A two-warehouse inventory model for non-instantaneous deteriorating items over stochastic planning horizon, Journal of Industrial and Production Engineering, 33 (2016), 516-532. doi: 10.1080/21681015.2016.1176964. Google Scholar [37] B. Mukherjee and A. Bansal, An Approach for Developing an Optimum Quantity Discount Policy of Deteriorating Items Inventory Transportation System, International Journal of Innovative Technology and Research, 5 (2017), 5811-5816.Google Scholar [38] B. Naik and R. Patel, Deteriorating items inventory model with different deterioration rates for imperfect quality items and shortages, International Journal of Computational and Applied Mathematics, 12 (2017), 273-284. Google Scholar [39] S. Pal, G. Mahapatra and G. Samanta, A production inventory model for deteriorating item with ramp type demand allowing inflation and shortages under fuzziness, Economic Modelling, 46 (2015), 334-345. doi: 10.1016/j.econmod.2014.12.031. Google Scholar [40] X. Pan and S. Li, Optimal control of a stochastic production-inventory system under deteriorating items and environmental constraints, International Journal of Production Research, 53 (2015), 607-628. doi: 10.1080/00207543.2014.961201. Google Scholar [41] S. Papachristos and K. Skouri, An optimal replenishment policy for deteriorating items with time-varying demand and partial-exponential type-backlogging, Operations Research Letters, 27 (2000), 175-184. doi: 10.1016/S0167-6377(00)00044-4. Google Scholar [42] P. Pathma Raja, A study on factors affecting quality deterioration in housing construction industry in Klang Valley, in Open University Malaysia.Google Scholar [43] M. Pervin, G. C. Mahata and S. K. Roy, An inventory model with declining demand market for deteriorating items under a trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251. doi: 10.1080/17509653.2015.1081082. Google Scholar [44] M. Pervin, S. K. Roy and G.-W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460. doi: 10.1007/s10479-016-2355-5. Google Scholar [45] M. Pervin, S. K. Roy and G.-W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002. Google Scholar [46] R. S. Rajan and R. Uthayakumar, Optimal pricing and replenishment policies for instantaneous deteriorating items with backlogging and trade credit under inflation, Journal of Industrial Engineering International, 13 (2017), 428-443. Google Scholar [47] A. Roy, Fuzzy inventory model for deteriorating items with price dependent demand, International Journal of Management Science and Engineering Management, 10 (2015), 237-241. doi: 10.1080/17509653.2014.959086. Google Scholar [48] G. Samanta, A production inventory model with deteriorating items and shortages, Yugoslav Journal of Operations Research, 14 (2016).Google Scholar [49] P. Samanta, J. Das and S. Indrajitsingha, Fuzzy Inventory Model for Two Parameter Weibull Deteriorating Items, (2017).Google Scholar [50] B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2017), 3138-3151. doi: 10.1016/j.apm.2012.07.026. Google Scholar [51] N. H. Shah and L. E. Cárdenas-Barrón, Retailer's decision for ordering and credit policies for deteriorating items when a supplier offers order-linked credit period or cash discount, Applied Mathematics and Computation, 259 (2015), 569-578. doi: 10.1016/j.amc.2015.03.010. Google Scholar [52] H. Shavandi, H. Mahlooji and N. E. Nosratian, A constrained multi-product pricing and inventory control problem, Applied Soft Computing, 12 (2012), 2454-2461. Google Scholar [53] S. W. Shinn, H. Hwang and S. P. Sung, Joint price and lot size determination under conditions of permissible delay in payments and quantity discounts for freight cost, European Journal of Operational Research, 91 (1996), 528-542. Google Scholar [54] M. Srichandan, Inventory Model of Deteriorating Items for Linear Holding Cost with Time Dependent Demand, (2015).Google Scholar [55] A. A. Taleizadeh, An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments, Applied Mathematical Modelling, 38 (2014), 5357-5366. doi: 10.1016/j.apm.2014.02.014. Google Scholar [56] A. A. Taleizadeh and M. Nematollahi, An inventory control problem for deteriorating items with back-ordering and financial considerations, Applied Mathematical Modelling, 38 (2014), 93-109. doi: 10.1016/j.apm.2013.05.065. Google Scholar [57] A. A. Taleizadeh, M. Noori-daryan and L. E. Cárdenas-Barrón, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295. doi: 10.1016/j.ijpe.2014.09.009. Google Scholar [58] R. Tat, A. A. Taleizadeh and M. Esmaeili, Developing economic order quantity model for non-instantaneous deteriorating items in vendor-managed inventory (VMI) system, International Journal of Systems Science, 46 (2015), 1257-1268. doi: 10.1080/00207721.2013.815827. Google Scholar [59] S. Tayal, S. Singh and R. Sharma, An inventory model for deteriorating items with seasonal products and an option of an alternative market, Uncertain Supply Chain Management, 3 (2015), 69-86. doi: 10.5267/j.uscm.2014.8.003. Google Scholar [60] J.-T. Teng, L. E. Cárdenas-Barrón, H.-J. Chang, J. Wu and Y. Hu, Inventory lot-size policies for deteriorating items with expiration dates and advance payments, Applied Mathematical Modelling, 40 (2016), 8605-8616. doi: 10.1016/j.apm.2016.05.022. Google Scholar [61] J.-T. Teng and C.-T. Chang, Optimal manufacturer's replenishment policies in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 195 (2009), 358-363. doi: 10.1016/j.ejor.2008.02.001. Google Scholar [62] J.-T. Teng, L.-Y. Ouyang and L.-H. Chen, A comparison between two pricing and lot-sizing models with partial backlogging and deteriorated items, International Journal of Production Economics, 105 (2007), 190-203. doi: 10.1016/j.ijpe.2006.03.003. Google Scholar [63] J. T. Teng, H. Yang and L. Ouyang, On an EOQ model for deteriorating items with time-varying demand and partial backlogging, Journal of the Operational Research Society, 54 (2003), 432-436. Google Scholar [64] J. T. Teng, M. S. Chern and H. L. Yang, An optimal recursive method for various inventory replenishment models with increasing demand and shortages, Naval Research Logistics, 44 (1997), 791-806. doi: 10.1002/(SICI)1520-6750(199712)44:8<791::AID-NAV6>3.0.CO;2-5. Google Scholar [65] S. Tiwari, L. E. Cárdenas-Barrón, A. Khanna and C. K. Jaggi, Impact of trade credit and inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-warehouse environment, International Journal of Production Economics, 176 (2016), 154-169. doi: 10.1016/j.ijpe.2016.03.016. Google Scholar [66] S. Tiwari, C. K. Jaggi, A. K. Bhunia, A. A. Shaikh and M. Goh, Two-warehouse inventory model for non-instantaneous deteriorating items with stock-dependent demand and inflation using particle swarm optimization, Annals of Operations Research, 254 (2017), 401-423. doi: 10.1007/s10479-017-2492-5. Google Scholar [67] W.-C. Wang, J.-T. Teng and K.-R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, European Journal of Operational Research, 232 (2014), 315-321. doi: 10.1016/j.ejor.2013.06.027. Google Scholar [68] X. Xu, Q. Bai and M. Chen, A comparison of different dispatching policies in two-warehouse inventory systems for deteriorating items over a finite time horizon, Applied Mathematical Modelling, 41 (2017), 359-374. doi: 10.1016/j.apm.2016.08.024. Google Scholar [69] H.-L. Yang, Two-warehouse partial backlogging inventory models for deteriorating items under inflation, International Journal of Production Economics, 103 (2006), 362-370. doi: 10.1016/j.ijpe.2005.09.003. Google Scholar [70] P.-C. Yang, Pricing strategy for deteriorating items using quantity discount when demand is price sensitive, European Journal of Operational Research, 157 (2004), 389-397. doi: 10.1016/S0377-2217(03)00241-8. Google Scholar [71] Q. Zhang, H. Dong, J. Luo and A. Segerstedt, Supply chain coordination with trade credit and quantity discount incorporating default risk, International Journal of Production Economics, 153 (2014), 352-360. doi: 10.1016/j.ijpe.2014.03.019. Google Scholar
Inventory level $(I)$ vs. Time.
Purchase cost vs. Ordering quantity$(Q_i)$
Sensitivity analysis for backlogging rate
Sensitivity analysis for deterioration rate
Sensitivity analysis for rework percentage
Sensitivity analysis for discount threshold
Review of previous works
 Number 1 2 3 4 5 6 7 8 9 10 Paper [49] [29] [12] [46] [48] [5] [9] [7] [37] [54] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ $\surd$ $\surd$ Three $\surd$ Levels Two $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes Inflation Discount $\surd$ $\surd$ and Inflation $\surd$ $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic Lost Sales Shortage Part. Backlog $\surd$ $\surd$ $\surd$ Com. Backlog $\surd$ $\surd$ $\surd$ Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ Lead Psitive Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Deter. Variable $\surd$ $\surd$ $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy Demand Prob. Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ Number 11 12 13 14 15 16 17 18 19 20 Paper [28] [2] [6] [10] [11] [14] [16] [17] [18] [21] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ $\surd$ $\surd$ Three $\surd$ $\surd$ $\surd$ Levels Two $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes $\surd$ Inflation Discount $\surd$ and Inflation $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two $\surd$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic
 Number 1 2 3 4 5 6 7 8 9 10 Paper [49] [29] [12] [46] [48] [5] [9] [7] [37] [54] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ $\surd$ $\surd$ Three $\surd$ Levels Two $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes Inflation Discount $\surd$ $\surd$ and Inflation $\surd$ $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic Lost Sales Shortage Part. Backlog $\surd$ $\surd$ $\surd$ Com. Backlog $\surd$ $\surd$ $\surd$ Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ Lead Psitive Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Deter. Variable $\surd$ $\surd$ $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy Demand Prob. Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ Number 11 12 13 14 15 16 17 18 19 20 Paper [28] [2] [6] [10] [11] [14] [16] [17] [18] [21] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ $\surd$ $\surd$ Three $\surd$ $\surd$ $\surd$ Levels Two $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes $\surd$ Inflation Discount $\surd$ and Inflation $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two $\surd$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic
Review of previous works
 Number 11 12 13 14 15 16 17 18 19 20 Paper [28] [2] [6] [10] [11] [14] [16] [17] [18] [21] Lost Sales $\surd$ Shortage Part. Backlog $\surd$ $\surd$ Com. Backlog $\surd$ Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Lead Psitive $\surd$ Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Deter. Variable $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy $\surd$ Demand Prob. Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ Number 21 22 23 24 25 26 27 28 29 30 Paper [24] [30] [33] [36] [38] [39] [40] [51] [13] [57] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Three $\surd$ Levels Two $\surd$ $\surd$ $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes Inflation Discount $\surd$ and Inflation $\surd$ $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two $\surd$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic Lost Sales Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $\surd$ Com. Backlog $\surd$ Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Lead Psitive $\surd$ Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Deter. Variable $\surd$ $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy Demand Prob. $\surd$ Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ Number 31 32 33 34 35 36 37 38 39 40 Paper [58] [60] [65] [66] [59] [27] [68] [3] [20] [23] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ $\surd$ Three Levels Two $\surd$ $\surd$ $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
 Number 11 12 13 14 15 16 17 18 19 20 Paper [28] [2] [6] [10] [11] [14] [16] [17] [18] [21] Lost Sales $\surd$ Shortage Part. Backlog $\surd$ $\surd$ Com. Backlog $\surd$ Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Lead Psitive $\surd$ Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Deter. Variable $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy $\surd$ Demand Prob. Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ Number 21 22 23 24 25 26 27 28 29 30 Paper [24] [30] [33] [36] [38] [39] [40] [51] [13] [57] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Three $\surd$ Levels Two $\surd$ $\surd$ $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes Inflation Discount $\surd$ and Inflation $\surd$ $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two $\surd$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic Lost Sales Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $\surd$ Com. Backlog $\surd$ Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Lead Psitive $\surd$ Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Deter. Variable $\surd$ $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy Demand Prob. $\surd$ Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ Number 31 32 33 34 35 36 37 38 39 40 Paper [58] [60] [65] [66] [59] [27] [68] [3] [20] [23] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ $\surd$ Three Levels Two $\surd$ $\surd$ $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Review of previous works
 Number 31 32 33 34 35 36 37 38 39 40 Paper [58] [60] [65] [66] [59] [27] [68] [3] [20] [23] Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes Inflation Discount $\surd$ and Inflation $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic Lost Sales Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Com. Backlog $\surd$ $\surd$ $\surd$ Not Allowed $\surd$ Lead Psitive $\surd$ $\surd$ Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Deter. Variable $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy Demand Prob. Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ $\surd$ $\surd$ Number 41 42 43 44 45 46 47 48 49 50 Paper [32] [34] [35] [47] [56] [63] [69] [55] [31] [8] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ Three Levels Two $\surd$ $\surd$ $\surd$ $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes Inflation Discount and Inflation $\surd$ $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two $\surd$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic Lost Sales Shortage Part. Backlog $\surd$ $\surd$ $\surd$ Com. Backlog $\surd$ $\surd$ $\surd$ $\surd$ Not Allowed $\surd$ $\surd$ $\surd$ Lead Psitive Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
 Number 31 32 33 34 35 36 37 38 39 40 Paper [58] [60] [65] [66] [59] [27] [68] [3] [20] [23] Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes Inflation Discount $\surd$ and Inflation $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic Lost Sales Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Com. Backlog $\surd$ $\surd$ $\surd$ Not Allowed $\surd$ Lead Psitive $\surd$ $\surd$ Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Deter. Variable $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy Demand Prob. Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ $\surd$ $\surd$ Number 41 42 43 44 45 46 47 48 49 50 Paper [32] [34] [35] [47] [56] [63] [69] [55] [31] [8] Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Type Production $\surd$ Three Levels Two $\surd$ $\surd$ $\surd$ $\surd$ one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Product Yes Inflation Discount and Inflation $\surd$ $\surd$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Time Infinity $\surd$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Ware Two $\surd$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Chain Cyclic Lost Sales Shortage Part. Backlog $\surd$ $\surd$ $\surd$ Com. Backlog $\surd$ $\surd$ $\surd$ $\surd$ Not Allowed $\surd$ $\surd$ $\surd$ Lead Psitive Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Review of previous works
 Number 41 42 43 44 45 46 47 48 49 50 Paper [32] [34] [35] [47] [56] [63] [69] [55] [31] [8] Deter. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy Demand Prob. Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ Number 51 52 53 54 Percentage(rounded) This Paper Paper [43] [45] [44] [41] $\%$ $^*$ Model Ordering $\surd$ $\surd$ $\surd$ $73\%$ $\surd$ Type Production $\surd$ $27\%$ Three $10\%$ Levels Two $\surd$ $26\%$ one $\surd$ $\surd$ $\surd$ $64\%$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $98\%$ $\surd$ Product Yes $2\%$ Inflation Discount $\surd$ $16\%$ $\surd$ and Inflation $\surd$ $24\%$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $68\%$ Time Infinity $27\%$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $73\%$ $\surd$ Ware Two $15\%$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $85\%$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $100\%$ Chain Cyclic $0\%$ $\surd$ Lost Sales $2\%$ Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $39\%$ $\surd$ Com. Backlog $22\%$ Not Allowed $\surd$ $37\%$ Lead Psitive $8\%$ Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $92\%$ $\surd$ Deter. Variable $\surd$ $32\%$ rate Fixed $\surd$ $\surd$ $\surd$ $68\%$ $\surd$ Fuzzy $2\%$ Demand Prob. $\surd$ $4\%$ Det. Variable $\surd$ $\surd$ $\surd$ $72\%$ $\surd$ Det. Fixed $22\%$
 Number 41 42 43 44 45 46 47 48 49 50 Paper [32] [34] [35] [47] [56] [63] [69] [55] [31] [8] Deter. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Fuzzy Demand Prob. Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Det. Fixed $\surd$ $\surd$ Number 51 52 53 54 Percentage(rounded) This Paper Paper [43] [45] [44] [41] $\%$ $^*$ Model Ordering $\surd$ $\surd$ $\surd$ $73\%$ $\surd$ Type Production $\surd$ $27\%$ Three $10\%$ Levels Two $\surd$ $26\%$ one $\surd$ $\surd$ $\surd$ $64\%$ $\surd$ Multi No $\surd$ $\surd$ $\surd$ $\surd$ $98\%$ $\surd$ Product Yes $2\%$ Inflation Discount $\surd$ $16\%$ $\surd$ and Inflation $\surd$ $24\%$ Discount Not Allowed $\surd$ $\surd$ $\surd$ $68\%$ Time Infinity $27\%$ Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $73\%$ $\surd$ Ware Two $15\%$ -houses One $\surd$ $\surd$ $\surd$ $\surd$ $85\%$ $\surd$ Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $100\%$ Chain Cyclic $0\%$ $\surd$ Lost Sales $2\%$ Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $39\%$ $\surd$ Com. Backlog $22\%$ Not Allowed $\surd$ $37\%$ Lead Psitive $8\%$ Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $92\%$ $\surd$ Deter. Variable $\surd$ $32\%$ rate Fixed $\surd$ $\surd$ $\surd$ $68\%$ $\surd$ Fuzzy $2\%$ Demand Prob. $\surd$ $4\%$ Det. Variable $\surd$ $\surd$ $\surd$ $72\%$ $\surd$ Det. Fixed $22\%$
Optimal solution of Numerical Example
 $n$ $TC$ $i$ $t_i$ $tw_i$ $s_i$ 6 12785.79 1 0 0.0310 0.0310 7 12279.73 2 0.0310 0.0314 0.0410 8 7978.37 3 0.0410 0.0410 0.0520 9 14197.62 4 0.0521 0.0522 0.0814 10 14886.37 5 0.0816 0.0817 0.194 11 21056.36 6 0.1965 0.1972 0.2075 7 0.2075 0.2086 0.2335 $n^*$=8 8 0.2345 0.2346 1.3667 $TC^*$=7978.37 9 - 1.3681 2
 $n$ $TC$ $i$ $t_i$ $tw_i$ $s_i$ 6 12785.79 1 0 0.0310 0.0310 7 12279.73 2 0.0310 0.0314 0.0410 8 7978.37 3 0.0410 0.0410 0.0520 9 14197.62 4 0.0521 0.0522 0.0814 10 14886.37 5 0.0816 0.0817 0.194 11 21056.36 6 0.1965 0.1972 0.2075 7 0.2075 0.2086 0.2335 $n^*$=8 8 0.2345 0.2346 1.3667 $TC^*$=7978.37 9 - 1.3681 2
Sensitivity analysis for backlogging rate
 $\beta$ 0(Complete backlogging) 10 20 40 $\infty$(No shortage) $TC$ 7874.34 7894.87 7978.37 8078.46 8213.75
 $\beta$ 0(Complete backlogging) 10 20 40 $\infty$(No shortage) $TC$ 7874.34 7894.87 7978.37 8078.46 8213.75
Sensitivity analysis for deterioration rate
 $\rho$ 0.01 0.016 0.02 0.024 0.03 $TC$ 7975.35 7977.15 7978.37 7979.56 7981.37
 $\rho$ 0.01 0.016 0.02 0.024 0.03 $TC$ 7975.35 7977.15 7978.37 7979.56 7981.37
Sensitivity analysis for rework percentage
 $\eta$ 0.1 0.16 0.2 0.24 0.3 $TC$ 7978.63 7978.47 7978.37 7978.25 7978.09
 $\eta$ 0.1 0.16 0.2 0.24 0.3 $TC$ 7978.63 7978.47 7978.37 7978.25 7978.09
Sensitivity analysis for discount threshold
 $M$ 50 200 400 600 $\infty$ $TC$ 12917 7978.37 10546 13848 11180
 $M$ 50 200 400 600 $\infty$ $TC$ 12917 7978.37 10546 13848 11180
 [1] Maryam Ghoreishi, Abolfazl Mirzazadeh, Gerhard-Wilhelm Weber, Isa Nakhai-Kamalabadi. Joint pricing and replenishment decisions for non-instantaneous deteriorating items with partial backlogging, inflation- and selling price-dependent demand and customer returns. Journal of Industrial & Management Optimization, 2015, 11 (3) : 933-949. doi: 10.3934/jimo.2015.11.933 [2] Po-Chung Yang, Hui-Ming Wee, Shen-Lian Chung, Yong-Yan Huang. Pricing and replenishment strategy for a multi-market deteriorating product with time-varying and price-sensitive demand. Journal of Industrial & Management Optimization, 2013, 9 (4) : 769-787. doi: 10.3934/jimo.2013.9.769 [3] Lizhao Yan, Fei Xu, Yongzeng Lai, Mingyong Lai. Stability strategies of manufacturing-inventory systems with unknown time-varying demand. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2033-2047. doi: 10.3934/jimo.2017030 [4] Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 21-50. doi: 10.3934/naco.2017002 [5] Prasenjit Pramanik, Sarama Malik Das, Manas Kumar Maiti. Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1289-1315. doi: 10.3934/jimo.2018096 [6] Vincent Choudri, Mathiyazhgan Venkatachalam, Sethuraman Panayappan. Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1153-1172. doi: 10.3934/jimo.2016.12.1153 [7] Javad Taheri-Tolgari, Mohammad Mohammadi, Bahman Naderi, Alireza Arshadi-Khamseh, Abolfazl Mirzazadeh. An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1317-1344. doi: 10.3934/jimo.2018097 [8] Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559 [9] Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653 [10] Roberta Fabbri, Russell Johnson, Sylvia Novo, Carmen Núñez. On linear-quadratic dissipative control processes with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 193-210. doi: 10.3934/dcds.2013.33.193 [11] Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150 [12] Robert G. McLeod, John F. Brewster, Abba B. Gumel, Dean A. Slonowsky. Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs. Mathematical Biosciences & Engineering, 2006, 3 (3) : 527-544. doi: 10.3934/mbe.2006.3.527 [13] Yangzi Hu, Fuke Wu. The improved results on the stochastic Kolmogorov system with time-varying delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1481-1497. doi: 10.3934/dcdsb.2015.20.1481 [14] Tingwen Huang, Guanrong Chen, Juergen Kurths. Synchronization of chaotic systems with time-varying coupling delays. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1071-1082. doi: 10.3934/dcdsb.2011.16.1071 [15] Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 [16] Konstantina Skouri, Ioannis Konstantaras. Two-warehouse inventory models for deteriorating products with ramp type demand rate. Journal of Industrial & Management Optimization, 2013, 9 (4) : 855-883. doi: 10.3934/jimo.2013.9.855 [17] Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Deteriorating inventory with preservation technology under price- and stock-sensitive demand. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-28. doi: 10.3934/jimo.2019019 [18] Sankar Kumar Roy, Magfura Pervin, Gerhard Wilhelm Weber. Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019032 [19] Mohsen Lashgari, Ata Allah Taleizadeh, Shib Sankar Sana. An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1091-1119. doi: 10.3934/jimo.2016.12.1091 [20] Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1345-1373. doi: 10.3934/jimo.2018098

2018 Impact Factor: 1.025

## Metrics

• PDF downloads (26)
• HTML views (181)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]

format: plain


text:
# Another interesting integral

• June 6th 2009, 07:50 PM
simplependulum
Another interesting integral
evaluate the integral $\int_{0}^{\frac{\pi}{2}} x \cot(x) ~dx$

In fact , it is not a challenge problem for an experienced integrator
but i believe that this may help remind us of some techniques (Nod)
• June 6th 2009, 08:41 PM
NonCommAlg
Quote:

Originally Posted by simplependulum
evaluate the integral $\int_{0}^{\frac{\pi}{2}} x \cot(x) ~dx$

In fact , it is not a challenge problem for an experienced integrator
but i believe that this may help remind us of some techniques (Nod)

let $x=u$ and $\cot x \ dx = dv.$ then integration by parts gives us: $I=\int_0^{\frac{\pi}{2}} x \cot x \ dx = -\int_0^{\frac{\pi}{2}} \ln(\sin x) \ dx.$ thus:
$-I=\frac{\pi}{2} \ln 2 + \int_0^{\frac{\pi}{2}} \ln(\sin(x/2)) \ dx + \int_0^{\frac{\pi}{2}} \ln(\cos(x/2)) \ dx$

$=\frac{\pi}{2} \ln 2 + \int_0^{\frac{\pi}{2}} \ln(\sin(x/2)) \ dx + \int_{\frac{\pi}{2}}^{\pi} \ln(\sin(x/2)) \ dx$

$=\frac{\pi}{2} \ln 2 + \int_0^{\pi} \ln(\sin(x/2)) \ dx=\frac{\pi}{2} \ln 2 + 2\int_0^{\frac{\pi}{2}} \ln(\sin x) \ dx$

$=\frac{\pi}{2} \ln 2 - 2I.$ therefore: $I=\frac{\pi}{2}\ln 2.$
• June 8th 2009, 02:03 AM
simplependulum
terrific !


text:
# 2009-08-26 Gimp 2.6 for Mac

Cool. I was looking at Isomage’s hexGIMP and found that the script didn’t work on my Gimp 2.2 on Mac OS 10.4. I groaned in fear of the upgrade hassle waiting for me. But lo and behold, I went to the Gimp download page, found a link to GIMP on OS X by Simone Lehmann. And it just worked!

All I had to do was copy my brushes and scripts from ~/.gimp-2.2 to ~/Library/Application Support/Gimp – and that was it. Awesome!

Thank you, Simone.

Tags:


text:
Electric Field Video Lessons

Concept

# Problem: Two very large sheets of charge are separated by a distance d. One sheet +o, -O has a surface charge density +o; the other, a surface charge density- A small region near the center of the sheets is shown. 1. Draw arrows on the diagram to indicate the direction of the electric field at points A, B, C, and D. 2. Compare the magnitudes of the electric fields at points A, B, C and D. Explain.

###### FREE Expert Solution

Electric field:

$\overline{){\mathbf{E}}{\mathbf{=}}{\mathbf{-}}\frac{\mathbf{∆}\mathbf{V}}{\mathbf{∆}\mathbf{x}}}$

1.

Electric field lines are directed from high potential to low potential.

The high potential surface is positively charged while the low potential surface is negatively charged.

90% (264 ratings)
###### Problem Details

Two very large sheets of charge are separated by a distance d. One sheet +o, -O has a surface charge density +o; the other, a surface charge density- A small region near the center of the sheets is shown.

1. Draw arrows on the diagram to indicate the direction of the electric field at points A, B, C, and D.

2. Compare the magnitudes of the electric fields at points A, B, C and D. Explain.


text:
# On the definition of adjoint maps on inner product spaces

Definition: Let $$A:V\to V$$ be linear, where $$V$$ is an inner product space over $$\mathbb{F}$$. Then $$A^*:V\to V$$ is defined by the equation: $$\langle y,A^*x\rangle = \langle Ay,x\rangle \text{ for all }x,y\in V$$ Here $$A^*$$ is called the adjoint operator with respect to $$A\in\mathcal{L}(V)$$.

So, it is clear that $$A^*$$ is defined on inner product spaces only. However, we see that the conjugate transpose of a matrix is defined regardless of whether $$V$$ is an inner product space or not (right?), i.e. if $$A\in M_n(\mathbb{F})$$, we define $$A^* = (\bar A)^T$$.

I suspect a possible correlation between the conjugate transpose defined for matrices, and the adjoint operator defined on inner product spaces, because of results such as:

Let $$\mathcal{B}$$ be an orthonormal ordered basis of $$V$$, then $$A:V\to V$$ is self adjoint (i.e. $$A^* = A$$) iff $$[A]^*_{\mathcal{B}} = [A]_{\mathcal{B}}$$ (matrices of linear maps w.r.t. basis $$\mathcal{B}$$).

However, it seems weird that $$A^*$$ (adjoint) in the context of linear maps is defined only on inner product spaces, while the conjugate transpose of a matrix $$A\in M_n(\mathbb{F})$$ is defined more generally. Am I missing something here? I was hoping for a stronger connection between the two definitions.

Thanks!

P.S.
As mentioned in the comments, it is required to assume $$\mathbb{F} = \mathbb{C}$$ for $$\bar A$$ to make sense.

• What's $\bar A$ over an arbitrary field $\mathbb F$? Dec 6 '20 at 12:47
• Fair point, I don't know. I think we are specializing to $\mathbb{F} = \mathbb{C}$ here. Dec 6 '20 at 13:18
• When $\mathbb F=\Bbb C$ you have $V=\Bbb C^n$, right? Dec 6 '20 at 13:28

The point is that $$\Bbb F^n$$ carries a natural inner product, when $$\Bbb F=\Bbb R$$ or $$\Bbb C$$, namely the ordinary one: $$\langle x,y\rangle:=\sum_k\overline{x_k}y_k\,.$$ And one can prove that $$A^*={\bar A}^T$$ using this inner product, when identifying a matrix $$A$$ with the linear map $$x\mapsto A\cdot x$$.
Let $$e_1,\dots,e_n$$ be the standard basis, then $$\langle e_i,Ae_j\rangle=A_{i,j}$$ is the $$i,j$$th matrix entry, so $$(A^*)_{i,j}\ =\ \langle e_i,A^*e_j\rangle\ =\ \overline{\langle A^*e_j,e_i\rangle}\ =\ \overline{\langle e_j,Ae_i\rangle}\ =\ \overline{A_{j,i}}\,.$$
Note that over $$\Bbb R$$, conjugation is the identity, we can omit it, so in that case the adjoint of $$A$$ is simply its transpose $$A^T$$.


text:
# Revision history [back]

### perspectiveTransform Error handler

I use http://docs.opencv.org/trunk/d7/dff/tutorial_feature_homography.html , but i changed SURF to SIFT, but it doesn't matter. Everything works good, but there are one problem: it hasn't error handler when img_object didn't find on img_scene. For example when images are really different - it's error in this:

perspectiveTransform(obj_corners, scene_corners, H);

So i need some error handler like:

if (**something**) {
printf("Didn't find!"); }

else { ...
perspectiveTransform(obj_corners, scene_corners, H);
...}

My questin is: What something ?? min_dist > 20 ? Or what condition i need?

P.S. Thank you for your answers, and sorry for my english :D

### perspectiveTransform Error handler

I use http://docs.opencv.org/trunk/d7/dff/tutorial_feature_homography.html , but i changed SURF to SIFT, but it doesn't matter. Everything works good, but there are one problem: it hasn't error handler when img_object didn't find on in img_scene. For example when images are really different - it's error in this:

perspectiveTransform(obj_corners, scene_corners, H);

So i need some error handler like:

if (**something**) {
printf("Didn't find!"); }

else { ...
perspectiveTransform(obj_corners, scene_corners, H);
...}

My questin is: What something ?? min_dist > 20 ? Or what condition i need?

P.S. Thank you for your answers, and sorry for my english :D


text:
# TN Keras Layers¶

TN Keras exists to simplify tensorization of existing TensorFlow models. These layers try to match the APIs for existing Keras layers closely. Please note these layers are currently intended for experimentation only, not production. These layers are in alpha and upcoming releases might include breaking changes.

APIs are listed here. An overview of these layers is available here.

 tensornetwork.tn_keras.layers.DenseDecomp(…) TN layer comparable to Dense that carries out matrix multiplication with 2 significantly smaller weight matrices instead of 1 large one. tensornetwork.tn_keras.layers.DenseMPO(…) Matrix Product Operator (MPO) TN layer. tensornetwork.tn_keras.layers.Conv2DMPO(…) 2D Convolutional Matrix Product Operator (MPO) TN layer. tensornetwork.tn_keras.layers.DenseCondenser(…) Condenser TN layer. tensornetwork.tn_keras.layers.DenseExpander(…) Expander TN layer. tensornetwork.tn_keras.layers.DenseEntangler(…) Entangler TN layer.


text:
Question-and-Answer Resource for the Building Energy Modeling Community
Get started with the Help page

# Revision history [back]

### Script for visualizing heat loss from surfaces

I find myself in the need of a diagnostic for visualizing heat transfer through various surfaces either on a design day, monthly or annual basis.

Anybody else on this forum come across a similar need? Any open source solutions?

My solution reads in the following variables and identifies appropriate surfaces (walls, ceilings, windows, doors, etc) from the SQLite database. It's a python script.

       'Surface Outside Face Conduction Heat Loss Rate [W]',
'Surface Outside Face Net Thermal Radiation Heat Gain Rate [W]',
'Surface Outside Face Convection Heat Gain Rate [W]',
'Zone Infiltration Total Heat Loss Energy [J]',
'Zone Infiltration Total Heat Gain Energy [J]',
'Surface Window Heat Loss Rate [W]', # Total heat transfer through window

The thing is I get strange results which I am not confident in. This could be due to several reasons such as not characterizing convective, conductive or radiative heat transfer in my outputs. Could be for other reasons.

Would like to compare solutions to others to build more confidence in these types of diagnostics.

### Script for visualizing heat loss from surfaces

I find myself in the need of a diagnostic for visualizing heat transfer through various surfaces either on a design day, monthly or annual basis.

Anybody else on this forum come across a similar need? Any open source solutions?

My solution reads in the following variables and identifies appropriate surfaces (walls, ceilings, windows, doors, etc) from the SQLite database. It's a python script.

       'Surface Outside Face Conduction Heat Loss Rate [W]',
'Surface Outside Face Net Thermal Radiation Heat Gain Rate [W]',
'Surface Outside Face Convection Heat Gain Rate [W]',
'Zone Infiltration Total Heat Loss Energy [J]',
'Zone Infiltration Total Heat Gain Energy [J]',
'Surface Window Heat Loss Rate [W]', # Total heat transfer through window

The thing is I get strange results which I am not confident in. This could be due to several reasons such as not characterizing convective, conductive or radiative heat transfer in my outputs. Could be for other reasons.

Would like to compare solutions to others to build more confidence in these types of diagnostics.

 3 retagged __AmirRoth__ 4388 ●5 ●16 http://bleedinggreenna...

### Script for visualizing heat loss from surfaces

I find myself in the need of a diagnostic for visualizing heat transfer through various surfaces either on a design day, monthly or annual basis.

Anybody else on this forum come across a similar need? Any open source solutions?

My solution reads in the following variables and identifies appropriate surfaces (walls, ceilings, windows, doors, etc) from the SQLite database. It's a python script.

       'Surface Outside Face Conduction Heat Loss Rate [W]',
'Surface Outside Face Net Thermal Radiation Heat Gain Rate [W]',
'Surface Outside Face Convection Heat Gain Rate [W]',
'Zone Infiltration Total Heat Loss Energy [J]',
'Zone Infiltration Total Heat Gain Energy [J]',
'Surface Window Heat Loss Rate [W]', # Total heat transfer through window

The thing is I get strange results which I am not confident in. This could be due to several reasons such as not characterizing convective, conductive or radiative heat transfer in my outputs. Could be for other reasons.

Would like to compare solutions to others to build more confidence in these types of diagnostics.

 4 retagged __AmirRoth__ 4388 ●5 ●16 http://bleedinggreenna...

### Script for visualizing heat loss from surfaces

I find myself in the need of a diagnostic for visualizing heat transfer through various surfaces either on a design day, monthly or annual basis.

Anybody else on this forum come across a similar need? Any open source solutions?

My solution reads in the following variables and identifies appropriate surfaces (walls, ceilings, windows, doors, etc) from the SQLite database. It's a python script.

       'Surface Outside Face Conduction Heat Loss Rate [W]',
'Surface Outside Face Net Thermal Radiation Heat Gain Rate [W]',
'Surface Outside Face Convection Heat Gain Rate [W]',
'Zone Infiltration Total Heat Loss Energy [J]',
'Zone Infiltration Total Heat Gain Energy [J]',
'Surface Window Heat Loss Rate [W]', # Total heat transfer through window

The thing is I get strange results which I am not confident in. This could be due to several reasons such as not characterizing convective, conductive or radiative heat transfer in my outputs. Could be for other reasons.

Would like to compare solutions to others to build more confidence in these types of diagnostics.


text:
# Sub and superscript on the left

I want to do something like this:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
Hom_{K\cap {}_a^bK}
\end{align*}
\end{document}

But there seems that the _a^b is on the \cap and not attached to the K.

How might I do this right?

-
–  Nicholas R. Peterson Jun 13 '13 at 14:14
@nrpeterson: I did take a look at that, but I can't get that to work. Can you show me in an answer how I might do what I am trying to do? –  Thomas Jun 13 '13 at 14:42

None of the mentioned packages would give a good result in this case, because TeX suppresses space around binary operations in subscripts.

A good solution for this particular case seems to be

\documentclass{article}
\usepackage{amsmath,leftidx}
\DeclareMathOperator{\Hom}{Hom}
\begin{document}
$\Hom_{K\cap \,\leftidx{_a^b}{\!K}{}}$
\end{document}

The \, pushes right the right operand a bit, so to separate it from the intersection symbol. The K would however be too detached from the prescripts, so I add a small correction.

-


text:
There is current flow I in the z direction (along z-axis) which will produce circular magnet field.

There is another current i along x direction initially ($i<), the direction of thos current can be adjusted with slider.

The magnet force between current segment, $i d\vec{\ell}$ in magnetic field $\vec{B}=\frac{\mu_o I}{2\pi r}\hat{\theta}$ is $d\vec{F}=i d \vec{\ell}\times \vec{B}$

The direction of current is represented with red arrow. Gray arrows are magnetic field, Blue arrows are interaction force.

/htdocs/ntnujava/ejsuser/2/users/ntnu/fkh/twolinesBfield_pkg/twolinesBfield.propertiesFull screen applet or Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list
Press the Alt key and the left mouse button to drag the applet off the browser and onto the desktop. This work is licensed under a Creative Commons Attribution 2.5 Taiwan License
Download EJS jar file(1553.7kB):double click downloaded file to run it. (23 times by 18 users) , Download EJS source (4 times by 3 users) View EJS source